PART 1: | Introduction and Centers X(1) - X(1000) |

PART 2: | Centers X(1001) - X(3000) |

PART 3: | Centers X(3001) - X(5000) |

PART 4: | Centers X(5001) - X(7000) |

PART 5: | Centers X(7001) - X(10000) |

PART 6: | Centers X(10001) - X(12000) |

PART 7: | Centers X(12001) - X(14000) |

PART 8: | Centers X(14001) - X(16000) |

PART 9: | Centers X(16001) - X(18000) |

PART 10: | Centers X(18001) - X(20000) |

PART 11: | Centers X(20001) - X(22000) |

PART 12: | Centers X(22001) - X(24000) |

PART 13: | Centers X(24001) - X(26000) |

PART 14: | Centers X(26001) - X(28000) |

PART 15: | Centers X(28001) - X(30000) |

PART 16: | Centers X(30001) - X(32000) |

PART 17: | Centers X(32001) - X(34000) |

PART 18: | Centers X(34001) - X(36000) |

PART 19: | Centers X(36001) - X(38000) |

PART 20: | Centers X(38001) - X(40000) |

Centuries passed, and someone * proved* that the three medians do indeed concur in a point, now called the centroid. The ancients found other points, too, now called the incenter, circumcenter, and orthocenter. More centuries passed, more special points were discovered, and a definition of triangle center emerged. Like the definition of continuous function, this definition is satisfied by infinitely many objects, of which only finitely many will ever be published. The Encyclopedia of Triangle Centers (ETC) extends a list of 400 triangle centers published in the 1998 book Triangle Centers and Central Triangles. For subsequent developments, click Links (one of the buttons atop this page). In particular, Eric Weisstein's MathWorld, covers much of classical and modern triangle geometry, including sketches and references.

A site in which triangle centers play a central role is Bernard Gibert's Cubics in the Triangle Plane. Special points and properties of 4-sided plane figures are closely associated with triangle centers; see Chris van Tienhoven's Encyclopedia of Quadri-Figures (EQF).

To determine if a possibly new center is already listed, click **Tables** at the top of this page and scroll to "Search 6.9.13". If you're unsure of a term, click **Glossary**. For visual constructions of selected centers with text, click **Sketches**. To learn about the triangle geometry interest group, Hyacinthos and other resources, or to view acknowledgments or supplementary encyclopedic material, click **Links**, **Thanks**, or **Tables**.

If you have The Geometer's Sketchpad, you can view sketches of many of the triangle centers. These are **dynamic** sketches, meaning that you can vary the shape of the reference triangle A, B, C by dragging these vertices. (For information on Sketchpad, click Sketchpad.) The sketches are also useful for making your own Sketchpad tools, so that you can quickly construct X-of-T for many choices of X and T. For example, starting with ABC and point P, you could efficiently construct center X of the four triangles ABC, BCP, CAP, ABP.

The algebraic definition of triangle center (MathWorld) admits points whose *geometric* interpretation for fixed numerical sidelengths a,b,c is not "central." Roger Smyth offers this example: on the domain of scalene triangles, define f(a,b,c) = 1 for a>b and a>c and f(a,b,c) = 0 otherwise; then f(a,b,c) : f(b,c,a) : f(c,a,b) is a triangle center which picks out the vertex opposite the longest side. Such centers turn out to be useful, as, for example, when distinguishing between the Fermat point and the 1st isogonic center; see the note at X(13).

The reference triangle is ABC, with sidelengths a,b,c. Each triangle center P has homogeneous trilinear coordinates, or simply trilinears, of the form x : y : z. This means that there is a nonzero function h of (a,b,c) such that

x = hx', y = hy', z = hz',

where x', y', z' denote the directed distances from P to sidelines BC, CA, AB. Likewise, u : v : w are **barycentrics** if there is a function k of (a,b,c) such that

u = ku', v = kv', w = kw',

where u', v', w' denote the signed areas of triangles PBC, PCA, PAB. Both coordinate systems are widely used; if trilinears for a point are x : y : z, then barycentrics are ax : by : cz.

In order that every center should have its own name, in cases where no particular name arises from geometrical or historical considerations, the name of a star is used instead. For example, X(770) is POINT ACAMAR. For a list of star names, visit SkyEye - (Un)Common Star Names.

If you wish to submit one or more triangles centers for possible inclusion in ETC, please click **Tables** at the top of this page, then scroll to and click **Search_13_6_9.** There, find **Writer**, to be used for proper formatting.

Many triangles are defined in the plane of a reference triangle ABC. Some of them have well-established names (e.g., medial, orthic, tangential), but many more have been discovered only recently.

The **Index** is authored and updated by César Lozada. You can access it here, and also from **Glossary** and **Tables**.

f(a,b,c) = G(a,b,c)*S^{2} + H(a,b,c)*S_{B}S_{C}.

For many choices of X, G(a,b,c) and H(a,b,c) are conveniently expressed in terms of the following:

E = (S_{B} + S_{C})(S_{C} + S_{A})(S_{A} + S_{B})/S^{2}, so that E = (abc/S)^{2} = 4R^{2}

F = S_{A}S_{B}S_{C}/S^{2}, so that F = (a^{2} + b^{2} + c^{2})/2 - 4R^{2} = S_{ω} - 4R^{2}

Examples:

X(2) has Shinagawa coefficients (1, 0); i.e., X(2) = 1*S^{2} + 0*S_{B}S_{C}

X(3) has Shinagawa coefficients (1, -1)

X(4) has Shinagawa coefficients (0, 1)

X(5) has Shinagawa coefficients (1, 1)

X(23) has Shinagawa coefficients (E + 4F, -4E - 4F)

X(1113) has Shinagawa coefficients (R - |OH|, -3R + |OH|)

A cyclic sum notation, $...$, is introduced here especially for use with Shinagawa coefficients. For example, $aS_{B}S_{C}$ abbreviates aS_{B}S_{C} + bS_{C}S_{A} + cS_{A}S_{B}.

Example: X(21) has Shinagawa coefficients ($aS_{A}$, abc - $aS_{A}$)

If a point X has Shinagawa coefficients (u,v) where u and v are real numbers (i.e, G(a,b,c) and H(a,b,c) are constants), then the segment joining X and X(2) is giving by |GX| = 2v|GO|/(3u + v), where |GO| = (E - 8F)^{1/2}/6. Then the equation |GX| = 2v|GO|/(3u + v) can be used to obtain these combos:

X = [(u + v)/2]*X(2) - (v/3)*X(3)

X = u*X(2) + (v/3)*X(4)

X = u*X(3) + [(u + v)/2]*X(4).

The function F is also given by these identities:

F = (4R^{2} - 36|GO|^{2})/8 and F = R^{2}( 1 - J^{2})/2, where J = |OH|/R.

Suppose that P and U are finite points having normalized barycentric coordinates (p,q,r) and (u,v,w). (Normalized means that p + q + r = 1 and u + v + w = 1.) Suppose that f = f(a,b,c) and g = g(a,b,c) are nonzero homogeneous functions having the same degree of homogeneity. Let x = fp + gu, y = fq + gv, z = fr + gw. **The (f,g) combo of P and U, denoted by f*P + g*U, is introduced here as the point X = x : y : z (homogeneous barycentric coordinates);
**
the normalized barycentric coordinates of X are (kx,ky,kz), where k=1/(x+y+z).

**Note 1.** If P and U are given by normalized trilinear coordinates (instead of barycentric), then f*P + g*U has homogeneous trilinears fp+gu : fq+gv : fr+gw, which is symbolically identical to the homogenous barycentrics for f*P + g*U. The normalized trilinear coordinates for X are (hx,hy,hz), where h=2*area(ABC))/(ax + by + cz).

**Note 2.** The definition of combo readily extends to finite sets of finite points. In particular, the (f,g,h) combo of P = (p,q,r), U = (u,v,w), J = (j,k,m) is given by fp + gu + hj : fq + gv + hk : fr + gw + hm and denoted by f*P + g*U + h*J.

**Note 3.** f*P + g*U is collinear with P and U, and its {P,Q}-harmonic conjugate is fp - gu : fq - gv : fr - gw.

**Note 4.** Suppose that f,g,h are homogeneous symmetric functions all of the same degree of homogeneity, and suppose that X, X', X" are triangle centers. Then f*X + g*X' + h*X'' is a triangle center.

**Note 5.** Suppose that X, X', X'', X''' are triangle centers and X', X'', X''' are not collinear. Then there exist f,g,h as in Note 4 such that X = f*X' + g*X'' + h*X'''. That is, loosely speaking, every triangle center is a linear combination of any other three noncollinear triangle centers.

**Note 6.** Continuing from Note 5, examples of f,g,h are conveniently given using Conway symbols for a triangle ABC with sidelengths a,b,c. Conway symbols and certain classical symbols are
identified here:

S = 2*area(ABC)

S_{A} = (b^{2} + c^{2} - a^{2})/2 = bc cos A

S_{B} = (c^{2} + a^{2} - b^{2})/2 = ca cos B

S_{C} = (a^{2} + b^{2} - c^{2})/2 = ab cos C

S_{ω} = S cot ω

s = (a + b + c)/2

s_{a} = (b + c - a)/2

s_{b} = (c + a - b)/2

s_{c} = (a + b - c)/2

r = inradius = S/(a + b + c)

R = circumradius = abc/(2S)

cot(ω) = (a^{2} + b^{2} + c^{2})/(2S), where ω is the Brocard angle

**Note 7.** The definition of combo along with many examples were developed by Peter Moses prior to November 1, 2011. After that combos have been further developed by Peter Moses, Randy Hutson, and Clark Kimberling.

Examples of two-point combos:**X(175) = 2s*X(1) - (r + 4R)*X(7)X(176) = 2s*X(1) + (r + 4R)*X(7)X(481) = s*X(1) - (r + 4R)*X(7)X(482) = s*X(1) + (r + 4R)*X(7)
**

Examples of three-point combos: see below at X(1), X(2), etc.

**Note 8.** Suppose that T is a (central) triangle with vertices A',B',C' given by normalized barycentrics. Then T is represented by a 3x3 matrix with row sums equal to 1. Let NT denote the set of these matrices and let * denote matrix multiplication. Then NT is closed under *. Also, NT is closed under matrix inversion, so that (NT, *) is a group. Once normalized, any central T can be used to produce triangle centers as combos of the form Xcom(nT); see the preambles to X(3663) and X(3739).

Barycentrics a : b : c

Barycentrics sin A : sin B : sin C

Tripolars Sqrt[b c (b + c - a)] : :

Tripolars sec A' : :, where A'B'C' is the excentral triangle

X(1) = X(1) = [A]/Ra + [B]/Rb + [C]/Rc - X(176)/Rs, where Ra, Rb, Rc = radii of Soddy circles, Rs = radius of inner Soddy circle, [A], [B], [C] are the vertices of ABC

X(1) = X(1) = [A]/Ra + [B]/Rb + [C]/Rc - X(175)/Rs', where Ra, Rb, Rc = radii of Soddy circles, Rs' = radius of outer Soddy circle, [A], [B], [C] are the vertices of ABC

X(1) = (sin A)*[A] + (sin B)*[B] + (sin C)*[C], where [A], [B], [C] are vertices of ABC

X(1) = a*[A] + b*[B] + c*[C], where [A], [B], [C] are vertices

X(1) is the point of concurrence of the interior angle bisectors of ABC; the point inside ABC whose distances from sidelines BC, CA, AB are equal. This equal distance, r, is the radius of the incircle, given by r = 2*area(ABC)/(a + b + c).

Three more points are also equidistant from the sidelines; they are given by these names and trilinears:

A-excenter = -1 : 1 : 1, B-excenter = 1 : -1 : 1, C-excenter = 1 : 1 : -1.

The radii of the excircles are 2*area(ABC)/(-a + b + c), 2*area(ABC)/(a - b + c), 2*area(ABC)/(a + b - c).

If you have The Geometer's Sketchpad, you can view **Incenter**.

If you have GeoGebra, you can view **Incenter**.

Writing the A-exradius as r(a,b,c), the others are r(b,c,a) and r(c,a,b). If these exradii are abbreviated as r_{a}, r_{b}, r_{c}, then 1/r = 1/r_{a} +1/r_{b} + 1/r_{c}. Moreover,

area(ABC) = sqrt(r*r_{a}*r_{b}*r_{c}) and r_{a} + r_{b} + r_{c} = r + 4R, where R denotes the radius of the circumcircle.

The incenter is the identity of the group of triangle centers under **trilinear multiplication** defined by (x : y : z)*(u : v : w) = xu : yv : zw.

A construction for * is easily obtained from the construction for "barycentric multiplication" mentioned in connection with X(2), just below.

The incenter and the other classical centers are discussed in these highly recommended books:

**Paul Yiu,** Introduction to the Geometry of the Triangle, 2002;**Nathan Altshiller Court,** *College Geometry,* Barnes & Noble, 1952;

**Roger A. Johnson,** *Advanced Euclidean Geometry,* Dover, 1960.

Let O_{A} be the circle tangent to side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define O_{B} and O_{C} cyclically. Let L_{A} be the external tangent to circles O_{B} and O_{C} that is nearest to O_{A}. Define L_{B} and L_{C} cyclically. Let A' = L_{B} ∩L_{C}, and define B' and C' cyclically. Then triangle A'B'C' is homothetic to the medial triangle, and the center of homothety is X(1); see the reference at X(1001).

Let A'B'C' and A"B"C" be the intouch and excentral triangles; X(1) is the radical center of the circumcircles of AA'A", BB'B", CC'C". (Randy Hutson, December 10, 2016)

X(1) lies on all Z-cubics (e.g., Thomson, Darboux, Napoleon, Neuberg) and these lines:

2,8 3,35 4,33 5,11
6,9 7,20 15,1251
16,1250 19,28 21,31
24,1061 25,1036 29,92
30,79 32,172 39,291
41,101 49,215 54,3460
60,110 61,203 62,202
64,1439 69,1245 71,579
74,3464 75,86 76,350
82,560 84,221 87,192
88,100 90,155 99,741
102,108 104,109 142,277
147,150 159,1486 163,293
164,258 166,1488 167,174
168,173 179,1142 181,970
182,983 184,1726 185,296
188,361 190,537 195,3467
196,207 201,212 204,1712
210,2334 224,377 227,1465 228,1730
229,267 256,511 257,385
280,1256 281,282 289,363
312,1089 318,1897 320,752
321,964 329,452 335,384
336,811 341,1050 344,1265
346,1219 357,1508 358,1507
364,365 371,1702 372,1703
376,553 378,1063 393,836
394,1711 399,3065 409,1247
410,1248 411,1254 442,1834
474,1339 475,1861 512,875
513,764 514,663 522,1459
528,1086 561,718 563,1820
564,1048 572,604 573,941
574,1571 594,1224 607,949
615,3300 631,1000 644,1280 647,1021
650,1643 651,1156 659,891
662,897 672,1002 689,719
704,1502 727,932 731,789
748,756 761,825 765,1052
810,1577 840,1308 905,1734
908,998 921,1800 939,1260
945,1875 947,1753 951,1435
969,1444 971,1419 989,1397
1013,1430 1037,1041 1053,1110
1057,1598 1059,1597 1073,3341
1075,1148 1106,1476 1157,3483
1168,1318 1170,1253 1185,1206
1197,1613 1292,1477 1333,1761
1342,1700 1343,1701 1361,1364
1389,1393 1399,1727 1406,1480
1409,1765 1437,1710 1472,1791
1719,1790 1855,1886 1859,1871
1872,1887 2120,3461 2130,3347
3183,3345 3342,3343 3344,3351
3346,3353 3348,3472 3350,3352
3354,3355 3462,3469

X(1) is the {X(2),X(8)}-harmonic conjugate of X(10). For a list of other harmonic conjugates of X(1), click **Tables** at the top of this page.

X(1) = midpoint of X(i) and X(j) for these (i,j): (3, 1482), (7,390), (8,145), (55,2099), (56,2098)

X(1) = reflection of X(i) in X(j) for these (i,j): (2,551), (3,1385), (4,946), (6,1386), (8,10), (9,1001), (10,1125), (11,1387), (36,1319), (40,3), (43,995), (46,56), (57,999), (63,993), (65,942), (72,960), (80,11), (100,214), (191,21), (200,997), (238,1279), (267,229), (291,1015), (355,5), (484,36), (984,37), (1046,58), (1054,106), (1478,226)

X(1) = isogonal conjugate of X(1)

X(1) = isotomic conjugate of X(75)

X(1) = cyclocevian conjugate of X(1029)

X(1) = circumcircle-inverse of X(36)

X(1) = Fuhrmann-circle-inverse of X(80)

X(1) = Bevan-circle-inverse of X(484)

X(1) = complement of X(8)

X(1) = anticomplement of X(10)

X(1) = anticomplementary conjugate of X(1330)

X(1) = complementary conjugate at X(1329)

X(1) = eigencenter of cevian triangle of X(i) for I = 1, 88, 100, 162, 190

X(1) = eigencenter of anticevian triangle of X(i) for I = 1, 44, 513

X(1) = exsimilicenter of inner and outer Soddy circles; insimilicenter is X(7)

X(1) = X(i)-Ceva conjugate of X(j) for these (i,j):

(2,9), (4,46), (6,43), (7,57), (8,40), (9,165), (10,191), (21,3), (29,4), (75,63), (77,223), (78,1490), (80,484), (81,6), (82,31), (85,169), (86,2), (88,44), (92,19), (100,513), (104,36), (105,238), (174,173), (188,164), (220,170), (259,503), (266,361), (280,84), (366,364), (508,362), (1492,1491)

X(1) = cevapoint of X(i) and X(j) for these (i,j):

(2,192), (6,55), (11,523), (15,202), (16,203), (19,204), (31,48), (34,207), (37,42), (50,215), (56,221), (65,73), (244,513)

X(1) = X(i)-cross conjugate of X(j) for these (i,j):

(2,87), (3,90), (6,57), (31,19), (33,282), (37,2), (38,75), (42,6), (44,88), (48,63), (55,9), (56,84), (58,267), (65,4), (73,3), (192,43), (207,1490), (221,40), (244,513), (259,258), (266,505), (354,7), (367,366), (500,35), (513,100), (517,80), (518,291), (1491,1492)

X(1) = crosspoint of X(i) and X(j) for these (i,j):

(2,7), (8,280), (21,29), (59,110), (75,92), (81,86)

X(1) = crosssum of X(i) and X(j) for these (i,j):

(2,192), (4,1148), (6,55), (11,523), (15,202), (16,203), (19,204), (31,48), (34,207), (37,42), (44,678), (50,215), (56,221), (57,1419), (65,73), (214,758), (244,513), (500,942), (512,1015), (774,820), (999,1480)

X(1) = crossdifference of every pair of points on line X(44)X(513)

X(1) = X(i)-Hirst inverse of X(j) for these (i,j): (2,239), (4,243), (6,238), (9,518), (19,240), (57,241), (105,294), (291,292)

X(1) = X(6)-line conjugate of X(44)

X(1) = X(i)-aleph conjugate of X(j) for these (i,j):

(1,1), (2,63), (4,920), (21,411), (29,412), (88,88), (100,100),
(162,162), (174,57), (188,40), (190,190), (266,978), (365,43), (366,9), (507,173), (508,169), (513,1052), (651, 651), (653,653), (655,655), (658,658), (660,660), (662,662), (673,673), (771,771), (799,799), (823,823), (897,897)

X(1) = X(i)-beth conjugate of X(j) for these (i,j): (1,56), (2,948), (8,8), (9,45), (21,1), (29,34), (55,869), (99,85), (100,1), (110,603), (162,208), (643,1), (644,1), (663,875), (664,1), (1043,78)

X(1) = insimilicenter of 1st & 2nd Johnson-Yff circles (the exsimilicenter is X(4))

X(1) = orthic-isogonal conjugate of X(46)

X(1) = excentral-isogonal conjugate of X(40)

X(1) = excentral-isotomic conjugate of X(2951)

X(1) = center of Conway circle

X(1) = center of Adams circle

X(1) = X(3) of polar triangle of Conway circle

X(1) = homothetic center of intangents triangle and reflection of extangents triangle in X(3)

X(1) = Hofstadter 1/2 point

X(1) = orthocenter of X(4)X(9)X(885)

X(1) = intersection of tangents at X(7) and X(8) to Lucas cubic K007

X(1) = trilinear product of vertices of 2nd mixtilinear triangle

X(1) = trilinear product of vertices of 2nd Sharygin triangle

X(1) = homothetic center of Mandart-incircle triangle and 2nd isogonal triangle of X(1); see X(36)

X(1) = trilinear pole of the antiorthic axis (which is also the Monge line of the mixtilinear excircles)

X(1) = pole wrt polar circle of trilinear polar of X(92) (line X(240)X(522))

X(1) = X(48)-isoconjugate (polar conjugate) of X(92)

X(1) = X(6)-isoconjugate of X(2)

X(1) = trilinear product of PU(i) for these i: 1, 17, 114, 115, 118, 119, 113

X(1) = barycentric product of PU(i) for these i: 6, 124

X(1) = vertex conjugate of PU(9)

X(1) = bicentric sum of PU(i) for these i: 28, 47, 51, 55, 64

X(1) = trilinear pole of PU(i) for these i: 33, 50, 57, 58, 74, 76, 78

X(1) = crossdifference of PU(i) for these i: 33, 50, 57, 58, 74, 76, 78

X(1) = midpoint of PU(i) for these i: 47, 51, 55

X(1) = PU(28)-harmonic conjugate of X(1023)

X(1) = PU(64)-harmonic conjugate of X(351)

X(1) = intersection of diagonals of trapezoid PU(6)PU(31)

X(1) = perspector circumconic centered at X(9)

X(1) = eigencenter of mixtilinear excentral triangle

X(1) = eigencenter of 2nd Sharygin triangle

X(1) = perspector of ABC and unary cofactor triangle of extangents triangle

X(1) = perspector of ABC and unary cofactor triangle of Feuerbach triangle

X(1) = perspector of ABC and unary cofactor triangle of Apollonius triangle

X(1) = perspector of ABC and unary cofactor triangle of 2nd mixtilinear triangle

X(1) = perspector of ABC and unary cofactor triangle of 4th mixtilinear triangle

X(1) = perspector of ABC and unary cofactor triangle of Apus triangle

X(1) = perspector of unary cofactor triangles of 6th and 7th mixtilinear triangles

X(1) = perspector of unary cofactor triangles of 2nd and 3rd extouch triangles

X(1) = perspector of 5th mixtilinear triangle and unary cofactor triangle of 2nd mixtilinear triangle

X(1) = perspector of 5th mixtilinear triangle and unary cofactor triangle of 4th mixtilinear triangle

X(1) = X(3)-of-reflection-triangle-of-X(1)

X(1) = X(1181)-of-2nd-extouch triangle

X(1) = perspector of ABC and orthic-triangle-of-2nd-circumperp-triangle

X(1) = X(4)-of-excentral triangle

X(1) = X(40)-of-Yff central triangle

X(1) = X(20)-of-1st circumperp triangle

X(1) = X(4)-of-2nd circumperp triangle

X(1) = X(4)-of-Fuhrmann triangle

X(1) = X(100)-of-X(1)-Brocard triangle

X(1) = antigonal image of X(80)

X(1) = trilinear pole wrt excentral triangle of antiorthic axis

X(1) = trilinear pole wrt incentral triangle of antiorthic axis

X(1) = Miquel associate of X(7)

X(1) = homothetic center of Johnson triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)

X(1) = homothetic center of 1st Johnson-Yff triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)

X(1) = homothetic center of 2nd Johnson-Yff triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)

X(1) = homothetic center of Mandart-incircle triangle and cross-triangle of ABC and 1st Johnson-Yff triangle

X(1) = homothetic center of medial triangle and cross-triangle of Aquila and anti-Aquila triangles

X(1) = homothetic center of outer Garcia triangle and cross-triangle of Aquila and anti-Aquila triangles

X(1) = X(8)-of-cross-triangle-of Aquila-and-anti-Aquila-triangles

X(1) = X(3)-of-Mandart-incircle-triangle

X(1) = X(100)-of-inner-Garcia-triangle

X(1) = Thomson-isogonal conjugate of X(165)

X(1) = X(8)-of-outer-Garcia-triangle

X(1) = X(486)-of-BCI-triangle

X(1) = X(164)-of-orthic-triangle if ABC is acute

X(1) = X(1593)-of-Ascella-triangle

X(1) = excentral-to-Ascella similarity image of X(1697)

X(1) = Dao image of X(1)

X(1) = X(40)-of-reflection of ABC in X(3)

X(1) = radical center of the tangent circles of ABC

X(1) = homothetic center of intangents triangle and anti-tangential midarc triangle

X(1) = K(X(15)) = K(X(16), as defined at X(174)

X(1) = X(3)-of-hexyl-triangle

X(1) = eigencenter of trilinear obverse triangle of X(2)

X(1) = hexyl-isogonal conjugate of X(40)

X(1) = inverse-in-polar-circle of X(1785)

X(1) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(5121)

X(1) = inverse-in-OI-inverter of X(1155)

X(1) = inverse-in-Steiner-circumellipse of X(239)

X(1) = inverse-in-MacBeath-circumconic of X(2323)

X(1) = inverse-in-circumconic-centered-at-X(9) of X(44)

X(1) = excentral-to-ABC barycentric image of X(40)

X(1) = excentral-to-ABC functional image of X(164)

X(1) = excentral-to-ABC trilinear image of X(164)

X(1) = orthic-to-ABC functional image of X(4), if ABC is acute

X(1) = orthic-to-ABC trilinear image of X(4), if ABC is acute

X(1) = intouch-to-ABC barycentric image of X(1)

X(1) = excentral-to-intouch similarity image of X(40)

X(1) = ABC-to-excentral barycentric image of X(8)

X(1) = X(1)-vertex conjugate of X(56)

X(1) = perspector of ABC and reflection triangle of intangents triangle

X(1) = perspector of pedal and anticevian triangles of X(40)

X(1) = perspector of hexyl triangle and antipedal triangle of X(40)

X(1) = perspector of hexyl triangle and anticevian triangle of X(57)

X(1) = X(4)-of-Pelletier-triangle

Trilinears bc : ca : ab

Trilinears csc A : csc B : csc C

Trilinears cos A + cos B cos C : cos B + cos C cos A : cos C + cos A cos B

Trilinears sec A + sec B sec C : sec B + sec C sec A : sec C + sec A sec B

Trilinears cos A + cos(B - C) : cos B + cos(C - A) : cos C + cos(A - B)

Trilinears cos B cos C - cos(B - C) : cos C cos A - cos(C - A) : cos A cos B - cos(A - B)

Trilinears tan(A/2) + cot(A/2) : :

Trilinears cot(B/2) + cot(C/2) : :

Trilinears 1 + csc A/2 sin B/2 sin C/2 : :

Barycentrics 1 : 1 : 1

Tripolars Sqrt[2(b^2 + c^2) - a^2] : :

X(2) = (3 + J) X(1113) + (3 - J) X(1114)

As a point on the Euler line, X(2) has Shinagawa coefficients (1, 0).

X(2) is the point of concurrence of the medians of ABC, situated 1/3 of the distance from each vertex to the midpoint of the opposite side. More generally, if L is any line in the plane of ABC, then the distance from X(2) to L is the average of the distances from A, B, C to L. An idealized triangular sheet balances atop a pin head located at X(2) and also balances atop any knife-edge that passes through X(2). The triangles BXC, CXA, AXB have equal areas if and only if X = X(2).

If you have The Geometer's Sketchpad, you can view Centroid.

If you have GeoGebra, you can view **Centroid**.

X(2) is the centroid of the set of 3 vertices, the centroid of the triangle including its interior, but not the centroid of the triangle without its interior; that centroid is X(10).

X(2) is the identity of the group of triangle centers under "barycentric multiplication" defined by

(x : y : z)*(u : v : w) = xu : yv : zw. X(2) is the unique point X (as a function of a,b,c) for which the vector-sum XA + XB + XC is the zero vector. The1,8 3,4 6,69 7,9 11,55 12,56 13,16 14,15 17,62 18,61 19,534 31,171 32,83 33,1040 34,1038 35,1479 36,535 37,75 38,244 39,76 40,946 44,89 45,88 51,262 52,1216 54,68 58,540 65,959 66,206 71,1246 72,942 74,113 77,189 80,214 85,241 92,273 94,300 95,97 98,110 99,111 101,116 102,117 103,118 104,119 106,121 107,122 108,123 109,124 112,127 128,1141 129,1298 130,1303 131,1300 133,1294 136,925 137,930 154,1503 165,516 169,1763 174,236 176,1659 178,188 187,316 196,653 201,1393 210,354 216,232 220,1170 222,651 231,1273 242,1851 243,1857 252,1166 253,1073 254,847 257,1432 261,593 265,1511 271,1034 272,284 280,318 283,580 290,327 292,334 294,949 308,702 311,570 314,941 319,1100 322,1108 330,1107 341,1219 351,804 355,944 360,1115 366,367 371,486 372,485 392,517 476,842 480,1223 489,1132 490,1131 495,956 496,1058 514,1022 523,1649 525,1640 561,716 568,1154 572,1746 573,1730 578,1092 585,1336 586,1123 588,1504 589,1505 594,1255 647,850 648,1494 650,693 664,1121 668,1015 670,1084 689,733 743,789 799,873 812,1635 846,1054 914,1442 918,1638 927,1566 954,1260 1073,1249 968,1738 1000,1145 1043,1834 1060,1870 1074,1785 1076,1838 1089,1224 1093,1217 1124,1378 1143,1489 1155,1836 1171,1509 1186,1207 1257,1265 1284,1403 1335,1377 1340,1349 1341,1348 1500,1574 1501,1691 1672,1681 1673,1680 1674,1679 1675,1678 1697,1706 3343,3344 3349,3350 3351,3352

X(2) is the {X(3),X(5)}-harmonic conjugate of X(4). For a list of other harmonic conjugates of X(2), click **Tables** at the top of this page.

X(2) = midpoint of X(i) and X(j) for these (i,j): {1,3679}, {3,381}, {4,376}, {5,549}, {6,599}, {7,6172}, {8,3241}, {9,6173}, {10,551}, {11,6174}, {13,5463}, {14,5464}, {20,3543}, {21,6175}, {32,7818}, {37,4688}, {39,9466}, {51,3917}, {69,1992}, {75,4664}, {76,7757}, {98,6054}, {99,671}, {110,9140}, {114,6055}, {115,2482}, {125,5642}, {126,9172}, {140,547}, {141,597}, {148,8591}, {154,1853}, {165,1699}, {190,903}, {192,4740}, {210,354}, {329,2094}, {351,9148}, {355,3655}, {373,5650}, {384,7924}, {385,7840}, {392,3753}, {428,7667}, {591,1991}, {618,5459}, {619,5460}, {620,5461}, {631,5071}, {648,1494}, {664,1121}, {668,3227}, {670,3228}, {858,7426}, {1003,7841}, {1086,4370}, {1125,3828}, {1635,4728}, {1638,1639}, {1641,1648}, {1644,1647}, {1649,8371}, {1650,1651}, {2454,2455}, {2479,2480}, {2487,4677}, {2966,5641}, {2976,6161}, {2979,3060}, {3034,3875}, {3034,7292}, {3251,4162}, {3268,9979}, {3448,9143}, {3524,3545}, {3534,3830}, {3576,5587}, {3616,4521}, {3617,3676}, {3623,4468}, {3628,10124}, {3654,3656}, {3681,3873}, {3739,4755}, {3740,3742}, {3817,10164}, {3819,5943}, {3845,8703}, {3929,4654}, {4025,4808}, {4108,5996}, {4120,4750}, {4364,10022}, {4373,4776}, {4379,4893}, {4430,4661}, {4643,4795}, {4730,6332}, {4763,4928}, {5054,5055}, {5108,9169}, {5309,7801}, {5466,9168}, {5485,9741}, {5569,8176}, {5603,5657}, {5640,7998}, {5692,5902}, {5858,5859}, {5860,5861}, {5862,5863}, {5883,10176}, {5891,9730}, {5892,10170}, {5927,10167}, {6032,9829}, {6039,6040}, {6189,6190}, {6545,6546}, {6656,6661}, {6784,6786}, {7615,7618}, {7617,7622}, {7734,10128}, {7753,7810}, {7811,7812}, {7817,7880}, {8010,8011}, {8352,8598}, {8356,8370}, {8360,8368}, {8597,9855}, {8667,9766}, {9185,9191}, {9200,9204}, {9201,9205}, {9761,9763}, {9774,10033}, {9778,9812}, {10162,10163}, {10165,10175}

X(2) = reflection of X(i) in X(j) for these (i,j): (1,551), (3,549), (4,381), (5,547), (6,597), (7,6173), (8,3679), (10,3828), (13,5459), (14,5460), (20,376), (23,7426), (37,4755), (51,5943), (69,599), (75,4688), (76,9466), (98,6055), (99,2482), (100,6174), (110,5642), (111,9172), (115,5461), (140,10124), (144,6172), (145,3241), (147,6054), (148,671), (154,10192), (165,10164), (182,10168), (190,4370), (192,4664), (193,1992), (194,7757), (210,3740), (315,7818), (352,9127), (353,10166), (354,3742), (356,5455), (376,3), (381,5), (384,6661), (547,3628), (549,140), (551,1125), (568,5946), (597,3589), (599,141), (616,5463), (617,5464), (648,3163), (671,115), (903,1086), (944,3655), (1003,8369), (1121,1146), (1278,4740), (1635,4763), (1651,402), (1699,3817), (1962,10180), (1992,6), (2094,57), (2475,6175), (2479,2454), (2480,2455), (2482,620), (2979,3917), (3034,2321), (3060,51), (3091,5071), (3146,3543), (3227,1015), (3228,1084), (3241,1), (3448,9140), (3524,5054), (3534,8703), (3543,4), (3545,5055), (3576,10165), (3617,4521), (3623,3676), (3655,1385), (3676,3616), (3679,10), (3681,210), (3742,3848), (3817,10171), (3828,3634), (3830,3845), (3839,3545), (3845,5066), (3873,354), (3877,392), (3917,3819), (3929,5325), (4240,1651), (4363,10022), (4370,4422), (4430,3873), (4440,903), (4453,1638), (4468,3617), (4521,1698), (4644,4795), (4661,3681), (4664,37), (4669,4745), (4677,4669), (4688,3739), (4728,4928), (4740,75), (4755,4698), (4776,3161), (4795,4670), (4808,3239), (4808,8834), (5066,10109), (5071,1656), (5309,7817), (5459,6669), (5460,6670), (5461,6722), (5463,618), (5464,619), (5466,8371), (5468,1641), (5569,1153), (5587,10175), (5603,5886), (5640,373), (5642,5972), (5692,10176), (5731,3576), (5860,591), (5861,1991), (5862,5858), (5863,5859), (5883,3833), (5890,9730), (5891,10170), (5902,5883), (5918,10178), (5919,10179), (5927,10157), (5943,6688), (6031,9829), (6032,10162), (6054,114), (6055,6036), (6161,2505), (6172,9), (6173,142), (6174,3035), (6175,442), (6546,10196), (6655,7924), (6661,7819), (6688,10219), (6792,9169), (7426,468), (7615,7617), (7618,7622), (7620,7615), (7622,7619), (7671,10177), (7757,39), (7779,7840), (7801,7880), (7811,7810), (7812,7753), (7818,626), (7833,8356), (7840,325), (7924,6656), (7998,5650), (8182,5569), (8353,8354), (8354,8358), (8356,8359), (8368,8365), (8369,8368), (8591,99), (8596,148), (8597,8352), (8860,3054), (9123,9125), (9140,125), (9143,110), (9144,5465), (9147,351), (9168,1649), (9172,6719), (9185,9189), (9263,3227), (9466,3934), (9485,9123), (9730,5892), (9778,165), (9779,7988), (9812,1699), (9829,10163), (9855,8598), (9909,10154), (9939,7811), (9965,2094), (9979,1637), (10022,4472), (10056,10197), (10072,10199), (10162,10173), (10166,10160), (10175,10172)

X(2) = isogonal conjugate of X(6)

X(2) = isotomic conjugate of X(2)

X(2) = cyclocevian conjugate of X(4)

X(2) = circumcircle-inverse of X(23)

X(2) = nine-point-circle-inverse of X(858)

X(2) = Brocard-circle-inverse of X(110)

X(2) = complement of X(2)

X(2) = anticomplement of X(2)

X(2) = anticomplementary conjugate of X(69)

X(2) = complementary conjugate of X(141)

X(2) = insimilicenter of incircle and Spieker circle

X(2) = insimilicenter of incircle and AC-incircle

X(2) = exsimilicenter of Spieker circle and AC-incircle

X(2) = insimilicenter of Conway circle and Spieker radical circle

X(2) = insimilicenter of polar circle and de Longchamps circle

X(2) = harmonic center of pedal circles of X(13) and X(14) (which are also the pedal circles of X(15) and X(16))

X(2) = X(99)-of -1st-Parry-triangle

X(2) = X(98)-of-2nd-Parry-triangle

X(2) = X(2)-of-1st-Brocard-triangle

X(2) = X(111)-of-4th-Brocard-triangle

X(2) = X(110)-of-X(2)-Brocard-triangle

X(2) = X(110)-of-orthocentroidal-triangle

X(2) = X(353)-of-circumsymmedial-triangle

X(2) = X(165)-of-orthic-triangle if ABC is acute

X(2) = X(51)-of-excentral-triangle

X(2) = inverse-in-polar-circle of X(468)

X(2) = inverse-in-de-Longchamps-circle of X(858)

X(2) = inverse-in-MacBeath-circumconic of X(323)

X(2) = inverse-in-Feuerbach-hyperbola of X(390)

X(2) = inverse-in-circumconic-centered-at-X(1) of X(3935)

X(2) = inverse-in-circumconic-centered-at-X(9) of X(3218)

X(2) = inverse-in-excircles-radical-circle of X(5212)

X(2) = inverse-in-Parry-isodynamic-circle of X(353)

X(2) = barycentric product of (real or nonreal) circumcircle intercepts of the de Longchamps line

X(2) = harmonic center of nine-point circle and Johnson circle

X(2) = pole wrt polar circle of trilinear polar of X(4) (orthic axis)

X(2) = polar conjugate of X(4)

X(2) = excentral-to-ABC functional image of X(165)

X(2) = excentral-to-ABC barycentric image of X(165)

X(2) = orthic-to-ABC functional image of X(51)

X(2) = orthic-to-ABC barycentric image of X(51)

X(2) = incentral-to-ABC functional image of X(1962)

X(2) = incentral-to-ABC barycentric image of X(1962)

X(2) = Feuerbach-to-ABC functional image of X(5947)

X(2) = Feuerbach-to-ABC barycentric image of X(5947)

X(2) = barycentric product of circumcircle intercepts of line X(325)X(523)

X(2) = perspector of orthic triangle and polar triangle of the complement of the polar circle

X(2) = trilinear pole, wrt orthocentroidal triangle, of Fermat axis

X(2) = trilinear pole, wrt 1st Parry triangle, of line X(1499)X(8598)

X(2) = pole of Brocard axis wrt Stammler hyperbola

X(2) = pole of de Longchamps line wrt the nine-point circle

X(2) = pole of de Longchamps line wrt the de Longchamps circle

X(2) = pole of orthic axis wrt polar circle

X(2) = crosspoint of X(3) and X(6) wrt both the excentral and tangential triangles

X(2) = intersection of tangents at X(1) and X(9) to the hyperbola passing through X(1), X(9) and the excenters (the Jerabek hyperbola of the excentral triangle)

X(2) = crosspoint of X(1) and X(9) wrt excentral triangle

X(2) = crosspoint of X(3) and X(6) wrt excentral triangle

X(2) = crosspoint of X(7) and X(8) wrt 2nd Conway triangle

X(2) = antipode of X(3228) in hyperbola {{A,B,C,X(2),X(6)}}

X(2) = antipode of X(1494) in hyperbola {{A,B,C,X(2),X(69)}}

X(2) = perspector of pedal and anticevian triangles of X(20)

X(2) = homothetic center of the 2nd pedal triangle of X(4) and the 3rd pedal triangle of X(3)

X(2) = perspector of ABC and the reflection in X(6) of the pedal triangle of X(6)

X(2) = perspector of orthic triangle and polar triangle of the complement of the polar circle

X(2) = Moses-radical-circle-inverse of X(34235)

X(2) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,192), (4,193), (6,194), (7,145), (8,144), (30,1494), (69,20), (75,8), (76,69), (83,6), (85,7), (86,1), (87,330), (95,3), (98,385), (99,523), (190,514), (264,4), (274,75), (276, 264), (287,401), (290,511), (308,76), (312,329), (325,147), (333,63), (348,347), (491,487), (492,488), (523,148), (626,1502)

X(2) = cevapoint of X(i) and X(j) for these (i,j): (1,9), (3,6), (5,216), (10,37), (11,650), (32,206), (39,141), (44,214), (57,223), (114,230), (115,523), (125,647), (128,231), (132,232), (140,233), (188,236), (5408,5409)

X(2) = X(i)-cross conjugate of X(j) for these (i,j):

(1,7), (3,69), (4,253), (5,264), (6,4), (9,8), (10,75), (32,66), (37,1), (39,6), (44,80), (57,189), (75,330), (114,325), (140,95), (141,76), (142,85), (178,508), (187,67), (206,315), (214,320), (216,3), (223,329), (226,92), (230,98), (233,5), (281,280), (395,14), (396,13), (440,306), (511,290), (514,190), (523,99)

X(2) = crosspoint of X(i) and X(j) for these (i,j): (1,87), (75,85), (76,264), (83,308), (86,274), (95,276)

X(2) = crosssum of X(i) and X(j) for these (i,j): (1,43), (2,194), (31,41), (32,184), (42,213), (51,217), (125,826), (649,1015), (688,1084), (902,1017), (1400,1409)

X(2) = crossdifference of every pair of points on line X(187)X(237)

X(2) = X(i)-Hirst inverse of X(j) for these (i,j):

(1,239), (3,401), (4,297), (6,385), (21,448), (27,447), (69,325), (75,350), (98,287), (115,148), (193,230), (291,335), (298,299), (449,452)

X(2) = X(3)-line conjugate of X(237) = X(316)-line conjugate of X(187)

X(2) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1045), (2,191), (86,2), (174,1046), (333,20), (366,846)

X(2) = X(i)-beth conjugate of X(j) for these (i,j): (2,57), (21,995) (190,2), (312,312), (333,2), (643,55), (645,2), (646,2), (648,196), (662,222)

X(2) = one of two harmonic traces of the power circles; the other is X(858)

X(2) = one of two harmonic traces of the McCay circles; the other is X(111)

X(2) = orthocenter of X(i)X(j)X(k) for these (i,j,k): ((4,6,1640), (4,10,4040)

X(2) = centroid of PU(1)X(76) (1st, 2nd, 3rd Brocard points)

X(2) = trilinear pole of PU(i) for these i: 24, 41

X(2) = crossdifference of PU(i) for these i: 2, 26

X(2) = barycentric product of PU(i) for these i 3, 35

X(2) = trilinear product of PU(i) for these i: 6,124

X(2) = bicentric sum of PU(i) for these i: 116, 117, 118, 119, 138, 148

X(2) = midpoint of PU(i) for these i: 116, 117, 118, 119, 127

X(2) = intersection of diagonals of trapezoid PU(11)PU(45) (lines P(11)P(45) and U(11)U(45))

X(2) = X(5182) of 6th Brocard triangle (see X(384))

X(2) = PU(148)-harmonic conjugate of X(669)

X(2) = bicentric difference of PU(147)

X(2) = eigencenter of 2nd Brocard triangle

X(2) = perspector of ABC and unary cofactor triangle of Lucas central triangle

X(2) = perspector of ABC and unary cofactor triangle of Lucas(-1) central triangle

X(2) = perspector of ABC and unary cofactor triangle of Lucas tangents triangle

X(2) = perspector of ABC and unary cofactor triangle of Lucas(-1) tangents triangle

X(2) = perspector of ABC and unary cofactor triangle of Lucas inner triangle

X(2) = perspector of ABC and unary cofactor triangle of Lucas(-1) inner triangle

X(2) = perspector of ABC and unary cofactor triangle of 1st anti-Brocard triangle

X(2) = perspector of ABC and unary cofactor triangle of 1st Sharygin triangle

X(2) = perspector of ABC and unary cofactor triangle of 2nd Sharygin triangle

X(2) = perspector of ABC and unary cofactor triangle of 1st Pamfilos-Zhou triangle

X(2) = perspector of ABC and unary cofactor triangle of Artzt triangle

X(2) = perspector of 1st Parry triangle and unary cofactor of 3rd Parry triangle

X(2) = X(6032) of 4th anti-Brocard triangle

X(2) = orthocenter of X(3)X(9147)X(9149)

X(2) = exsimilicenter of Artzt and anti-Artzt circles; the insimilicenter is X(183)

X(2) = perspector of ABC and cross-triangle of inner- and outer-squares triangles

X(2) = perspector of ABC and 2nd Brocard triangle of 1st Brocard triangle

X(2) = perspector of half-altitude triangle and cross-triangle of ABC and half-altitude triangle

X(2) = 4th-anti-Brocard-to-anti-Artzt similarity image of X(111)

X(2) = homothetic center of Aquila triangle and cross-triangle of Aquila and anti-Aquila triangles

X(2) = X(551)-of-cross-triangle-of Aquila-and-anti-Aquila-triangles

X(2) = harmonic center of polar circle and circle O(PU(4))

X(2) = Thomson-isogonal conjugate of X(3)

X(2) = Lucas-isogonal conjugate of X(20)

X(2) = X(3679)-of-outer-Garcia-triangle

X(2) = Dao image of X(13)

X(2) = Dao image of X(14)

X(2) = center of equilateral triangle X(3)PU(5)

X(2) = center of equilateral triangle formed by the circumcenters of BCF, CAF, ABF, where F = X(13)

X(2) = center of equilateral triangle formed by the circumcenters of BCF', CAF', ABF', where F' = X(14)

X(2) = trisector nearest X(5) of segment X(3)X(5)

X(2) = trisector nearest X(4) of segment X(4)X(20)

X(2) = pedal antipodal perspector of X(15)

X(2) = pedal antipodal perspector of X(16)

X(2) = K(X(3)), as defined at X(174)

X(2) = Ehrmann-mid-to-Johnson similarity image of X(381)

X(2) = Kiepert hyperbola antipode of X(671)

X(2) = antigonal conjugate of X(671)

X(2) = trilinear square of X(366)

X(2) = intersection of diagonals of trapezoid X(1)X(7)X(8)X(9)

X(2) = Danneels point of X(99)

X(2) = Danneels point of X(648)

X(2) = perspector of Spieker circle

X(2) = orthic-isogonal conjugate of X(193)

X(2) = X(154)-of-intouch-triangle

Trilinears a(b

Barycentrics sin 2A : sin 2B : sin 2C

Barycentrics tan B + tan C : tan C + tan A: tan A + tan B

Barycentrics S^2 - SB SC : :

Barycentrics 1 - cot B cot C : :

Tripolars 1 : 1 : 1

As a point on the Euler line, X(3) has Shinagawa coefficients (1, -1).

X(3) is the point of concurrence of the perpendicular bisectors of the sides of ABC. The lengths of segments AX, BX, CX are equal if and only if X = X(3). This common distance is the radius of the circumcircle, which passes through vertices A,B,C. Called the circumradius, it is given by R = a/(2 sin A) = abc/(4*area(ABC)).

The tangents at vertices of excentral triangle to the McCay cubic K003 concur in X(3). Also, the tangents at A,B,C to the orthocubic K006 concur in X(3). (Randy Hutson, November 18, 2015)

Let A'B'C' be the cevian triangle of X(4). Let A" be X(4)-of-AB'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(3). (Randy Hutson, June 27, 2018)

Let P be a point in the plane of ABC. Let P' be the isogonal conjugate of P. Let P" be the pedal antipodal perspector of P. X(3) is the unique point P for which P' = P". (Randy Hutson, June 27, 2018)

If you have The Geometer's Sketchpad, you can view **Circumcenter**.

If you have GeoGebra, you can view **Circumcenter**.

X(3) lies on the Thomson cubic, the Darboux cubic, the Napoleon cubic, the Neuberg cubic, the McCay cubic, then Darboux quintic, and these lines:

{1,35}, {2,4}, {6,15}, {7,943}, {8,100}, {9,84}, {10,197}, {11,499}, {12,498}, {13,17}, {14,18}, {19,1871}, {31,601}, {33,1753}, {34,1465}, {37,975}, {38,976}, {41,218}, {42,967}, {43,5247}, {47,1399}, {48,71}, {49,155}, {51,3527}, {54,97}, {60,1175}, {63,72}, {64,154}, {66,141}, {67,542}, {68,343}, {69,332}, {73,212}, {74,110}, {76,98}, {77,1410}, {80,5445}, {81,5453}, {83,262}, {85,5088}, {86,1246}, {90,1898}, {95,264}, {96,5392}, {101,103}, {102,109}, {105,277}, {106,1293}, {107,1294}, {108,1295}, {111,1296}, {112,1297}, {113,122}, {114,127}, {115,2079}, {119,123}, {125,131}, {128,1601}, {142,516}, {143,1173}, {144,5843}, {145,1483}, {147,2896}, {149,1484}, {158,243}, {161,1209}, {164,3659}, {169,910}, {172,2276}, {191,1768}, {193,1353}, {194,385}, {200,963}, {201,1807}, {207,1767}, {214,2800}, {217,3289}, {223,1035}, {225,1074}, {226,4292}, {227,1455}, {230,2549}, {232,1968}, {238,978}, {248,3269}, {252,930}, {256,987}, {269,939}, {295,2196}, {296,820}, {298,617}, {299,616}, {302,621}, {303,622}, {305,1799}, {315,325}, {323,3431}, {329,2096}, {345,1791}, {347,1119}, {348,1565}, {351,2780}, {352,353}, {356,3278}, {358,6120}, {373,3066}, {380,2257}, {388,495}, {390,1058}, {392,3420}, {393,1217}, {395,398}, {396,397}, {476,477}, {480,5223}, {485,590}, {486,615}, {489,492}, {490,491}, {496,497}, {501,5127}, {513,3657}, {518,3433}, {519,3654}, {523,5664}, {524,5486}, {525,878}, {528,3813}, {532,5859}, {533,5858}, {539,3519}, {541,5642}, {543,5569}, {551,3653}, {595,995}, {604,2269}, {607,1951}, {608,1950}, {609,5280}, {611,1469}, {612,5322}, {613,1428}, {614,5310}, {618,635}, {619,636}, {623,629}, {624,630}, {639,641}, {640,642}, {653,1148}, {659,2826}, {662,1098}, {667,1083}, {669,1499}, {690,6334}, {691,842}, {692,2807}, {695,1613}, {732,6308}, {741,6010}, {758,5884}, {759,6011}, {805,2698}, {840,2742}, {843,2709}, {846,2944}, {847,925}, {895,4558}, {901,953}, {902,1201}, {905,1946}, {915,2969}, {917,1305}, {920,1858}, {927,2724}, {929,2723}, {934,972}, {935,2697}, {938,3488}, {945,1457}, {947,5399}, {950,1210}, {951,1407}, {955,1170}, {960,997}, {962,1621}, {968,6051}, {974,5504}, {984,3497}, {1000,1476}, {1014,3945}, {1015,2241}, {1018,4513}, {1033,1249}, {1037,1066}, {1046,4650}, {1047,2636}, {1054,1283}, {1055,1334}, {1056,3600}, {1057,1450}, {1069,6238}, {1072,3011}, {1075,1941}, {1093,1105}, {1104,3752}, {1107,4386}, {1124,2066}, {1131,3316}, {1132,3317}, {1135,6121}, {1137,6122}, {1138,3471}, {1139,3370}, {1140,3397}, {1167,1413}, {1177,1576}, {1180,1627}, {1184,1194}, {1196,1611}, {1199,1994}, {1203,5313}, {1211,5810}, {1213,5816}, {1247,2640}, {1263,3459}, {1270,5874}, {1271,5875}, {1276,5240}, {1277,5239}, {1290,2687}, {1298,1303}, {1301,5897}, {1304,2693}, {1308,2717}, {1309,2734}, {1330,4417}, {1331,1797}, {1335,2067}, {1337,3489}, {1338,3490}, {1348,2040}, {1349,2039}, {1364,1795}, {1386,3941}, {1389,2320}, {1397,1682}, {1398,1870}, {1400,2268}, {1406,1464}, {1411,1772}, {1412,2213}, {1425,3561}, {1427,1448}, {1433,2188}, {1445,5728}, {1446,3188}, {1447,3673}, {1452,1905}, {1453,2999}, {1471,2293}, {1475,2280}, {1495,3426}, {1500,2242}, {1506,5475}, {1568,3521}, {1575,4426}, {1587,3068}, {1588,3069}, {1602,2550}, {1603,2551}, {1612,4000}, {1625,1987}, {1630,3197}, {1632,2790}, {1633,5698}, {1661,2883}, {1672,3238}, {1673,3237}, {1676,5403}, {1677,5404}, {1696,3731}, {1698,4413}, {1699,3624}, {1709,3683}, {1714,5721}, {1723,2264}, {1724,3216}, {1728,1864}, {1737,1837}, {1745,1935}, {1762,2939}, {1770,1836}, {1779,1780}, {1788,3486}, {1794,3173}, {1796,3690}, {1808,4173}, {1810,4587}, {1811,4571}, {1813,3270}, {1834,5292}, {1901,5747}, {1914,2275}, {1916,3406}, {1918,2274}, {1939,6181}, {1960,2821}, {1986,2904}, {2007,3235}, {2008,3236}, {2053,2108}, {2120,3463}, {2121,3482}, {2130,3343}, {2131,3350}, {2133,5670}, {2163,2334}, {2174,2911}, {2183,2267}, {2197,2286}, {2222,2716}, {2292,3724}, {2329,3501}, {2346,3296}, {2407,2452}, {2548,3815}, {2688,2690}, {2689,2695}, {2691,2752}, {2692,2758}, {2694,2766}, {2696,2770}, {2699,2703}, {2700,2702}, {2701,2708}, {2704,2711}, {2705,2712}, {2706,2713}, {2707,2714}, {2710,2715}, {2718,2743}, {2719,2744}, {2720,2745}, {2721,2746}, {2722,2747}, {2725,2736}, {2726,2737}, {2727,2738}, {2728,2739}, {2729,2740}, {2730,2751}, {2731,2757}, {2732,2762}, {2733,2765}, {2735,2768}, {2783,4436}, {2792,4655}, {2797,6130}, {2801,3678}, {2810,3939}, {2814,3960}, {2827,4491}, {2854,5505}, {2886,4999}, {2888,3448}, {2916,3456}, {2951,3646}, {2971,3563}, {3006,5300}, {3058,4309}, {3061,3496}, {3065,3467}, {3092,5413}, {3093,5412}, {3100,6198}, {3101,6197}, {3165,5669}, {3166,5668}, {3177,3732}, {3200,3205}, {3201,3206}, {3218,3418}, {3219,3876}, {3224,6234}, {3229,3360}, {3272,3334}, {3276,3280}, {3277,3282}, {3305,5927}, {3306,5439}, {3332,4648}, {3341,3347}, {3351,3354}, {3366,3391}, {3367,3392}, {3373,3387}, {3374,3388}, {3381,5402}, {3382,5401}, {3399,3407}, {3413,6178}, {3414,6177}, {3417,3869}, {3436,5552}, {3437,5224}, {3440,5674}, {3441,5675}, {3447,6328}, {3452,6259}, {3460,3465}, {3461,3483}, {3462,5667}, {3464,3466}, {3474,3485}, {3555,3870}, {3582,4330}, {3583,4324}, {3584,4325}, {3585,4316}, {3589,5480}, {3614,5326}, {3620,5921}, {3632,5288}, {3647,3652}, {3667,4057}, {3679,5258}, {3681,4420}, {3687,5814}, {3694,5227}, {3705,5015}, {3710,3977}, {3711,5531}, {3733,6003}, {3734,3934}, {3740,5302}, {3824,5715}, {3849,6232}, {3874,4973}, {3877,4881}, {3889,3957}, {3901,4880}, {3925,6253}, {4001,4101}, {4317,4995}, {4338,4870}, {4340,5323}, {4549,4846}, {4653,6176}, {4720,5372}, {4850,5262}, {4993,4994}, {5226,5714}, {5260,5818}, {5268,5345}, {5275,5277}, {5284,5550}, {5286,5305}, {5306,5319}, {5346,5355}, {5436,5437}, {5441,5442}, {5443,5444}, {5530,5725}, {5541,6264}, {5590,5594}, {5591,5595}, {5606,5951}, {5638,6141}, {5639,6142}, {5640,5643}, {5656,6225}, {5658,5811}, {5672,6191}, {5673,6192}, {5735,6173}, {5962,5963}, {5971,6031}, {6082,6093}, {6118,6250}, {6119,6251}, {6228,6230}, {6229,6231}, {6233,6323}, {6236,6325}, {6294,6295}, {6296,6298}, {6297,6299}, {6300,6302}, {6301,6303}, {6304,6306}, {6305,6307}, {6311,6313}, {6312,6314}, {6315,6317}, {6316,6318}, {6391,6461}, {6413,6458}, {6414,6457}, {6581,6582}}.

X(3) is the {X(2),X(4)}-harmonic conjugate of X(5). For a list of
other harmonic conjugates of X(3), click **Tables** at the top of
this page. If triangle ABC is acute, then X(3) is the incenter of the
tangential triangle and the Bevan point, X(40), of the orthic
triangle.

X(3) = midpoint of X(i) and X(j) for these (i,j): (1,40), (2,376), (4,20), (22,378), (74,110), (98,99), (100,104), (101,103), (102,109), (476,477)

X(3) = reflection of X(i) in X(j) for these (i,j): (1,1385), (2,549), (4,5), (5,140), (6,182), (20,550), (52,389), (110,1511), (114,620), (145,1483), (149,1484), (155,1147), (193,1353), (195,54), (265,125), (355,10), (381,2), (382,4), (399,110), (550,548), (576,575), (946,1125), (1351,6), (1352,141), (1482,1)

X(3) = isogonal conjugate of X(4)

X(3) = isotomic conjugate of X(264)

X(3) = inverse-in-nine-point-circle of X(2072)

X(3) = inverse-in-orthocentroidal-circle of X(5)

X(3) = inverse-in-1st-Lemoine-circle of X(2456)

X(3) = inverse-in-2nd-Lemoine-circle of X(1570)

X(3) = complement of X(4)

X(3) = anticomplement of X(5)

X(3) = complementary conjugate of X(5)

X(3) = anticomplementary conjugate of X(2888)

X(3) = eigencenter of the medial triangle

X(3) = eigencenter of the tangential triangle

X(3) = exsimilicenter of 1st and 2nd Kenmotu circles

X(3) = exsimilicenter of nine-point circle and tangential circle

X(3) = X(1)-of-Trinh-triangle if ABC is acute

X(3) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,6), (4,155), (5,195), (20,1498), (21,1), (22,159), (30,399), (63,219), (69,394), (77,222), (95,2), (96,68), (99,525), (100,521), (110,520), (250, 110), (283,255)

X(3) = cevapoint of X(i) and X(j) for these (i,j): (6,154), (48,212), (55,198), (71,228), (185,417), (216,418)

X(3) = X(i)-cross conjugate of X(j) for these (i,j): (48,222), (55,268), (71,63), (73,1), (184,6), (185,4), (212,219), (216,2), (228,48), (520,110)

X(3) = crosspoint of X(i) and X(j) for these (i,j): (1,90), (2,69), (4,254), (21,283), (54,96), (59,100), (63,77), (78,271), (81,272), (95,97), (99,249), (110,250), (485,486)

X(3) = crosssum of X(i) and X(j) for these (i,j):

(1,46), (2,193), (3,155), (5,52), (6,25), (11,513), (19,33), (30,113), (37,209), (39, 211), (51,53), (65,225), (114,511), (115,512), (116,514), (117, 515), (118,516), (119,517), (120,518), (121,519), (122,520), (123,521), (124,522), (125,523), (126,524), (127,525), (128,1154), (136,924), (184,571), (185,235), (371,372), (487,488)

X(3) = crossdifference of every pair of points on the line X(230)X(231)

X(3) = X(i)-Hirst inverse of X(j) for these (i,j): (2, 401), (4,450), (6,511), (21,416), (194, 385)

X(3) = X(2)-line conjugate of X(468)

X(3) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1046), (21,3), (188,191), (259,1045)

X(3) = X(i)-beth conjugate of X(j) for these (i,j): (3,603), (8,355), (21,56), (78,78), (100,3), (110,221), (271,84), (283,3), (333,379), (643,8)

X(3) = center of inverse-in-de-Longchamps-circle-of-anticomplementary-circle

X(3) = perspector of inner and outer Napoleon triangles

X(3) = Hofstadter 2 point

X(3) = trilinear product of vertices of 2nd Brocard triangle

X(3) = orthocenter of X(i)X(j)X(k) for these (i,j,k): (1,8,5556), (1,9,885), (2,6,1640), (2,10,4049), (3,6,879), (3,66,2435), (4,6,879), (7,8,885), (67,74,879), (6,64,2435), (4,66,2435)

X(3) = intersection of tangents at X(3) and X(4) to Orthocubic K006

X(3) = homothetic center of tangential triangle and 2nd isogonal triangle of X(4); see X(36)

X(3) = trilinear pole of line X(520)X(647)

X(3) = crossdifference of PU(4)

X(3) = trilinear product of PU(16)

X(3) = barycentric product of PU(22)

X(3) = midpoint of PU(i) for these i: 37, 44

X(3) = bicentric sum of PU(i) for these i: 37, 44, 63, 125

X(3) = vertex conjugate of PU(39)

X(3) = PU(63)-harmonic conjugate of X(351)

X(3) = PU(125)-harmonic conjugate of X(650)

X(3) = intersection of tangents to orthocentroidal circle at PU(5)

X(3) = X(3398) of 5th Brocard triangle (see X(32))

X(3) = X(182) of 6th Brocard triangle (see X(384))

X(3) = homothetic center of 1st anti-Brocard triangle and 6th Brocard triangle

X(3) = similitude center of antipedal triangles of the 1st and 2nd Brocard points (PU(1))

X(3) = inverse-in-polar-circle of X(403)

X(3) = inverse-in-{circumcircle, nine-point circle}-inverter of X(858)

X(3) = inverse-in-de-Longchamps-circle of X(3153)

X(3) = inverse-in-Steiner-circumellipse of X(401)

X(3) = inverse-in-Steiner-inellipse of X(441)

X(3) = inverse-in-MacBeath-circumconic of X(3284)

X(3) = radical trace of circumcircle and 8th Lozada circle

X(3) = perspector of medial triangle and polar triangle of the complement of the polar circle

X(3) = pole of line X(6)X(110) wrt Parry circle

X(3) = pole wrt polar circle of trilinear polar of X(2052) (line X(403)X(523))

X(3) = pole wrt {circumcircle, nine-point circle}-inverter of de Longchamps line

X(3) = polar conjugate of X(2052)

X(3) = X(i)-isoconjugate of X(j) for these (i,j): (6,92), (24,91), (25,75), (48,2052), (93,2964), (112,1577), (1101,2970), (2962,3518)

X(3) = X(30)-vertex conjugate of X(523)

X(3) = homothetic center of any 2 of {tangential, Kosnita, 2nd Euler} triangles

X(3) = X(5)-of-excentral-triangle

X(3) = X(26)-of-intouch-triangle

X(3) = antigonal image of X(265)

X(3) = X(2)-of-antipedal-triangle-of-X(6)

X(3) = perspector of the MacBeath Circumconic

X(3) = perspector of ABC and unary cofactor triangle of 5th Euler triangle

X(3) = intersection of trilinear polars of any 2 points on the MacBeath circumconic

X(3) = circumcevian isogonal conjugate of X(1)

X(3) = orthology center of ABC and orthic triangle

X(3) = orthology center of Fuhrmann triangle and ABC

X(3) = orthic isogonal conjugate of X(155)

X(3) = Miquel associate of X(2)

X(3) = X(40)-of-orthic-triangle if ABC is acute

X(3) = X(98)-of-1st-Brocard-triangle

X(3) = X(1380)-of-2nd-Brocard-triangle

X(3) = X(399)-of-orthocentroidal-triangle

X(3) = X(104)-of X(1)-Brocard-triangle

X(3) = X(74)-of X(2)-Brocard-triangle

X(3) = X(74)-of-X(4)-Brocard-triangle

X(3) = X(597)-of-antipedal-triangle-of-X(2)

X(3) = X(182)-of-1st-anti-Brocard-triangle

X(3) = X(381)-of-4th-anti-Brocard-triangle

X(3) = QA-P12 (Orthocenter of the QA-Diagonal Triangle)-of-quadrilateral X(98)X(99)X(110)X(111)

X(3) = orthocenter of X(2)X(9147)X(9149)

X(3) = perspector of ABC and 1st Brocard triangle of 6th Brocard triangle

X(3) = perspector of ABC and 1st Brocard triangle of circumorthic triangle

X(3) = perspector of ABC and 1st Brocard triangle of dual of orthic triangle

X(3) = perspector of ABC and cross-triangle of ABC and half-altitude triangle

X(3) = homothetic center of inner Yff triangle and cross-triangle of ABC and 1st Johnson-Yff triangle

X(3) = homothetic center of outer Yff triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle

X(3) = anti-Artzt-to-4th-anti-Brocard similarity image of X(6)

X(3) = Thomson-isogonal conjugate of X(2)

X(3) = Lucas-isogonal conjugate of X(2979)

X(3) = X(4)-of-2nd-anti-extouch triangle

X(3) = X(185)-of-A'B'C', as described in ADGEOM #2697 (8/26/2015, Tran Quang Hung)

X(3) = X(5)-of-3rd-anti-Euler-triangle

X(3) = X(5)-of-4th-anti-Euler-triangle

X(3) = X(671)-of-McCay-triangle

X(3) = Dao image of X(4)

X(3) = centroid of ABCX(20)

X(3) = Kosnita(X(20),X(2)) point

X(3) = centroid of incenter and excenters

X(3) = X(265)-of-Fuhrmann-triangle

X(3) = intersection of tangents to 2nd Lemoine circle at intersections with Brocard circle

X(3) = perspector of ABC and antipedal triangle of X(64)

X(3) = trisector nearest X(5) of segment X(5)X(20)

X(3) = Ehrmann-vertex-to-Ehrmann-side similarity image of X(4)

X(3) = Ehrmann-mid-to-ABC similarity image of X(4)

X(3) = Ehrmann-mid-to-Johnson similarity image of X(5)

X(3) = Johnson-to-Ehrmann-mid similarity image of X(20)

X(3) = center of inverse similitude of AAOA triangle and Ehrmann side-triangle

X(3) = X(5)-of-hexyl-triangle

X(3) = X(175)-of-Lucas-central-triangle

X(3) = reflection of X(2080) in the Lemoine axis

X(3) = excentral-isogonal conjugate of X(191)

X(3) = excentral-isotomic conjugate of X(2938)

X(3) = crosssum of foci of orthic inconic

X(3) = crosspoint of foci of orthic inconic

X(3) = similicenter of antipedal triangles of PU(1)

X(3) = excentral-to-ABC functional image of X(40)

X(3) = orthic-to-ABC barycentric image of X(4)

X(3) = orthic-to-ABC functional image of X(5)

X(3) = Feuerbach-to-ABC functional image of X(5)

X(3) = intouch-to-ABC functional image of X(1)

X(3) = ABC-to-excentral barycentric image of X(10)

X(3) = concurrence of Euler lines of intouch triangle and A-, B-, and C-extouch triangles

X(3) = perspector of hexyl triangle and cevian triangle of X(21)

X(3) = perspector of pedal and anticevian triangles of X(1498)

X(3) = perspector of ABC and medial triangle of pedal triangle of X(20)

X(3) = perspector of ABC and the reflection in X(6) of the antipedal triangle of X(6)

X(3) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,35,55), (1,36,56), (1,46,65), (1,55,3295), (1,56,999), (1,57,942), (1,165,40), (1,171,5711), (1,484,5903), (1,1038,1060), (1,1040,1062), (1,1754,5706), (1,2093,3340), (1,3333,5045), (1,3336,5902), (1,3338,354), (1,3361,3333), (1,3550,5255), (1,3576,1385), (1,3612,2646), (1,3746,3303), (1,5010,35), (1,5119,3057), (1,5131,3336), (1,5264,5710), (1,5563,3304), (1,5697,2098), (1,5903,2099), (2,4,5), (2,5,1656), (2,20,4), (2,21,405), (2,22,25), (2,23,1995), (2,24,6642), (2,25,5020), (2,140,3526), (2,186,6644), (2,377,442), (2,381,5055), (2,382,3851), (2,401,458), (2,404,474), (2,411,3149), (2,418,6638), (2,452,5084), (2,464,440), (2,546,5079), (2,548,1657), (2,549,5054), (2,550,382), (2,631,140), (2,858,5094), (2,859,4245), (2,1010,2049), (2,1113,1344), (2,1114,1345), (2,1370,427), (2,1599,1583), (2,1600,1584), (2,1656,5070), (2,1657,3843), (2,2071,378), (2,2475,2476), (2,2478,4187), (2,2554,2570), (2,2555,2571), (2,2675,2676), (2,3090,3628), (2,3091,3090), (2,3146,3091), (2,3151,469), (2,3152,5125), (2,3522,20), (2,3523,631), (2,3524,549), (2,3525,632), (2,3528,550), (2,3529,546), (2,3534,3830), (2,3543,3545), (2,3545,547), (2,3546,3548), (2,3547,3549), (2,3548,6640), (2,3549,6639), (2,3552,384), (2,3627,5072), (2,3832,5056), (2,3839,5071), (2,4184,1011), (2,4188,404), (2,4189,21), (2,4190,377), (2,4210,4191), (2,4216,859), (2,4226,1316), (2,5046,4193), (2,5056,5067), (2,5059,3832), (2,5189,5169), (2,6636,22), (4,5,381), (4,21,3560), (4,24,25), (4,25,1598), (4,140,1656), (4,186,24), (4,376,20), (4,378,1593), (4,381,3843), (4,382,3830), (4,548,3534), (4,549,3526), (4,550,1657), (4,631,2), (4,632,5079), (4,1006,405), (4,1593,1597), (4,1656,3851), (4,1657,5073), (4,1658,2070), (4,2937,5899), (4,3088,1595), (4,3089,1596), (4,3090,3091), (4,3091,546), (4,3146,3627), (4,3147,3542), (4,3515,3517), (4,3520,378), (4,3522,550), (4,3523,140), (4,3524,631), (4,3525,3090), (4,3526,5055), (4,3528,376), (4,3529,3146), (4,3530,5054), (4,3533,5056), (4,3541,427), (4,3542,235), (4,3543,3853), (4,3545,3832), (4,3548,2072), (4,3627,5076), (4,3628,5072), (4,3832,3845), (4,3839,3861), (4,3855,3839), (4,5054,5070), (4,5056,3850), (4,5067,3545), (4,5068,3858), (4,5071,3855), (4,6353,3089), (4,6621,6624), (4,6622,6623), (5,20,382), (5,26,25), (5,140,2), (5,376,1657), (5,381,3851), (5,382,3843), (5,427,5576), (5,546,3091), (5,547,5056), (5,548,20), (5,549,140), (5,631,3526), (5,632,3628), (5,1656,5055), (5,1657,3830), (5,1658,24), (5,3090,5079), (5,3091,5072), (5,3522,3534), (5,3523,5054), (5,3526,5070), (5,3529,5076), (5,3530,631), (5,3534,5073), (5,3627,546), (5,3628,3090), (5,3845,3850), (5,3850,3545), (5,3853,3832), (5,3858,5066), (5,3861,3855), (5,5066,5068), (5,5498,6143), (5,6642,5020), (5,6644,6642), (6,182,5050), (6,187,1384), (6,371,3311), (6,372,3312), (6,574,5024), (6,1151,371), (6,1152,372), (6,1351,5093), (6,1620,1192), (6,2076,5017), (6,3053,32), (6,3311,6417), (6,3312,6418), (6,3592,6419), (6,3594,6420), (6,4252,58), (6,4255,386), (6,4258,4251), (6,5013,39), (6,5022,4253), (6,5023,3053), (6,5085,182), (6,5102,5097), (6,5210,187), (6,5585,5210), (6,6200,6221), (6,6221,6199), (6,6396,6398), (6,6398,6395), (6,6409,1151), (6,6410,1152), (6,6411,6200), (6,6412,6396), (6,6417,6500), (6,6418,6501), (6,6419,6427), (6,6420,6428), (6,6425,3592), (6,6426,3594), (6,6433,6437), (6,6434,6438), (6,6451,6445), (6,6452,6446), (6,6455,6407), (6,6456,6408), (7,3487,6147), (7,5703,3487), (8,100,5687), (8,2975,956), (8,5657,5690), (8,5731,944), (9,936,5044), (9,1490,5777), (9,5438,936), (10,355,5790), (10,993,958), (10,5267,993), (10,5745,5791), (11,5433,499), (11,6284,1479), (12,5432,498), (15,16,6), (15,62,61), (15,3364,371), (15,3365,372), (15,5237,62), (15,5352,5238), (16,61,62), (16,3389,371), (16,3390,372), (16,5238,61), (16,5351,5237), (20,21,1012), (20,140,381), (20,186,26), (20,376,550), (20,381,5073), (20,404,3149), (20,417,6638), (20,549,1656), (20,550,3534), (20,631,5), (20,1006,3560), (20,1656,3830), (20,1658,2937), (20,2060,3079), (20,3090,3627), (20,3091,3146), (20,3146,3529), (20,3522,376), (20,3523,2), (20,3524,140), (20,3525,546), (20,3526,3843), (20,3528,548), (20,3530,3526), (20,3533,3845), (20,3543,5059), (20,3628,5076), (20,5054,3851), (20,5056,3543), (20,5067,3853), (21,404,2), (21,411,4), (21,416,1982), (21,1816,29), (21,1817,28), (21,3658,3109), (21,4188,474), (21,4203,4195), (21,4225,859), (22,24,26), (22,26,2937), (22,381,5899), (22,426,6638), (22,631,6642), (22,1599,3155), (22,1600,3156), (22,1995,23), (22,6644,2070), (23,1995,25), (24,25,3517), (24,26,2070), (24,186,3515), (24,378,4), (24,1593,1598), (24,1657,5899), (24,3516,1597), (24,3520,1593), (25,378,1597), (25,426,6617), (25,1593,4), (25,3515,24), (25,3516,1593), (26,140,6642), (26,378,382), (26,382,5899), (26,6642,3517), (26,6644,24), (28,4219,4), (29,412,4), (32,39,6), (32,182,3398), (32,187,3053), (32,574,39), (32,3053,1384), (32,5171,2080), (32,5206,187), (33,1753,1872), (35,36,1), (35,56,3295), (35,5010,5217), (35,5204,999), (35,5563,3746), (35,5584,6244), (36,55,999), (36,165,3428), (36,2078,5126), (36,3746,5563), (36,5010,55), (36,5217,3295), (39,187,32), (39,574,5013), (39,5008,5041), (39,5013,5024), (39,5023,1384), (39,5206,3053), (40,57,5709), (40,165,3579), (40,1385,1482), (40,3576,1), (41,672,218), (46,3612,1), (48,71,219), (50,566,6), (52,389,568), (52,569,6), (55,56,1), (55,165,6244), (55,3303,3746), (55,3304,3303), (55,5204,56), (55,5217,35), (55,5584,40), (56,1466,57), (56,3303,3304), (56,3304,5563), (56,5204,36), (56,5217,55), (56,5584,3428), (57,942,5708), (57,1420,1467), (57,3601,1), (58,386,6), (58,580,5398), (58,4256,386), (58,4257,4252), (58,4276,4267), (58,4278,3286), (61,62,6), (61,5238,15), (61,5351,16), (61,5864,1351), (62,5237,16), (62,5352,15), (62,5865,1351), (63,72,3927), (63,78,72), (63,3984,3951), (63,4652,3916), (63,4855,78), (63,5440,3940), (64,154,1498), (65,1155,46), (65,2646,1), (69,3926,3933), (69,6337,3926), (71,1818,3781), (72,78,3940), (72,3916,63), (72,5440,78), (73,255,3157), (73,603,222), (74,1511,399), (74,1614,6241), (76,99,1975), (76,1078,183), (78,1259,1260), (78,3916,3927), (78,3951,3984), (78,4652,63), (78,4855,5440), (84,936,5777), (84,5044,5779), (84,5438,5720), (99,1078,76), (99,5152,5989), (100,2975,8), (100,5303,2975), (101,3730,220), (104,5657,956), (110,1614,156), (140,376,382), (140,381,5070), (140,382,5055), (140,546,3628), (140,549,631), (140,550,4), (140,631,5054), (140,632,3525), (140,1368,3548), (140,1657,3851), (140,1658,6644), (140,3146,5079), (140,3522,1657), (140,3528,3534), (140,3529,5072), (140,3530,549), (140,3534,3843), (140,3627,3090), (140,3628,632), (140,3845,5067), (140,3853,547), (140,5428,1006), (140,6636,2937), (143,5946,3567), (155,1147,3167), (157,160,159), (165,5010,2077), (165,6282,3587), (171,5329,1460), (182,576,575), (182,578,569), (182,1160,6418), (182,1161,6417), (182,1350,1351), (182,5092,5085), (182,5171,32), (183,1975,76), (184,185,1181), (184,394,3167), (184,1092,1147), (184,1147,49), (184,1204,185), (184,3917,394), (184,5562,155), (185,1092,155), (185,3917,5562), (186,376,22), (186,378,25), (186,550,2937), (186,1593,3517), (186,3516,1598), (186,3520,4), (186,3651,2915), (187,574,6), (187,2021,1691), (187,5162,2076), (187,5188,5171), (187,5206,5023), (191,6326,5694), (198,1436,610), (199,1011,25), (199,3145,2915), (212,603,255), (212,4303,3157), (216,577,6), (216,3284,5158), (220,3207,101), (230,5254,3767), (232,1968,2207), (235,468,3542), (235,1885,4), (237,3148,25), (243,1940,158), (255,4303,222), (283,1790,1437), (284,579,6), (371,372,6), (371,1151,6221), (371,1152,3312), (371,1350,1161), (371,2459,6423), (371,3103,6422), (371,3311,6199), (371,3312,6417), (371,3594,6427), (371,6200,1151), (371,6395,6500), (371,6396,372), (371,6398,6418), (371,6409,6449), (371,6410,6398), (371,6411,6455), (371,6412,6450), (371,6419,3592), (371,6420,6419), (371,6425,6447), (371,6426,6428), (371,6449,6407), (371,6450,6395), (371,6452,6408), (371,6453,6425), (371,6454,6420), (371,6455,6445), (371,6481,6432), (371,6484,6429), (371,6486,6480), (371,6497,6446), (372,1151,3311), (372,1152,6398), (372,1350,1160), (372,2460,6424), (372,3102,6421), (372,3311,6418), (372,3312,6395), (372,3592,6428), (372,6199,6501), (372,6200,371), (372,6221,6417), (372,6396,1152), (372,6409,6221), (372,6410,6450), (372,6411,6449), (372,6412,6456), (372,6419,6420), (372,6420,3594), (372,6425,6427), (372,6426,6448), (372,6449,6199), (372,6450,6408), (372,6451,6407), (372,6453,6419), (372,6454,6426), (372,6456,6446), (372,6480,6431), (372,6485,6430), (372,6487,6481), (372,6496,6445), (376,549,381), (376,631,4), (376,1006,1012), (376,3090,3529), (376,3522,548), (376,3523,5), (376,3524,2), (376,3525,3146), (376,3526,5073), (376,3528,3522), (376,3530,1656), (376,5054,3830), (376,5067,5059), (378,2070,3830), (378,2937,5073), (378,3515,1598), (378,3520,3516), (378,6644,381), (381,382,4), (381,1656,5), (381,1657,382), (381,2070,25), (381,3526,1656), (381,5054,2), (381,5072,3091), (381,5079,5072), (382,631,5070), (382,1656,381), (382,3526,5), (382,3534,1657), (382,5054,1656), (382,5076,3627), (382,5079,546), (384,3552,1003), (384,5999,4), (386,573,970), (386,581,5396), (386,991,581), (386,4256,4255), (386,4257,58), (386,5752,5754), (388,3085,495), (388,5218,3085), (389,578,6), (394,1181,155), (394,3796,184), (394,5406,5408), (394,5407,5409), (404,1006,140), (404,4189,405), (404,6636,2915), (405,474,2), (405,1012,3560), (405,2915,25), (405,3149,5), (408,4189,6638), (411,1006,5), (411,3523,474), (411,4189,1012), (417,1593,6617), (418,6641,25), (426,3148,441), (426,6641,2), (427,3575,4), (428,1907,4), (454,3548,6617), (465,466,2), (468,1885,235), (474,1012,5), (474,3560,1656), (485,5418,590), (485,6560,3070), (486,5420,615), (486,6561,3071), (487,488,69), (489,492,637), (490,491,638), (497,3086,496), (498,1478,12), (498,4299,1478), (499,1479,11), (499,4302,1479), (500,582,6), (500,5396,581), (546,549,3525), (546,550,3529), (546,632,3090), (546,3090,5072), (546,3091,381), (546,3146,5076), (546,3525,1656), (546,3529,382), (546,3627,4), (546,3628,5), (546,5079,3851), (547,3543,381), (547,3845,3545), (547,3850,5), (547,3853,3850), (547,5067,1656), (548,549,4), (548,550,376), (548,631,382), (548,632,3529), (548,3523,381), (548,3524,1656), (548,3530,5), (548,5054,5073), (549,550,5), (549,1657,5070), (549,3522,382), (549,3528,1657), (549,3530,3523), (549,3534,5055), (549,3627,632), (549,3853,3533), (549,6636,2070), (550,631,381), (550,632,3627), (550,1656,5073), (550,1658,22), (550,3523,1656), (550,3524,3526), (550,3525,5076), (550,3526,3830), (550,3530,2), (550,3628,3146), (550,3850,5059), (550,5054,3843), (550,5498,3153), (551,5493,4301), (567,568,6), (567,3581,568), (568,6243,52), (569,578,567), (570,571,6), (572,573,6), (572,3430,581), (573,579,5755), (573,581,5752), (574,5171,3095), (574,5206,32), (574,5210,1384), (575,576,6), (577,578,2055), (577,5158,3284), (579,991,5751), (579,5751,5753), (580,581,6), (580,3430,5752), (581,991,500), (582,5398,580), (583,584,6), (590,3070,485), (595,995,1191), (601,602,31), (615,3071,486), (616,628,634), (617,627,633), (620,626,3788), (627,633,298), (628,634,299), (631,1657,5055), (631,3090,3525), (631,3091,632), (631,3146,3628), (631,3523,549), (631,3524,3523), (631,3528,20), (631,3529,3090), (631,3534,3851), (631,3545,3533), (631,3651,3149), (631,5059,547), (631,6636,26), (631,6643,3548), (632,3091,1656), (632,3146,5072), (632,3525,3526), (632,3529,381), (632,3627,5), (632,3628,2), (632,5079,5070), (800,5065,6), (902,1201,3915), (910,1212,169), (936,1490,5720), (936,5732,1490), (936,5777,5780), (938,4313,3488), (940,5706,5707), (942,5709,2095), (943,3487,954), (944,5657,8), (946,1125,5886), (950,1210,5722), (950,3911,1210), (956,5687,8), (958,1376,10), (962,3616,5603), (965,5776,5778), (970,5396,5754), (980,5337,940), (997,1158,5887), (999,3295,1), (1006,3651,4), (1011,4191,2), (1012,3149,4), (1030,5096,5132), (1030,5124,6), (1038,1040,1), (1060,1062,1), (1074,1076,225), (1092,1181,3167), (1092,5562,394), (1106,1253,1496), (1125,5248,1001), (1147,1216,394), (1150,5767,5769), (1151,1152,6), (1151,3312,6199), (1151,3592,6425), (1151,3594,3592), (1151,6200,6449), (1151,6221,6407), (1151,6396,3312), (1151,6398,6417), (1151,6408,6500), (1151,6409,6200), (1151,6410,372), (1151,6411,6409), (1151,6412,1152), (1151,6419,6447), (1151,6425,6453), (1151,6426,6419), (1151,6429,6480), (1151,6431,6437), (1151,6433,6484), (1151,6437,6429), (1151,6438,6431), (1151,6446,6501), (1151,6449,6445), (1151,6450,6418), (1151,6453,6519), (1151,6454,6427), (1151,6456,6395), (1151,6497,6408), (1152,3311,6395), (1152,3592,3594), (1152,3594,6426), (1152,6200,3311), (1152,6221,6418), (1152,6396,6450), (1152,6398,6408), (1152,6407,6501), (1152,6409,371), (1152,6410,6396), (1152,6411,1151), (1152,6412,6410), (1152,6420,6448), (1152,6425,6420), (1152,6426,6454), (1152,6430,6481), (1152,6432,6438), (1152,6434,6485), (1152,6437,6432), (1152,6438,6430), (1152,6445,6500), (1152,6449,6417), (1152,6450,6446), (1152,6453,6428), (1152,6454,6522), (1152,6455,6199), (1152,6496,6407), (1155,2646,65), (1160,3311,1351), (1161,3312,1351), (1180,1627,5359), (1191,3052,595), (1193,4300,1064), (1210,4304,950), (1216,5447,3917), (1312,1313,2072), (1319,3057,1), (1333,4261,6), (1340,1341,182), (1340,1380,6), (1341,1379,6), (1342,1343,182), (1342,1670,6), (1343,1671,6), (1344,1345,381), (1350,5023,5171), (1350,5085,6), (1350,5092,5050), (1351,5050,6), (1368,3547,1656), (1384,5024,6), (1385,6585,1617), (1388,2098,1), (1399,2361,47), (1420,1697,1), (1428,3056,613), (1469,2330,611), (1470,5172,1617), (1479,4302,6284), (1495,5650,5651), (1504,1505,1570), (1504,5062,6), (1505,5058,6), (1532,4187,5), (1578,1579,6), (1583,1584,2), (1583,3156,5020), (1584,3155,5020), (1589,1590,2), (1593,3515,25), (1593,3516,378), (1593,6642,381), (1594,6240,4), (1597,1598,4), (1597,3517,1598), (1597,5020,381), (1598,3517,25), 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(6449,6497,6398), (6449,6522,6419), (6450,6451,371), (6450,6452,6456), (6450,6455,3311), (6450,6456,6396), (6450,6496,6221), (6450,6497,6410), (6450,6519,6420), (6451,6452,6), (6451,6455,6409), (6451,6456,3311), (6451,6497,3312), (6451,6522,6425), (6452,6455,3312), (6452,6456,6410), (6452,6496,3311), (6452,6519,6426), (6453,6454,6), (6453,6482,6480), (6453,6484,6482), (6453,6519,6407), (6454,6483,6481), (6454,6485,6483), (6454,6522,6408), (6455,6456,6), (6455,6496,6451), (6455,6497,372), (6456,6496,371), (6456,6497,6452), (6465,6466,6467), (6468,6469,6), (6468,6471,6470), (6469,6470,6471), (6470,6471,6), (6472,6473,6), (6472,6474,6221), (6473,6475,6398), (6474,6475,6), (6476,6477,6), (6478,6479,6), (6480,6481,6), (6480,6484,1151), (6480,6486,6484), (6480,6487,6432), (6481,6485,1152), (6481,6486,6431), (6481,6487,6485), (6482,6483,6), (6482,6487,6420), (6483,6486,6419), (6484,6485,6), (6484,6486,6433), (6485,6487,6434), (6486,6487,6), (6488,6489,6), (6490,6491,6), (6492,6493,6), (6494,6495,6), (6494,6499,6435), (6495,6498,6436), (6496,6497,6), (6496,6522,6519), (6497,6519,6522), (6498,6499,6), (6500,6501,6), (6519,6522,6), (6566,6567,1570), (6639,6640,2)

Trilinears cos A - sin B sin C : cos B - sin C sin A : cos C - sin A sinB

Trilinears cos A - cos(B - C) : cos B - cos(C - A) : cos C - cos(A - B)

Trilinears sin B sin C - cos(B - C) : sin C sin A - cos(C - A) : sin A sin B - cos(A - B)

Trilinears csc A tan 3A - 2 sec 3A : :

Trilinears 4 cos A - cos(B - C) - 3 sin B sin C : :

Trilinears cos A + cos(B - C) + 5 cos B cos C - 2 sin B sin C : :

Barycentrics 1/SA : 1/SB : 1/SC

Barycentrics tan A : tan B : tan C

Barycentrics 1/(b

Tripolars |cos A| : :

Tripolars |a(b^2 + c^2 - a^2)| : :

X(4) = (1 + J) X(1113) + (1 - J) X(1114)

X(4) = (tan A)*[A] + (tan B)*[B] + (tan C)*[C], where A, B, C are the angles and [A], [B], [C] are the vertices

As a point on the Euler line, X(4) has Shinagawa coefficients (0, 1).

X(4) is the point of concurrence of the altitudes of ABC.

The tangents at A,B,C to the McCay cubic K003 concur in X(4). Also, the tangents at A,B,C to the Lucas cubic K007 concur in X(4). (Randy Hutson, November 18, 2015)

Let P be a point in the plane of ABC. Let Oa be the circumcenter of BCP, and define Ob and Oc cyclically. Let Q be the circumcenter of OaObOc. P = Q only when P = X(4). (Randy Hutson, June 27, 2018)

If you have The Geometer's Sketchpad, you can view Orthocenter.

If you have GeoGebra, you can view **Orthocenter**.

X(4) and the vertices A,B,C comprise an orthocentric system, defined as a set of four points, one of which is the orthocenter of the triangle of the other three (so that each is the orthocenter of the other three). The incenter and excenters also form an orthocentric system. If ABC is acute, then X(4) is the incenter of the orthic triangle.

Suppose P is not on a sideline of ABC. Let A'B'C' be the cevian triangle of P. Let D,E,F be the circles having diameters AA', BB',CC'. The radical center of D,E,F is X(4). (Floor van Lamoen, Hyacinthos #214, Jan. 24, 2000.)

Let A2B2C2 be the 2nd Conway triangle. Let A' be the crosspoint of B2 and C2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(4). (Randy Hutson, December 10, 2016)

**Ross Honsberger,** *Episodes in Nineteenth and Twentieth
Century Euclidean Geometry,* Mathematical Association of America,
1995. Chapter 2: The Orthocenter.

X(4) lies on the Thomson, Darboux, Napoleon, Lucas, McCay, and Neuberg cubics, and the Darboux septic, and on these lines:

{1,33}, {2,3}, {6,53}, {7,273}, {8,72}, {9,10}, {11,56}, {12,55}, {13,61}, {14,62}, {15,17}, {16,18}, {31,3072}, {32,98}, {35,498}, {36,499}, {37,1841}, {39,232}, {41,2202}, {42,1860}, {46,90}, {48,1881}, {49,156}, {50,9220}, {51,185}, {52,68}, {54,184}, {57,84}, {58,5292}, {63,5709}, {64,459}, {65,158}, {66,9969}, {67,338}, {69,76}, {74,107}, {75,12689}, {78,908}, {79,1784}, {80,1825}, {81,5707}, {83,182}, {85,4872}, {93,562}, {94,143}, {95,8797}, {96,231}, {99,114}, {100,119}, {101,118}, {102,124}, {103,116}, {105,5511}, {106,5510}, {109,117}, {110,113}, {111,1560}, {120,1292}, {121,1293}, {122,1294}, {123,1295}, {126,1296}, {127,1289}, {128,930}, {129,1303}, {130,1298}, {131,135}, {137,933}, {141,1350}, {142,5732}, {144,2894}, {145,149}, {147,148}, {150,152}, {151,2818}, {154,8888}, {155,254}, {157,5593}, {160,3613}, {162,270}, {165,1698}, {171,601}, {175,10905}, {176,10904}, {177,8095}, {181,9553}, {183,3785}, {187,7607}, {189,5908}, {191,2949}, {193,1351}, {195,399}, {200,6769}, {201,7069}, {204,1453}, {210,7957}, {212,3074}, {214,12119}, {215,9652}, {216,8799}, {218,294}, {230,3053}, {233,10979}, {236,8128}, {238,602}, {240,256}, {250,1553}, {251,8879}, {252,1487}, {255,1935}, {276,327}, {279,1565}, {280,2968}, {282,3345}, {284,5747}, {290,6528}, {298,5864}, {299,5865}, {312,7270}, {325,1975}, {333,5788}, {339,10749}, {341,12397}, {345,7283}, {346,3695}, {347,6356}, {348,5088}, {354,3296}, {371,485}, {372,486}, {373,11465}, {385,7823}, {386,2051}, {390,495}, {391,2322}, {394,1217}, {477,1304}, {484,3460}, {487,489}, {488,490}, {493,8212}, {494,8213}, {496,999}, {512,879}, {518,6601}, {519,3680}, {523,1552}, {524,5485}, {525,2435}, {527,5735}, {528,3913}, {529,3813}, {532,5862}, {533,5863}, {535,8666}, {538,7758}, {539,9936}, {541,9140}, {542,576}, {543,5503}, {544,10710}, {551,9624}, {566,9221}, {567,7578}, {569,1179}, {572,1474}, {574,1506}, {575,598}, {579,1713}, {580,1714}, {584,8818}, {590,1151}, {595,8750}, {603,3075}, {604,7120}, {608,1518}, {615,1152}, {616,627}, {617,628}, {618,5473}, {619,5474}, {620,7862}, {625,3788}, {626,3734}, {635,3643}, {636,3642}, {639,5590}, {640,5591}, {641,12124}, {642,12123}, {651,3157}, {653,1156}, {674,12587}, {684,2797}, {685,2682}, {690,11005}, {693,8760}, {695,3981}, {754,7751}, {758,5693}, {774,1254}, {800,13380}, {801,1092}, {842,935}, {885,3309}, {912,3868}, {916,2997}, {936,3452}, {937,1534}, {940,1396}, {941,1880}, {953,1309}, {958,2886}, {960,5794}, {970,9534}, {972,5514}, {973,6145}, {974,7729}, {983,5255}, {990,4000}, {991,4648}, {993,11012}, {1000,3057}, {1015,9651}, {1029,2906}, {1032,5910}, {1034,5911}, {1036,1065}, {1037,1067}, {1038,1076}, {1039,1096}, {1040,1074}, {1041,2263}, {1043,4417}, {1046,2648}, {1060,4296}, {1062,3100}, {1073,2130}, {1078,5171}, {1089,3974}, {1104,3772}, {1111,4056}, {1123,7133}, {1125,3576}, {1131,3311}, {1132,3312}, {1138,2132}, {1139,3368}, {1140,3395}, {1157,2120}, {1160,1162}, {1161,1163}, {1164,3595}, {1165,3593}, {1175,5320}, {1177,5622}, {1192,3532}, {1209,4549}, {1216,2979}, {1248,2660}, {1251,1832}, {1260,5687}, {1317,12763}, {1319,7704}, {1327,6419}, {1328,6420}, {1329,1376}, {1336,2362}, {1340,1348}, {1341,1349}, {1342,1676}, {1343,1677}, {1353,5093}, {1379,2040}, {1380,2039}, {1383,8791}, {1384,8778}, {1385,1538}, {1389,2099}, {1392,3241}, {1393,7004}, {1399,5348}, {1420,4311}, {1430,1468}, {1435,3333}, {1440,7053}, {1441,4329}, {1445,3358}, {1448,7365}, {1469,12589}, {1483,3623}, {1484,12773}, {1495,3431}, {1499,1550}, {1500,9650}, {1510,10412}, {1511,12121}, {1521,7115}, {1562,6529}, {1566,2724}, {1609,9722}, {1621,10267}, {1670,5404}, {1671,5403}, {1682,9552}, {1689,2010}, {1690,2009}, {1691,3406}, {1697,7160}, {1715,1730}, {1716,1721}, {1717,1718}, {1726,1782}, {1729,8558}, {1764,10479}, {1768,3065}, {1773,2961}, {1781,2955}, {1798,13323}, {1840,4876}, {1903,2262}, {1942,2790}, {1957,5247}, {1970,1971}, {1973,2201}, {1987,3269}, {1994,2904}, {2077,3814}, {2080,7793}, {2092,3597}, {2093,4848}, {2095,9965}, {2098,10944}, {2121,3481}, {2131,3349}, {2133,8440}, {2181,4642}, {2217,3417}, {2275,9597}, {2276,9596}, {2278,5397}, {2287,5778}, {2331,3755}, {2332,4251}, {2353,3425}, {2355,3579}, {2361,7299}, {2393,5486}, {2456,10349}, {2457,3667}, {2477,9653}, {2482,12117}, {2536,2540}, {2537,2541}, {2574,2592}, {2575,2593}, {2646,4305}, {2651,2907}, {2679,2698}, {2687,2766}, {2697,10423}, {2734,10017}, {2752,10101}, {2770,10098}, {2771,9803}, {2778,10693}, {2783,10769}, {2784,11599}, {2787,10768}, {2791,4516}, {2793,9180}, {2801,3254}, {2802,12641}, {2814,3762}, {2817,13532}, {2822,4466}, {2823,4858}, {2826,10773}, {2827,10774}, {2828,10775}, {2830,10779}, {2831,10780}, {2840,4939}, {2889,6101}, {2896,6194}, {2900,3189}, {2905,6625}, {2908,7139}, {2917,8146}, {2929,2935}, {2972,10745}, {2975,5841}, {2995,8048}, {3023,12185}, {3024,12374}, {3027,12184}, {3028,12373}, {3054,5210}, {3056,12588}, {3058,3303}, {3062,3339}, {3094,3399}, {3096,3098}, {3101,8251}, {3120,3924}, {3162,5359}, {3164,9290}, {3172,3424}, {3180,5873}, {3181,5872}, {3184,6716}, {3190,3191}, {3212,7261}, {3216,5400}, {3218,5770}, {3255,5883}, {3270,11461}, {3304,5434}, {3305,3587}, {3306,7171}, {3314,7885}, {3320,12945}, {3329,7864}, {3338,7284}, {3340,3577}, {3342,3347}, {3344,3348}, {3352,3354}, {3356,3637}, {3364,3391}, {3365,3392}, {3366,3389}, {3367,3390}, {3369,3397}, {3370,3396}, {3371,3387}, {3372,3388}, {3373,3385}, {3374,3386}, {3379,5402}, {3380,5401}, {3381,3394}, {3382,3393}, {3398,3407}, {3413,3558}, {3414,3557}, {3416,3714}, {3426,13093}, {3430,3454}, {3438,3443}, {3439,3442}, {3440,5682}, {3441,5681}, {3461,7165}, {3463,5683}, {3466,3469}, {3479,3489}, {3480,3490}, {3495,8866}, {3497,7351}, {3499,8925}, {3500,7350}, {3502,8867}, {3521,5946}, {3527,8796}, {3580,11472}, {3582,4325}, {3584,4330}, {3589,5085}, {3590,6221}, {3591,6398}, {3601,4304}, {3611,11460}, {3614,5217}, {3617,5690}, {3620,7879}, {3621,5844}, {3622,5901}, {3624,7987}, {3629,5102}, {3632,4900}, {3633,11224}, {3634,10164}, {3648,3652}, {3668,8809}, {3671,5665}, {3679,4866}, {3701,5300}, {3704,5695}, {3706,10371}, {3738,10771}, {3741,10476}, {3746,4309}, {3753,9800}, {3812,5880}, {3815,5013}, {3819,13348}, {3820,6244}, {3822,5248}, {3825,10200}, {3826,11495}, {3829,11194}, {3841,7688}, {3847,6691}, {3849,7615}, {3870,5534}, {3871,10528}, {3877,7700}, {3885,12648}, {3887,10772}, {3911,6705}, {3916,5744}, {3917,7999}, {3925,5584}, {3933,7776}, {3934,5188}, {3940,5763}, {3947,4314}, {3972,7828}, {4008,12723}, {4045,7808}, {4048,5103}, {4277,4646}, {4308,7743}, {4313,5226}, {4316,7280}, {4317,5563}, {4324,5010}, {4339,5266}, {4355,10980}, {4357,10444}, {4423,7958}, {4425,8235}, {4444,6002}, {4512,10268}, {4645,7155}, {4654,11518}, {4658,5733}, {4692,4894}, {4721,4805}, {4723,12693}, {4768,9525}, {4846,5462}, {4847,12527}, {4863,12692}, {5007,5309}, {5008,5346}, {5032,11405}, {5038,11170}, {5044,10157}, {5045,5558}, {5050,5395}, {5092,7859}, {5097,7894}, {5119,7162}, {5121,11512}, {5123,13528}, {5173,12677}, {5204,5433}, {5206,6781}, {5221,10308}, {5223,12777}, {5249,10884}, {5253,10269}, {5265,10593}, {5273,5791}, {5278,9958}, {5281,10592}, {5377,6074}, {5418,6200}, {5420,6396}, {5424,5441}, {5435,5704}, {5437,9841}, {5438,6700}, {5439,9776}, {5440,5748}, {5447,7998}, {5449,7689}, {5461,10153}, {5505,10752}, {5513,9085}, {5533,10074}, {5535,6597}, {5536,6763}, {5542,6744}, {5550,11230}, {5553,7702}, {5556,10977}, {5557,12005}, {5559,5697}, {5561,11552}, {5597,8196}, {5598,8203}, {5599,11822}, {5600,11823}, {5601,8200}, {5602,8207}, {5606,5950}, {5609,5655}, {5623,8446}, {5624,8456}, {5627,6070}, {5670,8487}, {5671,8494}, {5672,8444}, {5673,8454}, {5674,8495}, {5675,8496}, {5676,8486}, {5677,7329}, {5678,8491}, {5679,8492}, {5680,7164}, {5685,8480}, {5688,12698}, {5689,12697}, {5705,5745}, {5708,12684}, {5795,9623}, {5848,10759}, {5853,6765}, {5854,13271}, {5860,6278}, {5861,6281}, {5874,11917}, {5875,11916}, {5885,10266}, {5892,11451}, {5933,10362}, {5934,8079}, {5935,7593}, {5943,9729}, {5951,5952}, {5965,7877}, {5984,7766}, {6020,12955}, {6032,12506}, {6036,7857}, {6055,9166}, {6073,11607}, {6075,10428}, {6082,6092}, {6114,9750}, {6115,9749}, {6128,8749}, {6130,9409}, {6147,11036}, {6196,8927}, {6204,8957}, {6217,6266}, {6218,6267}, {6219,6276}, {6220,6277}, {6224,6265}, {6233,13234}, {6235,8705}, {6238,10055}, {6285,7049}, {6292,7935}, {6323,12494}, {6326,6596}, {6339,10981}, {6407,9542}, {6409,8253}, {6410,8252}, {6453,9681}, {6462,8220}, {6463,8221}, {6467,12283}, {6519,9692}, {6680,7844}, {6704,9751}, {6735,12534}, {6752,8795}, {6777,11602}, {6778,11603}, {7017,7141}, {7028,8127}, {7059,7345}, {7060,7344}, {7149,8811}, {7161,11010}, {7264,7272}, {7320,9785}, {7325,8449}, {7326,8459}, {7327,8432}, {7352,10071}, {7587,8379}, {7588,8086}, {7589,8382}, {7595,12681}, {7603,11669}, {7605,13339}, {7617,8182}, {7618,8176}, {7666,10272}, {7676,7679}, {7677,7678}, {7693,13363}, {7703,11454}, {7712,10610}, {7720,7725}, {7721,7726}, {7723,12219}, {7730,7731}, {7739,7753}, {7757,7858}, {7769,7782}, {7777,7783}, {7778,7789}, {7779,7900}, {7786,7847}, {7792,7851}, {7794,7818}, {7796,7809}, {7798,7838}, {7799,7814}, {7801,7821}, {7804,7834}, {7813,7903}, {7815,7830}, {7820,7867}, {7822,7853}, {7829,7902}, {7831,7910}, {7832,7934}, {7835,7899}, {7836,7912}, {7839,7921}, {7845,7855}, {7846,7919}, {7854,7873}, {7856,12150}, {7863,7888}, {7875,7923}, {7883,10302}, {7889,7913}, {7891,7925}, {7905,7926}, {7906,7941}, {7932,10583}, {8068,10058}, {8069,10321}, {8075,8087}, {8076,8088}, {8077,8085}, {8080,8092}, {8099,9793}, {8100,9795}, {8105,8426}, {8106,8427}, {8107,8380}, {8108,8381}, {8109,8377}, {8110,8378}, {8117,8123}, {8118,8124}, {8125,8129}, {8126,8130}, {8141,9536}, {8144,9538}, {8172,8447}, {8173,8457}, {8193,9911}, {8197,12458}, {8204,12459}, {8222,11828}, {8223,11829}, {8224,8230}, {8225,8228}, {8372,12674}, {8431,8443}, {8433,8483}, {8434,8484}, {8435,8481}, {8436,8482}, {8437,8497}, {8438,8498}, {8445,8458}, {8448,8455}, {8450,8461}, {8451,8460}, {8452,8463}, {8453,8462}, {8488,8527}, {8489,8532}, {8490,8533}, {8501,8509}, {8502,8508}, {8515,8536}, {8516,8535}, {8517,8534}, {8538,11416}, {8582,10860}, {8583,10863}, {8588,10185}, {8591,8724}, {8596,12355}, {8674,10767}, {8679,12586}, {8719,10155}, {8864,8921}, {8868,8872}, {8878,10340}, {8983,9583}, {9147,11615}, {9300,9607}, {9530,10718}, {9627,9629}, {9628,9630}, {9638,10535}, {9646,9660}, {9647,9661}, {9648,9662}, {9649,9663}, {9658,9672}, {9659,9673}, {9705,13482}, {9783,12488}, {9787,12489}, {9789,12490}, {9791,9959}, {9845,12577}, {9857,12497}, {9874,12139}, {9897,11280}, {9919,13171}, {9934,13198}, {9942,10391}, {9967,12220}, {9973,13622}, {10042,10050}, {10043,10057}, {10052,10073}, {10088,12896}, {10187,10646}, {10188,10645}, {10202,11220}, {10264,10620}, {10293,12099}, {10305,11023}, {10309,12676}, {10313,10316}, {10363,10369}, {10415,10422}, {10434,10887}, {10435,12547}, {10455,10464}, {10529,10680}, {10546,10564}, {10547,10548}, {10627,13340}, {10634,11420}, {10635,11421}, {10707,11240}, {10791,12197}, {10797,10799}, {10798,12835}, {10831,10833}, {10873,10877}, {10882,10886}, {10897,11417}, {10898,11418}, {10912,13463}, {10915,12703}, {10916,12704}, {10923,10927}, {10924,10928}, {10956,10965}, {10957,10966}, {10958,11502}, {11082,11135}, {11087,11136}, {11171,11272}, {11177,11632}, {11270,11468}, {11402,11426}, {11408,11485}, {11409,11486}, {11423,13366}, {11449,12038}, {11557,11560}, {11587,13558}, {11646,13330}, {11649,11663}, {11698,12331}, {11703,12165}, {11755,11759}, {11764,11768}, {11773,11777}, {11782,11786}, {11792,13508}, {11800,12284}, {11869,11873}, {11870,11874}, {11891,12491}, {11900,12696}, {11905,11909}, {11930,11947}, {11931,11948}, {11990,11992}, {12006,13364}, {12061,12063}, {12120,12864}, {12146,12849}, {12166,12309}, {12168,12310}, {12169,12311}, {12170,12312}, {12171,12313}, {12172,12314}, {12175,12316}, {12223,12603}, {12224,12604}, {12226,12606}, {12271,12272}, {12273,12280}, {12350,12354}, {12369,13495}, {12387,12394}, {12388,12393}, {12507,13249}, {12515,12619}, {12516,12620}, {12517,12621}, {12518,12622}, {12519,12623}, {12520,12609}, {12521,12612}, {12522,12613}, {12523,12614}, {12524,12615}, {12556,13089}, {12624,13238}, {12739,12743}, {12837,13077}, {12859,12863}, {12941,13075}, {12942,13076}, {12944,13078}, {12946,13079}, {12947,13080}, {12948,13081}, {12949,13082}, {13007,13023}, {13008,13024}, {13009,13039}, {13010,13040}, {13321,13451}, {13353,13470}, {13418,13423}

X(4) is the {X(3),X(5)}-harmonic conjugate of X(2). For a list of
other harmonic conjugates of X(4), click **Tables** at the top of
this page.

X(4) = midpoint of X(i) and X(j) for these (i,j): (3,382), (146,3448), (147,148), (149,153), (150,152)

X(4) = reflection of X(i) in X(j) for these (i,j): (1,946), (2,381), (3,5), (5,546), (8,355), (20,3), (24,235), (25,1596), (40,10), (69,1352), (74,125), (98,115), (99,114), (100,119), (101,118), (102,124), (103,116), (104,11), (107,133), (109,117), (110,113), (112,132), (145,1482), (185,389), (186,403), (193,1351), (376,2), (378,427), (550,140), (917,5190), (925,131), (930,128), (944,1), (1113,1312), (1114,1313), (1141,137), (1292,120), (1293,121), (1294,122), (1295,123), (1296,126), (1297,127), (1298,130), (1299,135), (1300,136), (1303, 129), (1350,141), (1593,1595)

X(4) = isogonal conjugate of X(3)

X(4) = isotomic conjugate of X(69)

X(4) = cyclocevian conjugate of X(2)

X(4) = circumcircle-inverse of X(186)

X(4) = nine-point-circle-inverse of X(403)

X(4) = complement of X(20)

X(4) = anticomplement of X(3)

X(4) = complementary conjugate of X(2883)

X(4) = anticomplementary conjugate of X(20)

X(4) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1249), (7,196), (27,19), (29,1), (92,281), (107,523), (264,2), (273,278), (275,6), (286,92), (393, 459)

X(4) = cevapoint of X(i) and X(j) for these (i,j):

(1,46), (2,193), (3,155), (5,52), (6,25), (11,513), (19,33), (30,113),
(34,208), (37,209), (39,211), (51,53), (65,225), (114,511), (115,512), (116,514), (117,515), (118,516), (119,517), (120,518), (121,519), (122,520), (123,521), (124,522), (125,523), (126,524), (127,525), (185,235), (371,372), (487,488)

X(4) = X(i)-cross conjugate of X(j) for these (i,j):

(3,254), (6,2), (19,278), (25,393), (33,281), (51,6), (52,24), (65,1), (113,403), (125,523), (185,3), (193,459), (225,158), (389,54), (397,17), (398,18), (407,225), (427,264), (512,112), (513,108), (523,107)

X(4) = crosspoint of X(i) and X(j) for these (i,j): (2,253), (7,189), (27,286), (92,273)

X(4) = crosssum of X(i) and X(j) for these (i,j):

(4,1075), (6,154), (25,1033), (48,212), (55,198), (56,1035), (71,228), (184,577), (185,417), (216,418)

X(4) = crossdifference of every pair of points on line X(520)X(647)

X(4) = X(i)-Hirst inverse of X(j) for these (i,j):

(1,243), (2,297), (3,350), (19,242), (21,425), (24,421), (25,419), (27,423), (28,422), (29,415), (420,427), (424,451), (459,460), (470,471), (1249,1503)

X(4) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1047), (29,4)

X(4) = X(i)-beth conjugate of X(j) for these (i,j):

(4,34), (8,40), (29,4), (162,56), (318,318), (811,331)

X(4) = intersection of tangents at X(3) and X(4) to McCay cubic K003

X(4) = intersection of tangents at X(4) and X(69) to Lucas cubic K007

X(4) = exsimilicenter of 1st & 2nd Johnson-Yff circles; the insimilicenter is X(1)

X(4) = trilinear pole of PU(4) (the orthic axis)

X(4) = trilinear pole wrt orthic triangle of orthic axis

X(4) = trilinear pole wrt intangents triangle of orthic axis

X(4) = trilinear pole wrt circumsymmedial triangle of orthic axis

X(4) = trilinear product of PU(15)

X(4) = barycentric product of PU(i) for these i: 21, 45

X(4) = bicentric sum of PU(i) for these i: 126, 131

X(4) = PU(126)-harmonic conjugate of X(652)

X(4) = midpoint of PU(131)

X(4) = crosspoint of polar conjugates of PU(4)

X(4) = cevapoint of foci of orthic inconic

X(4) = QA-P33 (Centroid of the Orthocenter Quadrangle) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/61-qa-p33.html)

X(4) = Hofstadter -1 point

X(4) = X(4)-of X(i)X(j)X(k) for these {i,j,k}: {1,8,5556}, {1,9,885}, {2,6,1640}, {2,10,4049}, {3,6,879}, {3,66,2435}, {7,8,885}

X(4) = homothetic center of these triangles: orthic, X(13)-Ehrmann, X(14)-Ehrmann (see X(25))

X(4) = perspector of anticomplementary circle

X(4) = pole wrt polar circle of trilinear polar of X(2) (line at infinity)

X(4) = pole wrt {circumcircle, nine-point circle}-inverter of Lemoine axis

X(4) = X(48)-isoconjugate (polar conjugate) of X(2)

X(4) = X(i)-isoconjugate of X(j) for these (i,j): (6,63), (75,184), (91,1147), (92,577), (1101,125), (2962,49), (2964,3519)

X(4) = X(1342)-vertex conjugate of X(1343)

X(4) = Zosma transform of X(1)

X(4) = X(1352) of 1st anti-Brocard triangle

X(4) = centroid of the union of X(8) and its 3 extraversions

X(4) = X(5) of extraversion triangle of X(8)

X(4) = homothetic center of orthic triangle and reflection of tangential triangle in X(5)

X(4) = homothetic center of 2nd circumperp and 3rd Euler triangles

X(4) = trilinear product of vertices of half-altitude triangle

X(4) = trilinear product of vertices of orthocentroidal triangle

X(4) = trilinear product of vertices of reflection triangle

X(4) = trilinear product of vertices of 4th Brocard triangle

X(4) = center of conic that is the locus of orthopoles of lines passing through X(4)

X(4) = perspector of circumanticevian triangle of X(4) and unary cofactor triangle of circumanticevian triangle of X(3)

X(4) = X(3)-of-2nd-extouch-triangle

X(4) = perspector of ABC and 2nd and 3rd extouch triangles

X(4) = perspector of ABC and 1st Brocard triangle of anticomplementary triangle

X(4) = perspector of ABC and 1st Brocard triangle of Johnson triangle

X(4) = perspector of ABC and mid-triangle of 2nd and 3rd extouch triangles

X(4) = perspector of extouch triangle and cross-triangle of ABC and 2nd extouch triangle

X(4) = perspector of 2nd Hyacinth triangle and cross-triangle of ABC and 2nd Hyacinth triangle

X(4) = homothetic center of 2nd Johnson-Yff triangle and cross-triangle of ABC and 1st Johnson-Yff triangle

X(4) = homothetic center of 1st Johnson-Yff triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle

X(4) = X(1)-of-orthic-triangle if ABC is acute, and an excenter of orthic triangle otherwise

X(4) = X(52)-of-excentral triangle

X(4) = X(65)-of-tangential-triangle if ABC is acute

X(4) = X(155)-of-intouch-triangle

X(4) = X(110)-of-Fuhrmann-triangle

X(4) = X(147)-of-1st-Brocard-triangle

X(4) = X(1296)-of-4th-Brocard-triangle

X(4) = X(74)-of-orthocentroidal-triangle

X(4) = X(110)-of-X(4)-Brocard-triangle

X(4) = harmonic center of circle O(PU(4)) and orthoptic circle of Steiner inellipse

X(4) = Thomson-isogonal conjugate of X(154)

X(4) = Lucas-isogonal conjugate of X(11206)

X(4) = perspector of ABC and cross-triangle of 1st and 2nd Neuberg triangles

X(4) = perspector of circumconic centered at X(1249)

X(4) = center of circumconic that is locus of trilinear poles of lines passing through X(1249)

X(4) = circumcevian isogonal conjugate of X(4)

X(4) = orthic-isogonal conjugate of X(4)

X(4) = X(1)-of-circumorthic-triangle if ABC is acute

X(4) = isogonal conjugate wrt half-altitude triangle of X(185)

X(4) = Miquel associate of X(4)

X(4) = crosspoint of X(3) and X(155) wrt both the excentral and tangential triangles

X(4) = crosspoint of X(487) and X(488) wrt both the excentral and anticomplementary triangles

X(4) = X(3)-of-Ehrmann-mid-triangle

X(4) = X(110)-of-X(3)-Fuhrmann-triangle

X(4) = barycentric product X(112)*X(850)

X(4) = Kosnita(X(20),X(20)) point

X(4) = perspector of ABC and the reflection of the excentral triangle in X(10)

X(4) = pedal antipodal perspector of X(3)

X(4) = Ehrmann-side-to-Ehrmann-vertex similarity image of X(3)

X(4) = Ehrmann-vertex-to-orthic similarity image of X(4)

X(4) = Ehrmann-side-to-orthic similarity image of X(3)

X(4) = Ehrmann-mid-to-ABC similarity image of X(5)

X(4) = perspector of hexyl triangle and cevian triangle of X(27)

X(4) = perspector of hexyl triangle and anticevian triangle of X(19)

X(4) = perspector of ABC and medial triangle of pedal triangle of X(64)

X(4) = perspector of ABC and the reflection in X(2) of the antipedal triangle of X(2)

X(4) = perspector of hexyl triangle and tangential triangle wrt excentral triangle of the excentral-hexyl ellipse

X(4) = inverse-in-Steiner-circumellipse of X(297)

X(4) = {X(2479),X(2480)}-harmonic conjugate of X(297)

X(4) = symgonal of every point on the nine-point circle

X(4) = center of bianticevian conic of PU(4) (this conic being the polar circle)

X(4) = orthoptic-circle-of-Steiner-inellipse inverse of X(468)

X(4) = de-Longchamps-circle inverse of X(2071)

X(4) = center of inverse-in-de-Longchamps-circle of circumcircle

X(4) = inner-Napoleon circle-inverse of X(32460)

X(4) = outer-Napoleon circle-inverse of X(32461)

X(4) = excentral-to-ABC functional image of X(1)

X(4) = orthic-to-ABC functional image of X(52)

X(4) = incentral-to-ABC functional image of X(500)

X(4) = Feuerbach-to-ABC functional image of X(5948)

X(4) = excentral-to-intouch similarity image of X(84)

X(4) = barycentric product of circumcircle intercepts of line X(297)(525)

Trilinears cos A + 2 cos B cos C : :

Trilinears cos A - 2 sin B sin C

Trilinears bc[a

Trilinears 2 sec A + sec B sec C : :

Trilinears cos B cos C + sin B sin C : :

Trilinears cos^2(B/2 - C/2) - sin^2(B/2 - C/2) : :

Trilinears cos A' : :, where A' is the angle formed by internal tangents to incircle and A-excircle

Trilinears cos A" : : , where A" is the angle formed by external tangents to B- and C-excircles

Barycentrics a cos(B - C) : b cos(C - A) : c cos(A - B)

Barycentrics a

Barycentrics S^2 + SB SC : :

Barycentrics 1 + cot B cot C : :

Tripolars Sqrt[a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2+3 a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6] : :

X(5) = 3*X(2) + X(4) = 3*X(2) - X(3) = X(3) + X(4)

X(5) = (2 + J) X(1113) + (2 - J) X(1114)

As a point on the Euler line, X(5) has Shinagawa coefficients (1, 1).

X(5) is the center of the nine-point circle. Euler showed in that this circle passes through the midpoints of the sides of ABC and the feet of the altitudes of ABC, hence six of the nine points. The other three are the midpoints of segments A-to-X(4), B-to-X(4), C-to-X(4). The radius of the nine-point circle is one-half the circumradius.

**Dan Pedoe,** *Circles: A Mathematical View,* Mathematical
Association of America, 1995.

Let Pa be the parabola with focus A and directrix BC. Let La be the polar of X(3) wrt Pa. Define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C; cyclically. Then X(5) = X(597)-of-A'B'C'. (Randy Hutson, December 10, 2016)

Let A'B'C' be the half-altitude triangle. Let A" be the trilinear pole, wrt A'B'C', of line BC, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(5). (Randy Hutson, December 10, 2016)

Let A' be the nine-point center of BCX(13), and define B' and C' cyclically. Let A" be the nine-point center of BCX(14), and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(5). (Randy Hutson, December 10, 2016)

Let A'B'C' be the Euler triangle. Let A" be the centroid of AB'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(5). (Randy Hutson, December 10, 2016)

Let A'B'C' be any equilateral triangle inscribed in the circumcircle of ABC. The Simson lines of A', B', C' form an equilateral triangle with center X(5). If A'B'C' is the circumtangential triangle, the Simson lines of A', B', C' concur in X(5). (Randy Hutson, December 10, 2016)

If you have The Geometer's Sketchpad, you can view these sketches: Nine-point center, Euler Line, Roll Circle, MacBeath Inconic

If you have GeoGebra, you can view

Let A'B'C' be the Feuerbach triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(5). (Randy Hutson, July 20, 2016)

Let A'B'C' be the reflection triangle. Let A" be the trilinear pole of line B'C', and define B"and C" cyclically. The lines AA", BB", CC" concur in X(5). (Randy Hutson, July 20, 2016)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). X(5) = X(6146)-of-A'B'C'.

Let A'B'C' be the cevian triangle of X(5). Let A" be X(5)-of-AB'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(5). (Randy Hutson, June 27, 2018)

Let Na, Nb, Nc be the nine-point centers of BCF, CAF, ABF, resp., where F = X(13). Let Na', Nb', Nc' be the nine-point centers of BCF', CAF', ABF', resp., where F' = X(14). The lines NaNa', NbNb', NcNc' concur in X(5). (Randy Hutson, June 27, 2018)

Let Na, Nb, Nc be the nine-point centers of BCX, CAX, ABX, resp., where X = X(17). Let Na', Nb', Nc' be the nine-point centers of BCX', CAX', ABX', resp., where X' = X(18). The lines NaNa', NbNb', NcNc' concur in X(5). (Randy Hutson, June 27, 2018)

X(5) lies on the Napoleon cubic (also known as the Feuerbach cubic) and these lines:

{1,11}, {2,3}, {6,68}, {7,5704}, {8,1389}, {9,1729}, {10,517}, {13,18}, {14,17}, {15,2913}, {16,2912}, {19,8141}, {32,230}, {33,1062}, {34,1060}, {35,3583}, {36,3585}, {39,114}, {40,1698}, {46,1836}, {47,5348}, {49,54}, {51,52}, {53,216}, {55,498}, {56,499}, {57,1728}, {60,5397}, {64,4846}, {65,1737}, {67,9970}, {69,1351}, {72,908}, {74,3521}, {76,262}, {78,3419}, {79,1749}, {83,98}, {84,5437}, {85,1565}, {93,6344}, {94,9221}, {96,1166}, {97,4994}, {99,5966}, {100,10738}, {101,10739}, {102,10740}, {103,10741}, {104,5253}, {105,10743}, {106,10744}, {107,10745}, {108,10746}, {109,10747}, {111,10748}, {112,10749}, {113,125}, {116,118}, {117,124}, {120,5511}, {121,2885}, {122,133}, {126,5512}, {127,132}, {128,137}, {129,130}, {131,136}, {141,211}, {142,971}, {145,10247}, {146,10620}, {147,3329}, {148,7783}, {149,3871}, {153,12773}, {154,9833}, {156,184}, {165,7965}, {169,6506}, {171,3073}, {181,10407}, {182,206}, {183,315}, {187,3054}, {191,5535}, {193,5093}, {194,7777}, {195,1994}, {214,6246}, {217,1625}, {222,8757}, {225,1465}, {226,912}, {236,8130}, {238,3072}, {252,1157}, {264,1093}, {273,6356}, {275,2055}, {276,6528}, {298,634}, {299,633}, {302,622}, {303,621}, {311,1225}, {312,3695}, {316,1078}, {318,2968}, {324,6663}, {329,2095}, {339,1235}, {354,13407}, {356,3608}, {371,590}, {372,615}, {385,7762}, {386,1834}, {388,999}, {389,5448}, {390,7678}, {392,1512}, {394,10982}, {399,3448}, {484,5445}, {487,12313}, {488,12314}, {491,637}, {492,638}, {493,8220}, {494,8221}, {497,3085}, {515,1125}, {516,3579}, {518,10916}, {519,3813}, {523,6757}, {524,576}, {528,8715}, {529,8666}, {538,7764}, {539,1493}, {542,575}, {543,9771}, {551,5882}, {566,9220}, {568,3567}, {570,1879}, {572,2126}, {573,1213}, {574,3055}, {577,6748}, {578,1147}, {579,1901}, {580,5127}, {581,5453}, {582,1754}, {583,8818}, {598,7607}, {599,11477}, {601,750}, {602,748}, {611,12589}, {613,12588}, {616,13103}, {617,13102}, {618,629}, {619,630}, {620,6721}, {641,6250}, {642,6251}, {671,7608}, {698,8149}, {754,7780}, {758,5694}, {804,11615}, {842,1287}, {920,1454}, {925,2383}, {930,6592}, {938,3487}, {944,3616}, {950,13411}, {956,3436}, {958,10526}, {962,5657}, {986,3944}, {993,4999}, {997,5794}, {1001,10198}, {1007,3926}, {1056,5261}, {1058,5274}, {1069,10055}, {1071,5249}, {1073,1217}, {1087,2599}, {1089,3703}, {1090,1091}, {1092,5651}, {1111,3665}, {1112,12358}, {1117,3470}, {1131,3317}, {1132,3316}, {1139,3393}, {1140,3370}, {1145,7704}, {1151,5418}, {1152,5420}, {1155,1770}, {1158,5880}, {1160,5590}, {1161,5591}, {1173,1487}, {1181,1899}, {1199,3410}, {1211,5752}, {1212,5179}, {1214,1838}, {1249,8888}, {1270,11917}, {1271,11916}, {1297,12918}, {1327,6426}, {1328,6425}, {1350,3763}, {1376,10525}, {1393,7069}, {1420,9613}, {1441,3007}, {1447,4911}, {1490,5787}, {1495,11572}, {1498,1853}, {1499,11182}, {1511,5972}, {1519,3753}, {1537,7705}, {1538,8582}, {1539,2777}, {1587,3069}, {1588,3068}, {1601,3432}, {1614,5012}, {1621,11491}, {1697,9614}, {1706,12700}, {1709,12679}, {1714,4383}, {1724,5398}, {1750,8726}, {1768,7701}, {1788,4295}, {1843,9967}, {1848,1871}, {1861,1872}, {1916,3399}, {1935,3075}, {1936,3074}, {1975,6390}, {1986,7723}, {1990,5158}, {1992,11482}, {2052,13599}, {2066,9646}, {2067,9661}, {2077,11826}, {2086,9490}, {2098,12647}, {2099,10573}, {2120,2121}, {2241,9665}, {2242,9650}, {2482,9880}, {2486,2783}, {2549,5013}, {2550,9709}, {2551,9708}, {2595,3460}, {2601,2602}, {2607,2957}, {2635,4303}, {2646,10572}, {2682,10568}, {2771,5883}, {2781,6698}, {2792,4672}, {2794,6036}, {2797,8552}, {2800,3754}, {2801,12005}, {2802,10284}, {2826,3837}, {2829,5450}, {2883,5892}, {2887,3831}, {2896,7616}, {2963,2965}, {2971,2974}, {2975,5080}, {2979,7999}, {3006,3701}, {3035,5840}, {3053,7737}, {3057,10039}, {3058,3584}, {3060,6243}, {3096,7934}, {3098,7914}, {3120,5492}, {3157,10071}, {3167,6193}, {3258,11749}, {3272,3609}, {3284,6749}, {3303,10056}, {3304,10072}, {3306,13226}, {3314,7912}, {3333,5290}, {3338,10404}, {3357,5893}, {3359,12705}, {3368,5401}, {3381,3382}, {3406,3407}, {3434,5552}, {3462,3463}, {3468,3469}, {3488,5703}, {3576,3624}, {3581,10545}, {3582,5270}, {3586,3601}, {3617,8148}, {3618,5050}, {3619,10519}, {3622,7967}, {3626,11278}, {3629,5097}, {3630,7882}, {3631,7896}, {3636,13607}, {3649,5693}, {3654,7991}, {3656,3679}, {3670,3782}, {3673,7179}, {3687,5295}, {3705,4385}, {3734,3788}, {3737,8819}, {3739,12490}, {3742,12675}, {3785,9752}, {3812,3838}, {3819,5447}, {3823,12393}, {3833,6701}, {3867,11574}, {3874,6583}, {3880,10915}, {3911,4292}, {3913,11235}, {3917,10625}, {3931,5530}, {3947,5045}, {3972,7857}, {4004,10273}, {4030,4894}, {4045,6683}, {4293,5229}, {4294,5218}, {4297,10165}, {4299,5204}, {4302,5217}, {4311,5126}, {4323,11041}, {4354,9629}, {4413,10310}, {4417,10449}, {4420,5178}, {4425,9959}, {4511,5086}, {4550,7689}, {4662,12612}, {4668,11224}, {4855,9945}, {4861,5176}, {4885,8760}, {5007,5306}, {5010,5326}, {5015,7081}, {5024,7738}, {5038,11646}, {5041,5355}, {5082,7080}, {5092,6704}, {5099,13162}, {5119,12701}, {5131,5442}, {5171,7761}, {5181,8263}, {5188,6249}, {5221,11544}, {5223,6067}, {5224,10446}, {5233,9534}, {5237,5350}, {5238,5349}, {5248,5842}, {5251,11012}, {5257,10445}, {5259,6253}, {5286,7736}, {5309,7772}, {5334,11485}, {5335,11486}, {5339,10654}, {5340,10653}, {5354,10339}, {5395,7612}, {5412,10897}, {5413,10898}, {5422,7592}, {5471,6783}, {5472,6782}, {5544,5656}, {5550,5731}, {5597,8200}, {5598,8207}, {5599,8196}, {5600,8203}, {5601,11875}, {5602,11876}, {5643,5655}, {5658,9799}, {5745,12572}, {5814,11679}, {5890,11451}, {5895,10606}, {5908,5909}, {5913,6032}, {5925,8567}, {5944,6689}, {5947,5948}, {5950,5952}, {5961,8146}, {5962,13557}, {6043,11992}, {6054,7827}, {6055,10991}, {6118,8180}, {6119,8184}, {6130,9517}, {6150,10615}, {6152,12606}, {6153,11692}, {6174,10993}, {6179,7812}, {6191,7345}, {6192,7344}, {6194,7938}, {6221,6459}, {6223,12684}, {6224,12747}, {6225,13093}, {6237,11435}, {6238,11436}, {6241,10574}, {6256,10200}, {6291,12603}, {6329,12007}, {6361,9812}, {6398,6460}, {6406,12604}, {6407,9692}, {6417,7582}, {6418,7581}, {6419,8960}, {6433,12819}, {6434,12818}, {6449,9541}, {6462,11949}, {6463,11950}, {6515,9777}, {6523,10002}, {6599,12660}, {6662,13409}, {6669,6694}, {6670,6695}, {6671,6673}, {6672,6674}, {6692,6705}, {6703,13323}, {6735,10914}, {6736,13600}, {6769,8580}, {7013,10400}, {7028,8129}, {7160,12856}, {7198,7272}, {7280,7294}, {7596,8228}, {7615,11184}, {7620,11165}, {7691,11016}, {7693,12307}, {7694,9756}, {7703,11439}, {7709,7864}, {7743,9957}, {7754,7774}, {7758,9766}, {7760,7858}, {7766,7921}, {7768,7809}, {7771,7802}, {7778,7795}, {7779,7941}, {7784,7800}, {7786,7790}, {7787,7806}, {7793,7823}, {7794,7821}, {7801,7888}, {7803,7851}, {7810,7873}, {7811,7860}, {7818,7854}, {7820,7874}, {7822,7867}, {7826,7845}, {7830,7842}, {7831,7911}, {7832,7899}, {7835,7940}, {7836,7925}, {7846,7942}, {7852,7889}, {7855,7903}, {7856,7878}, {7859,7919}, {7875,7932}, {7877,7926}, {7893,7900}, {7898,7904}, {7935,8722}, {7998,13340}, {8014,11555}, {8015,11556}, {8069,10320}, {8085,8087}, {8086,8088}, {8121,8123}, {8122,8124}, {8158,8165}, {8212,8222}, {8213,8223}, {8280,8855}, {8281,8854}, {8351,8379}, {8377,8380}, {8378,8381}, {8538,8541}, {8591,12355}, {8725,9751}, {8798,13157}, {8800,8905}, {8909,8966}, {8918,10218}, {8919,10217}, {8929,11581}, {8930,11582}, {8961,8963}, {8985,8990}, {9159,11639}, {9172,10162}, {9512,11061}, {9535,9566}, {9538,9642}, {9539,9641}, {9542,9691}, {9543,9690}, {9544,9704}, {9545,9703}, {9782,9809}, {9786,9815}, {9862,10583}, {9874,12872}, {9919,13203}, {9964,12528}, {10037,10832}, {10038,10874}, {10040,10925}, {10041,10926}, {10046,10831}, {10047,10873}, {10048,10923}, {10049,10924}, {10053,12185}, {10054,12351}, {10058,12764}, {10059,12860}, {10060,12950}, {10061,12951}, {10062,12952}, {10063,12836}, {10064,12954}, {10065,12374}, {10066,12956}, {10067,12958}, {10068,12959}, {10069,12184}, {10070,12350}, {10074,12763}, {10075,12859}, {10076,12940}, {10077,12941}, {10078,12942}, {10079,12837}, {10080,12944}, {10081,12373}, {10082,12946}, {10083,12948}, {10084,12949}, {10085,12678}, {10086,13183}, {10087,13274}, {10088,12904}, {10089,13182}, {10090,13273}, {10091,12903}, {10168,11645}, {10187,12816}, {10188,12817}, {10266,12919}, {10278,10279}, {10311,10316}, {10312,10317}, {10524,10530}, {10528,10596}, {10529,10597}, {10546,11464}, {10575,11381}, {10584,10785}, {10585,10786}, {10586,10805}, {10587,10806}, {10628,11557}, {10634,10641}, {10635,10642}, {10733,12121}, {10797,10802}, {10798,10801}, {10984,11550}, {11236,12513}, {11264,13366}, {11392,11399}, {11393,11398}, {11411,11431}, {11425,12118}, {11429,12428}, {11449,12278}, {11455,12279}, {11456,11457}, {11501,11508}, {11502,11507}, {11536,12234}, {11576,12363}, {11649,12061}, {11671,13512}, {11746,12236}, {11754,11755}, {11763,11764}, {11772,11773}, {11781,11782}, {11869,11879}, {11870,11880}, {11871,11877}, {11872,11878}, {11905,11913}, {11906,11912}, {11930,11953}, {11931,11954}, {11932,11951}, {11933,11952}, {12099,12827}, {12308,12317}, {12309,12318}, {12310,12319}, {12311,12320}, {12312,12321}, {12316,12325}, {12383,12902}, {12384,13115}, {12494,13234}, {12599,12864}, {12600,13089}, {12613,12621}, {12614,12622}, {12615,12623}, {12624,13249}, {12849,13126}, {12945,13117}, {12947,13129}, {12955,13116}, {12957,13128}, {13023,13025}, {13024,13026}, {13039,13051}, {13040,13052}, {13219,13310}, {13296,13312}, {13297,13311}, {13348,13570}, {13507,13508}

X(5) is the {X(2),X(4)}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(5), click **Tables** at the top of this page.

X(5) = midpoint of X(i) and X(j) for these (i,j):

(1,355), (2,381), (3,4), (11,119), (20,382), (68,155), (110,265), (113,125), (114,115), (116,118), (117,124), (122,133), (127,132), (128,137), (129,130), (131,136), (399,3448)

X(5) = reflection of X(i) in X(j) for these (i,j): (2,547), (3,140), (4,546), (20,548), (52,143), (549,2), (550,3), (1263,137), (1353,6), (1385,1125), (1483,1), (1484,11)

X(5) = isogonal conjugate of X(54)

X(5) = isotomic conjugate of X(95)

X(5) = circumcircle-inverse of X(2070)

X(5) = orthocentroidal-circle-inverse of X(3)

X(5) = complement of X(3)

X(5) = anticomplement of X(140)

X(5) = complementary conjugate of X(3)

X(5) = eigencenter of anticevian triangle of X(523)

X(5) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,216), (4,52), (110,523), (264, 324), (265,30), (311,343),
(324,53)

X(5) = cevapoint of X(i) and X(j) for these (i,j): (3,195), (51,216)

X(5) = X(i)-cross conjugate of X(j) for these (i,j): (51,53), (54, 2121), (216,343), (233,2)

X(5) = crosspoint of X(i) and X(j) for these (i,j): (2,264), (311,324)

X(5) = crosssum of X(i) and X(j) for these (i,j): (3,1147), (6,184)

X(5) = crossdifference of every pair of points on line X(50)X(647)

X(5) = X(1)-aleph conjugate of X(1048)
X(5) = radical center of Stammler circles

X(5) = center of inverse-in-circumcircle-of-tangential-circle

X(5) = harmonic center of 1st & 2nd Hutson circles

X(5) = homothetic center of circumorthic triangle and 2nd isogonal triangle of X(4); see X(36)

X(5) = X(3)-of-X(4)-Brocard-triangle

X(5) = X(4)-of-Schroeter-triangle

X(5) = X(5)-of-Fuhrmann-triangle

X(5) = X(5)-of-complement-of-excentral-triangle (or extraversion triangle of X(10))

X(5) = X(114)-of-1st-Brocard-triangle

X(5) = X(143)-of-excentral-triangle

X(5) = X(156)-of-intouch-triangle

X(5) = X(1511)-of-orthocentroidal-triangle

X(5) = bicentric sum of PU(i) for these i: 5, 7, 38, 65, 173

X(5) = midpoint of PU(i) for these i: 5, 7, 38

X(5) = trilinear product of PU(69)

X(5) = PU(65)-harmonic conjugate of X(351)

X(5) = perspector of circumconic centered at X(216)

X(5) = center of circumconic that is locus of trilinear poles of lines passing through X(216)

X(5) = trilinear pole of line X(2081)X(2600)

X(5) = pole wrt polar circle of trilinear polar of X(275) (line X(186)X(523))

X(5) = X(48)-isoconjugate (polar conjugate) of X(275)

X(5) = X(252)-isoconjugate of X(2964)

X(5) = homothetic center of medial triangle and Euler triangle

X(5) = homothetic center of ABC and the triangle obtained by reflecting X(3) in the points A, B, C

X(5) = radical center of the Stammler circles

X(5) = centroid of {A, B, C, X(4)}

X(5) = antigonal image of X(1263)

X(5) = crosspoint of X(627) and X(628) wrt both the excentral and anticomplementary triangles

X(5) = intersection of tangents to Evans conic at X(15) and X(16)

X(5) = polar-circle-inverse of X(186)

X(5) = inverse-in-{circumcircle, nine-point circle}-inverter of X(23)

X(5) = inverse-in-Kiepert-hyperbola of X(39)

X(5) = inverse-in-Steiner-inellipse of X(297)

X(5) = {X(2009),X(2010)}-harmonic conjugate of X(39)

X(5) = {X(2454),X(2455)}-harmonic conjugate of X(297)

X(5) = perspector of medial triangles of ABC, orthic and half-altitude triangles

X(5) = X(6)-isoconjugate of X(2167)

X(5) = orthic-isogonal conjugate of X(52)

X(5) = Thomson-isogonal conjugate of X(6030)

X(5) = X(1)-of-submedial triangle if ABC is acute

X(5) = harmonic center of circumcircles of Euler and anti-Euler triangles

X(5) = perspector of Feuerbach triangle and cross-triangle of ABC and Feuerbach triangle

X(5) = Kosnita(X(4),X(2)) point

X(5) = Kosnita(X(4),X(3)) point

X(5) = Kosnita(X(4),X(20)) point

X(5) = X(4)-of-Ehrmann-mid-triangle

X(5) = homothetic center of Ehrmann vertex-triangle and Kosnita triangle

X(5) = homothetic center of Ehrmann side-triangle and circumorthic triangle

X(5) = perspector of Ehrmann mid-triangle and submedial triangle

X(5) = Ehrmann-side-to-orthic similarity image of X(4)

X(5) = Johnson-to-Ehrmann-mid similarity image of X(3)

X(5) = excentral-to-ABC functional image of X(3)

X(5) = QA-P32 center (Centroid of the Circumcenter Quadrangle) of quadrangle ABCX(4) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/60-qa-p32.html)

Trilinears sin A : sin B : sin C

Trilinears (1 + cos A)(1 - cos A + cos B + cos C) : :

Trilinears cot B/2 + cot C/2 : :

Barycentrics a

Barycentrics SB + SC : :

Barycentrics SA - SW : :

Barycentrics cot A - cot ω : :

Barycentrics cot B + cot C - cot A + cot ω : :

Tripolars b c Sqrt[2(b^2 + c^2) - a^2] : :

X(6) = (1 + sqrt(3)*tan(ω))*X(13) + (1 - sqrt(3)*tan(ω))*X(14)

X(6) = (1 + sqrt(3)*tan(ω))*X(15) + (1 - sqrt(3)*tan(ω))*X(16)

X(6) = (3 + 5*sqrt(3)*tan(ω))*X(17) + (3 - 5*sqrt(3)*tan(ω))*X(18)

(The above four combos for X(6) found by Peter Moses, November, 2011)

X(6) is the point of concurrence of the symmedians (i.e., reflections of medians in corresponding angle bisectors). X(6) is the point which, when given by *actual* trilinear distances x,y,z, minimizes x^{2} + y^{2} + z^{2}.

X(6) in Navigation: **A talk about the symmedian point, by William Lionheart.**

If you have The Geometer's Sketchpad, you can view Symmedian point.

If you have GeoGebra, you can view **Symmedian point**.

Let A'B'C' be the pedal triangle of an arbitrary point X, and let S(X) be the vector sum XA' + XB' + XC'. Then

S(X) = (0 vector) if and only if X = X(6).

The "if" implication is equivalent to the well known fact that X(6) is the centroid of its pedal triangle, and the converse was proved by Barry Wolk (Hyacinthos #19, Dec. 23, 1999).

X(6) is the radical trace of the 1st and 2nd Lemoine circles. (Peter J. C. Moses, 8/24/03)

X(6) is the perspector of ABC and the medial triangle of the orthic triangle of ABC. (Randy Hutson, 8/23/2011)

**Ross Honsberger,** *Episodes in Nineteenth and Twentieth Century Euclidean Geometry,* Mathematical Association of America, 995. Chapter 7: The Symmedian Point.

Let A' be the center of the conic through the contact points of the incircle and the A-excircle with the sidelines of ABC. Define B' and C' cyclically. Let A" be the center of the conic through the contact points of the B- and C- excircles with the sidelines of ABC. Define B" and C" cyclically. The triangles A'B'C' and A"B"C" are perspective at X(6). See also X(25), X(218), X(222), X(940), X(1743). (Randy Hutson, July 23, 2015)

The tangents at A,B,C to the Thomson cubic K002 concur in X(6). Let Ha be the foot of the A-altitude. Let Ba, Ca be the feet of perpendiculars from Ha to CA, AB, resp. Let A' be the orthocenter of HaBaCa, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(6). (Randy Hutson, November 18, 2015)

Let Ja, Jb, Jc be the excenters and I the incenter of ABC. Let Ka be the symmedian point of JbJcI, and define Kb and Kc cyclically. Then KaKbKc is perspective to JaJbJc at X(6). (Randy Hutson, February 10, 2016)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung); then X(6) = X(6467)-of-A'B'C'. (Randy Hutson, June 27, 2018)

X(6) is the perspector of every pair of these triangles: anticevian triangle of X(3), submedial triangle, unary cofactor triangle of submedial triangle, unary cofactor triangle of the intangents triangle, unary cofactor triangle of the extangents triangle. (Randy Hutson, June 27, 2018)

Let A'B'C' be the tangential triangle of the Jerabek hyperbola. Let A" be the pole wrt circumcircle of line B'C', and define B", C" cyclically. The lines A'A", B'B", C'C" concur in X(6). (Randy Hutson, November 30, 2018)

Let A'B'C' be the half-altitude (midheight) triangle. Let L_{A} be the line through A parallel to B'C', and define L_{B} and L_{C} cyclically. Let A" = L_{B}∩L_{C}, and define B", C" cyclically. The lines A'A", B'B", C'C" concur in X(6). (Randy Hutson, November 30, 2018)

X(6) is the unique point that is the centroid of its pedal triangle. (Randy Hutson, June 7, 2019)

Let A'B'C' be any one of {Lucas(t) central triangle, Lucas(t) tangents triangle, Lucas(t) inner triangle} (for arbitrary t). Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(6). (Randy Hutson, July 11, 2019)

Let A'B'C' be the medial triangle, and A"B"C" the orthic triangle. Let A* be the centroid of AA'A", and define B* and C* cyclically. A*B*C* is inversely similar to ABC, and the lines A'A*, B'B*, C'C* concur in X(6). (Randy Hutson, July 11, 2019)

X(6) is the intersection of the isotomic conjugate of the polar conjugate of the Euler line (i.e., line X(2)X(6)), and the polar conjugate of the isotomic conjugate of the Euler line (i.e., line X(4)X(6)). (Randy Hutson, July 11, 2019)

X(6) lies on the Thomson cubic and these lines:

1,9 2,69 3,15 4,53
5,68 7,294 8,594
10,1377 13,14 17,18
19,34 21,941 22,251
23,353 24,54 25,51
26,143 27,1246 31,42
33,204 36,609 40,380
41,48 43,87 57,222
60,1169 64,185 66,427
67,125 70,1594 74,112
75,239 76,83 77,241
88,89 98,262 99,729
100,739 101,106 105,1002
110,111 145,346 157,248
160,237 162,1013 169,942
181,197 190,192 194,384
210,612 226,1751 256,1580
264,287 274,1218 279,1170
281,1146 282,1256 291,985
292,869 297,317 305,1241
314,981 330,1258 344,1332
354,374 442,1714 493,1583
494,1584 513,1024 517,998
519,996 523,879 560,1631
561,720 588,1599 589,1600
593,1171 595,1126 598,671
603,1035 644,1120 657,1459
662,757 688,882 689,703
691,843 692,1438 694,1084
706,1502 717,789 750,899
753,825 755,827 840,919
846,1051 893,1403 909,1415
911,1461 939,1802 943,1612
947,1622 959,961 963,1208
967,1790 971,990 986,1046
1073,3343 1096,1859 1112,1177
1131,1132 1139,1140 1166,1601
1173,1614 1174,1617 1195,1399
1201,1696 1214,1708 1327,1328
1362,1416 1398,1425 1423,1429
1718,1781 1826,1837 1836,1839
1854,1858 3342,3351 3344,3350

X(6) is the {X(15),X(16)}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(6), click **Tables** at the top of this page.

X(6) = midpoint of X(i) and X(j) for these (i,j): (32,5028), (39,5052), (69,193), (125,5095), (187,5107), (1689, 1690)

X(6) = reflection of X(i) in X(j) for these (i,j): (1,1386), (2,597), (3,182), (67,125), (69,141), (159,206), (182,575), (592,2), (694,1084), (1350,3), (1351,576), (1352,5)

X(6) = isogonal conjugate of X(2)

X(6) = isotomic conjugate of X(76)

X(6) = cyclocevian conjugate of X(1031)

X(6) = circumcircle-inverse of X(187)

X(6) = orthocentroidal-circle-inverse of X(115)

X(6) = 1st-Lemoine-circle-inverse of X(1691)

X(6) = complement of X(69)

X(6) = anticomplement of X(141)

X(6) = anticomplementary conjugate of X(1369)

X(6) = complementary conjugate of X(1368)

X(6) = crossdifference of every pair of points on line X(30)X(511)

X(6) = insimilicenter of 1st and 2nd Kenmotu circles

X(6) = exsimilicenter of circumcircle and (1/2)-Moses circle

X(6) = harmonic center of circumcircle and Gallatly circle

X(6) = perspector of polar circle wrt Schroeter triangle

X(6) = X(i)-Ceva conjugate of X(j) for these (i,j):

(1,55), (2,3), (3,154), (4,25), (7,1486), (8,197), (9,198), (10,199), (54,184), (57, 56), (58,31), (68,161), (69,159), (74,1495), (76,22), (81,1), (83,2), (88,36), (95,160), (98,237), (110,512), (111,187), (222,221), (249,110), (251,32), (264,157), (275,4), (284,48), (287,1503), (288,54), (323,399), (394,1498), (1613,3360)

X(6) = cevapoint of X(i) and X(j) for these (i,j): (1,43), (2,194), (31,41), (32,184), (42,213), (51,217)

X(6) = X(i)-cross conjugate of X(j) for these (i,j): (25,64), (31,56), (32,25), (39,2), (41,55), (42,1), (51,4), (184,3), (187,111), (213,31), (217,184), (237,98), (512,110)

X(6) = crosspoint of X(i) and X(j) for these (i,j):

(1,57), (2,4), (9,282), (54,275), (58,81), (83, 251), (110,249), (266,289)

X(6) = crosssum of X(i) and X(j) for these (i,j): (1,9), (3,6), (4,1249), (5,216), (10,37), (11,650), (32,206), (39,141), (44,214), (56,478), (57,223), (114,230), (115,523), (125,647), (128,231), (132,232), (140,233), (142,1212), (188,236), (226,1214), (244,661), (395,619), (396, 618), (512,1084), (513,1015), (514,1086), (522,1146), (570,1209), (577,1147), (590,641), (615,642), (1125,1213), (1196,1368)

X(6) = X(i)-Hirst inverse of X(j) for these (i,j): (1,238), (2,385), (3,511), (15,16), (25,232), (56,1458), (58,1326), (523,1316), (1423,1429)

X(6) = X(i)-line conjugate of X(j) for these (i,j): (1,518), (2,524), (3,511)

X(6) = X(i)-aleph conjugate of X(j) for these (i,j): (1,846), (81,6), (365,1045), (366,191), (509,1046)

X(6) = X(i)-beth conjugate of X(j) for these (i,j): (6,604), (9,9), (21,1001), (101,6), (284,6), (294,6), (644,6), (645,6),
(651,6), (652,7), (666,6)

X(6) = exsimilicenter of circumcircle and (1/2)-Moses circle; the insimilicenter is X(5013)

X(6) = homothetic center of outer Napoleon triangle and pedal triangle of X(15)

X(6) = homothetic center of inner Napoleon triangle and pedal triangle of X(16)

X(6) = trilinear product of vertices of Thomson triangle

X(6) = orthocenter of X(i)X(j)X(k) for these (i,j,k): (2,4,1640), (3,4,879), (3,64,2435)

X(6) = intersection of tangents at X(3) and X(4) to Darboux cubic K004

X(6) = radical trace of circumcircle and Ehrmann circle

X(6) = one of two harmonic traces of Ehrmann circles; the other is (X(23)

X(6) = X(3734)-of-1st anti-Brocard-triangle

X(6) = X(182)-of-anti-McCay triangle

X(6) = intersection of tangents to 2nd Brocard circle at PU(1) (i.e., pole of line X(39)X(512) wrt 2nd Brocard circle)

X(6) = intersection of diagonals of trapezoid PU(1)PU(39)

X(6) = intersection of diagonals of trapezoid PU(6)PU(33)

X(6) = intersection of diagonals of trapezoid PU(31)PU(33)

X(6) = the point in which the extended legs P(6)U(31) and U(6)P(31) of the trapezoid PU(6)PU(31) meet

X(6) = trilinear pole of PU(i) for these i: 2, 26

X(6) = crosssum of PU(4)

X(6) = trilinear product of PU(8)

X(6) = barycentric product of PU(i) for these i: 1, 17, 113, 114, 115, 118, 119

X(6) = crossdifference of PU(i) for these i: 24, 41

X(6) = midpoint of PU(i) for these i: 45, 46, 54

X(6) = bicentric sum of PU(i) for these i: 45, 46, 54, 62

X(6) = crosssum of X(5408) and X(5409)

X(6) = Zosma transform of X(19)

X(6) = trilinear square of X(365)

X(6) = radical center of {circumcircle, Parry circle, Parry isodynamic circle}; see X(2)

X(6) = PU(62)-harmonic conjugate of X(351)

X(6) = vertex conjugate of PU(118)

X(6) = eigencenter of orthocentroidal triangle

X(6) = eigencenter of Stammler triangle

X(6) = eigencenter of outer Grebe triangle

X(6) = eigencenter of inner Grebe triangle

X(6) = eigencenter of submedial triangle

X(6) = perspector of unary cofactor triangles of every pair of homothetic triangles

X(6) = perspector of ABC and unary cofactor triangle of any triangle homothetic to ABC

X(6) = perspector of Stammler triangle and unary cofactor triangle of circumtangential triangle

X(6) = perspector of Stammler triangle and unary cofactor triangle of circumnormal triangle

X(6) = perspector of submedial triangle and unary cofactor triangle of orthic triangle

X(6) = perspector of unary cofactor triangles of extraversion triangles of X(7) and X(9)

X(6) = X(3)-of-reflection-triangle-of-X(2)

X(6) = center of the orthic inconic

X(6) = orthic isogonal conjugate of X(25)

X(6) = center of bicevian conic of X(371) and X(372)

X(6) = center of bicevian conic of X(6) and X(25)

X(6) = perspector of ABC and mid-triangle of Mandart-incircle and Mandart-excircles triangles

X(6) = X(381)-of-anti-Artzt-triangle

X(6) = homothetic center of medial triangle and cross-triangle of ABC and inner Grebe triangle

X(6) = homothetic center of medial triangle and cross-triangle of ABC and outer Grebe triangle

X(6) = 4th-anti-Brocard-to-anti-Artzt similarity image of X(3)

X(6) = perspector of pedal and anticevian triangles of X(3)

X(6) = X(9)-of-orthic-triangle if ABC is acute

X(6) = X(7)-of-tangential-triangle if ABC is acute

X(6) = X(53)-of-excentral-triangle

X(6) = Thomson-isogonal conjugate of X(376)

X(6) = perspector of ABC and mid-triangle of 1st and 2nd anti-Conway triangles

X(6) = X(193)-of-3rd-tri-squares-central-triangle

X(6) = X(193)-of-4th-tri-squares-central-triangle

X(6) = X(6)-of-circumsymmedial-triangle

X(6) = X(6)-of-inner-Grebe-triangle

X(6) = X(6)-of-outer-Grebe-triangle

X(6) = X(157)-of-intouch-triangle

X(6) = perspector, wrt Schroeter triangle, of polar circle

X(6) = center of the perspeconic of these triangles: ABC and Ehrmann vertex

X(6) = barycentric square of X(1)

X(6) = pole, wrt circumcircle, of Lemoine axis

X(6) = pole wrt polar circle of trilinear polar of X(264) (line X(297)X(525))

X(6) = polar conjugate of X(264)

X(6) = X(i)-isoconjugate of X(j) for these {i,j}: {1,2}, {6,75}, {31,76}, {91,1993}, {110, 1577}, {338,1101}, {1994,2962}

X(6) = inverse-in-2nd-Brocard-circle of X(39)

X(6) = inverse-in-Steiner-inellipse of X(230)

X(6) = inverse-in-Steiner-circumellipse of X(385)

X(6) = inverse-in-Kiepert-hyperbola of X(381)

X(6) = inverse-in-circumconic-centered-at-X(9) of X(238)

X(6) = perspector of medial triangle and half-altitude triangle

X(6) = intersection of tangents to Kiepert hyperbola at X(2) and X(4)

X(6) = antigonal conjugate of X(67)

X(6) = vertex conjugate of foci of Steiner inellipse

X(6) = X(99)-of-1st-Brocard-triangle

X(6) = X(1379)-of-2nd-Brocard-triangle

X(6) = X(6)-of-4th-Brocard-triangle

X(6) = X(6)-of-orthocentroidal-triangle

X(6) = reflection of X(2453) in the Euler line

X(6) = similitude center of ABC and orthocentroidal triangle

X(6) = similitude center of 4th Brocard and circumsymmedial triangles

X(6) = tangential isogonal conjugate of X(22)

X(6) = tangential isotomic conjugate of X(1498)

X(6) = barycentric product of (nonreal) circumcircle intercepts of the line at infinity

X(6) = eigencenter of anti-orthocentroidal triangle

X(6) = perspector of Aquarius conic

X(6) = trilinear pole wrt tangential triangle of Lemoine axis

X(6) = trilinear pole wrt symmedial triangle of Lemoine axis

X(6) = trilinear pole wrt circumsymmedial triangle of Lemoine axis

X(6) = crosspoint of X(2) and X(194) wrt both the excentral and anticomplementary triangles

X(6) = pedal antipodal perspector of X(5004) and of X(5005)

X(6) = vertex conjugate of Jerabek hyperbola intercepts of Lemoine axis

X(6) = hyperbola {{A,B,C,X(2),X(6)}} antipode of X(694)

X(6) = perspector of orthic triangle and tangential triangle, wrt orthic triangle, of the circumconic of the orthic triangle centered at X(4) (the bicevian conic of X(4) and X(459))

X(6) = perspector of excentral triangle and extraversion triangle of X(9)

X(6) = excentral-to-ABC functional image of X(9)

X(6) = orthic-to-ABC functional image of X(53)

X(6) = intouch-to-ABC functional image of X(7)

X(6) = QA-P23 (Inscribed Square Axes Crosspoint) of quadrangle ABCX(2); see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/51-qa-p23.html

Trilinears sec

Trilinears 1/(tan(B/2) + tan(C/2)) : 1/(tan(C/2) + tan(A/2)) : 1/(tan(A/2) + tan(B/2))

Trilinears (bc - S

Trilinears (1 - cos A)a

Barycentrics 1/(b + c - a) : 1/(c + a - b) : 1/(a + b - c)

Barycentrics tan A/2 : tan B/2 : tan C/2

Barycentrics bc - S

Barycentrics r

Tripolars (b+c-a) Sqrt[a (a-b-c) (a^2+a b-2 b^2+a c+4 b c-2 c^2)] : :

X(7) = [A]/Ra + [B]/Rb + [C]/Rc, where Ra, Rb, Rc = radii of Soddy circles

Let A', B', C' denote the points in which the incircle meets the sides BC, CA, AB, respectively. The lines AA', BB', CC' concur in X(7).

If you have The Geometer's Sketchpad, you can view Gergonne point.

If you have GeoGebra, you can view **Gergonne point**.

X(7) lies on the Lucas cubic and these lines:

1,20 2,9 3,943 4,273
6,294 8,65 11,658
12,1268 21,56 27,81
37,241 33,1041 34,1039 55,2346
58,272 59,1275 72,443
73,1246 76,1479 80,150
92,189 100,1004 104,934
108,1013 109,675 145,1266
171,983 174,234 177,555
190,344 192,335 193,239
218,277 220,1223 225,969
238,1471 253,280 256,982
274,959 281,653 286,331
310,314 330,1432 349,1269
354,479 404,1259 452,1467
464,1214 480,1376 492,1267
513,885 517,1000 528,664
554,1082 594,599 604,1429
757,1414 840,927 857,1901
870,1431 940,1407 941,1427
944,1389 952,1159 986,1254
987,1106 1002,1362 1020,1765
1061,1870 1210,3091 1354,1367 1365,1366
1386,1456 1419,1449 1435,1848
1486,1602 1617,1621 2475,2893

X(7) is the {X(69),X(75)}-harmonic conjugate of X(8). For a list of
other harmonic conjugates of X(7), click **Tables** at the top of
this page.

X(7) = reflection of X(i) in X(j) for these (i,j): (9,142), (144,9), (390,1), (673,1086), (1156,11)

X(7) = isogonal conjugate of X(55)

X(7) = isotomic conjugate of X(8)

X(7) = cyclocevian conjugate of X(7)

X(7) = circumcircle-inverse of (32624)

X(7) = incircle-inverse of (1323)

X(7) = complement of X(144)

X(7) = anticomplement of X(9)

X(7) = complementary conjugate of X(2884)

X(7) = anticomplementary conjugate of X(329)

X(7) = X(i)-Ceva conjugate of X(j) for these (i,j): (75,347), (85,2), (86,77), (286,273), (331,278)

X(7) = cevapoint of X(i) and X(j) for these (i,j):

(1,57), (2,145), (4,196), (11,514), (55,218), (56,222), (63,224),
(65,226), (81,229), (177,234)

X(7) = X(i)-cross conjugate of X(j) for these (i,j):

(1,2), (4,189), (11,514), (55,277), (56,278), (57,279), (65,57), (177,174), (222,348), (226,85), (354,1), (497,8), (517,88)

X(7) = crosspoint of X(i) and X(j) for these (i,j): (75,309), (86,286)

X(7) = crosssum of X(i) and X(j) for these (i,j): (41,1253), (42,228)

X(7) = crossdifference of every pair of points on line X(657)X(663)

X(7) = X(57)-Hirst inverse of X(1447)

X(7) = insimilicenter of inner and outer Soddy circles; the exsimilicenter is X(1)

X(7) = X(i)-beth conjugate of X(j) for these (i,j):

(7,269), (21,991), (75,75), (86,7), (99,7), (190,7), (290,7), (314,69), (645,344), (648,7), (662,6), (664,7), (666,7), (668,7), (670,7), (671,7), (811,264), (886,7), (889,7), (892,7), (903,7)

X(7) = vertex conjugate of foci of inellipse that is isotomic conjugate of isogonal conjugate of incircle (centered at X(2886))

X(7) = trilinear product of vertices of Hutson-extouch triangle

X(7) = orthocenter of X(4)X(8)X(885)

X(7) = trilinear cube of X(506)

X(7) = barycentric product of PU(47)

X(7) = trilinear product of PU(94)

X(7) = vertex conjugate of PU(95)

X(7) = bicentric sum of PU(120)

X(7) = perspector of ABC and its intouch triangle.
X(7) = perspector of ABC and the reflection in X(57) of the pedal triangle of X(57)

X(7) = perspector of AC-incircle

X(7) = X(6)-of-extraversion triangle-of-X(8)

X(7) = X(6)-of-intouch-triangle; X(7) is the only point X inside ABC Such that X(ABC) = X(A'B'C'), where A'B'C' = cevian triangle of X

X(7) = {X(2),X(63)}-harmonic conjugate of X(5273)

X(7) = {X(9),X(57)}-harmonic conjugate of X(1445)

X(7) = {X(1371),X(1372)}-harmonic conjugate of X(1)

X(7) = {X(1373),X(1374)}-harmonic conjugate of X(1)

X(7) = trilinear pole of Gergonne line

X(7) = trilinear pole, wrt intouch triangle, of Gergonne line

X(7) = pole of Gergonne line wrt incircle

X(7) = pole wrt polar circle of trilinear polar of X(281) (line X(3064)X(3700))

X(7) = X(48)-isoconjugate (polar conjugate) of X(281)

X(7) = X(6)-isoconjugate of X(9)

X(7) = X(75)-isoconjugate of X(2175)

X(7) = X(1101)-isoconjugate of X(4092)

X(7) = perspector of circumconic centered at X(3160)

X(7) = center of circumconic that is locus of trilinear poles of lines passing through X(3160)

X(7) = X(2)-Ceva conjugate of X(3160)

X(7) = antigonal image of X(1156)

X(7) = homothetic center of intouch triangle and anticomplement of the excentral triangle

X(7) = X(6)-of-intouch-triangle; X(7) is the only point X inside ABC such that X(ABC) = X(A'B'C'), where A'B'C' = cevian triangle of X

X(7) = perspector of ABC and cross-triangle of inner and outer Soddy triangles

X(7) = perspector of excentral triangle and cross-triangle of ABC and Honsberger triangle

X(7) = perspector of inverse-in-excircles triangle and cross-triangle of ABC and inverse-in-excircles triangle

X(7) = perspector of inverse-in-incircle triangle and cross-triangle of ABC and inverse-in-incircle triangle

X(7) = X(1843)-of-excentral-triangle

X(7) = Cundy-Parry Phi transform of X(943)

X(7) = Cundy-Parry Psi transform of X(942)

X(7) = {X(1),X(1742)}-harmonic conjugate of X(2293)

X(7) = barycentric square of X(508)

X(7) = perspector of ABC and cross-triangle of ABC and Gemini triangle 40

X(7) = barycentric product of vertices of Gemini triangle 40

X(7) = excentral-to-intouch similarity image of X(9)

X(7) = endo-homothetic center of Ehrmann vertex-triangle and Ehrmann mid-triangle; the homothetic center is X(3818)

Trilinears csc

Trilinears (bc + S

Trilinears (1 + cos A)a

Trilinears (r/R) - sin B sin C : :

Barycentrics b + c - a : c + a - b : a + b - c

Barycentrics cot A/2 : cot B/2 : cot C/2

Barycentrics square of semi-minor axis of A-Soddy ellipse : :

Barycentrics bc + S

Tripolars Sqrt[a (a^2-a b-2 b^2-a c+4 b c-2 c^2)] : :

X(8) = [A]*Ra + [B]*Rb + [C]*Rc, where Ra, Rb, Rc = radii of Soddy circles

Let A'B'C' be the points in which the A-excircle meets BC, the B-excircle meets CA, and the C-excircle meets AB, respectively. The lines AA', BB', CC' concur in X(8). Another construction of A' is to start at A and trace around ABC half its perimeter, and similarly for B' and C'. Also, X(8) is the incenter of the anticomplementary triangle.

X(8) = perspector of ABC and the intouch triangle of the medial triangle of ABC. (Randy Hutson, 9/23/2011)

Let Ab, Ac, Bc, Ba, Ca, Cb be as in the construction of the Conway circle (http://mathworld.wolfram.com/ConwayCircle.html). Let Oa be the circumcircle of ABcCb, and define Ob, Oc cyclically. X(8) is the radical center of Oa, Ob, Oc. see also X(21) and X(274). (Randy Hutson, April 9, 2016)

Let A'B'C' be Triangle T(-2,1). Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(519). The lines A'A", B'B", C'C" concur in X(8). (Randy Hutson, November 18, 2015)

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(8) = X(1)-of-IaIbIc. (Randy Hutson, September 14, 2016)

Let A_{28}B_{28}C_{28} be the Gemini triangle 28. Let L_{A} be the line through A_{28} parallel to BC, and define L_{B} and L_{C} cyclically. Let A'_{28} = L_{B}∩L_{C}, and define B'_{28}, C'_{28} cyclically. Triangle A'_{28}B'_{28}C'_{28} is homothetic to ABC at X(8). (Randy Hutson, November 30, 2018)

If you have The Geometer's Sketchpad, you can view Nagel point.

If you have GeoGebra, you can view **Nagel point**.

For a construction of X(8), see Dasari Naga Vijay Krishna, "On A Simple Construction of Triangle Centers X(8), X(197), X(K) & X(594)", Scientific Inquiry and Review, Vol. 2, Issue 3, July 2018.

X(8) lies on these curves:

Feuerbach hyperbola, anticomplementary Feuerbach hyperbola, Mandart hyperbola, Fuhrmann circle, K007, K013, K028, K033, K034, K154, K157, K199, K200, K201, K259, K308, K311, K338, K366, K386, K387, K455, K461, K506, K521, K623, K651, K654, K680, K692, K696, K697, K702, K744, K767, Q045

X(8) lies on the Lucas cubic and these lines:

{1,2}, {3,100}, {4,72}, {5,1389}, {6,594}, {7,65}, {9,346}, {11,1320}, {12,2099}, {19,1891}, {20,40}, {21,55}, {22,8193}, {23,8185}, {25,7718}, {29,219}, {30,3578}, {31,987}, {32,5291}, {33,1039}, {34,1041}, {35,993}, {36,4188}, {37,941}, {38,986}, {39,7976}, {41,2329}, {44,4217}, {45,3943}, {46,3218}, {56,404}, {57,1219}, {58,996}, {60,7058}, {76,668}, {79,758}, {80,149}, {81,1010}, {86,2334}, {90,2994}, {101,1311}, {109,2370}, {113,7978}, {114,7970}, {115,7983}, {119,6941}, {125,7984}, {140,1483}, {141,3242}, {142,3243}, {144,516}, {147,9864}, {150,1930}, {151,2817}, {153,2800}, {165,3522}, {171,1468}, {172,4386}, {175,1270}, {176,1271}, {177,556}, {178,236}, {181,959}, {188,2090}, {190,528}, {191,3065}, {192,256}, {193,894}, {194,730}, {197,1603}, {201,1937}, {210,312}, {213,981}, {214,5445}, {215,9701}, {220,294}, {221,651}, {224,914}, {226,3340}, {238,983}, {244,3976}, {253,307}, {274,1002}, {277,1280}, {278,1257}, {279,7273}, {291,330}, {304,3263}, {313,2997}, {314,1264}, {315,760}, {326,1442}, {344,480}, {348,664}, {350,3789}, {354,3698}, {376,3579}, {381,8148}, {392,1000}, {394,3562}, {405,943}, {406,1061}, {411,3428}, {442,495}, {443,942}, {474,999}, {475,1063}, {479,7182}, {484,4299}, {491,1267}, {492,5391}, {496,3820}, {514,4546}, {521,4397}, {522,4474}, {523,4774}, {524,4363}, {527,4454}, {529,3474}, {535,3245}, {536,4419}, {537,4440}, {573,3588}, {595,1724}, {596,4674}, {599,1086}, {618,7975}, {619,7974}, {631,1385}, {637,7595}, {641,7981}, {642,7980}, {643,1098}, {645,4092}, {646,3271}, {663,4147}, {672,3501}, {673,4437}, {704,8264}, {712,4805}, {726,1278}, {860,1068}, {885,3900}, {901,2757}, {908,946}, {912,5553}, {961,1460}, {965,2256}, {971,9961}, {982,4457}, {1015,1574}, {1016,1083}, {1018,3730}, {1019,4807}, {1034,1895}, {1036,1183}, {1046,4418}, {1054,9457}, {1071,6916}, {1104,3744}, {1106,9363}, {1107,2276}, {1120,3445}, {1124,1377}, {1126,6539}, {1147,9933}, {1159,6147}, {1191,4383}, {1209,7979}, {1211,1834}, {1212,3693}, {1215,4865}, {1229,6601}, {1237,4485}, {1251,5239}, {1266,4346}, {1279,6687}, {1281,1282}, {1309,2745}, {1312,2103}, {1313,2102}, {1317,1388}, {1319,6049}, {1331,2988}, {1332,8759}, {1335,1378}, {1361,3042}, {1364,3040}, {1386,3618}, {1392,5048}, {1397,2985}, {1420,3911}, {1449,4982}, {1453,5294}, {1467,8732}, {1500,1573}, {1512,5720}, {1575,2275}, {1656,5901}, {1672,1680}, {1673,1681}, {1674,1678}, {1675,1679}, {1682,9564}, {1699,3832}, {1738,3620}, {1739,3953}, {1743,4058}, {1748,6197}, {1757,3923}, {1759,5011}, {1783,8743}, {1812,3193}, {1836,3962}, {1857,1896}, {1869,5307}, {1897,7358}, {1914,4426}, {1943,4296}, {1953,3949}, {1959,7379}, {1992,3758}, {1997,3816}, {2007,2013}, {2008,2014}, {2053,8851}, {2077,5450}, {2093,4001}, {2122,2123}, {2170,3061}, {2175,4157}, {2176,2238}, {2242,5277}, {2310,9365}, {2318,2654}, {2320,2646}, {2335,3694}, {2363,6043}, {2399,8058}, {2463,2467}, {2464,2468}, {2477,9702}, {2482,9884}, {2564,2568}, {2565,2569}, {2647,4332}, {2650,4938}, {2785,4088}, {2787,4730}, {2796,8596}, {2801,5696}, {2810,3888}, {2883,7973}, {2891,3754}, {2893,2897}, {2894,6839}, {2896,9857}, {2901,3995}, {2943,9355}, {3021,3039}, {3022,3041}, {3038,6018}, {3056,4110}, {3058,3715}, {3068,7969}, {3069,7968}, {3090,5886}, {3096,9997}, {3152,6360}, {3158,3601}, {3174,7675}, {3177,4712}, {3247,5257}, {3254,4858}, {3304,4413}, {3305,5129}, {3306,3333}, {3309,4462}, {3336,4317}, {3339,4298}, {3361,4315}, {3427,6836}, {3452,3680}, {3467,4309}, {3475,3925}, {3496,5282}, {3523,3576}, {3524,3655}, {3545,3656}, {3583,3899}, {3585,4067}, {3619,3844}, {3629,7227}, {3631,7232}, {3647,5441}, {3649,6175}, {3663,4452}, {3664,4924}, {3666,4646}, {3670,3987}, {3671,5290}, {3672,3755}, {3683,5302}, {3716,4895}, {3721,3959}, {3731,3950}, {3735,3954}, {3738,4768}, {3739,4648}, {3740,3983}, {3742,4731}, {3746,5248}, {3760,6381}, {3762,3887}, {3772,4952}, {3775,4085}, {3814,5154}, {3817,5068}, {3823,4864}, {3826,4966}, {3829,7173}, {3841,5425}, {3879,3945}, {3881,3918}, {3884,3992}, {3891,4972}, {3892,3968}, {3894,3919}, {3896,3931}, {3898,3956}, {3901,4084}, {3904,4528}, {3907,4041}, {3928,5128}, {3929,7285}, {3947,5726}, {3963,9052}, {3967,4005}, {3977,4304}, {3978,6382}, {3986,4898}, {3993,4704}, {4002,5045}, {4003,4706}, {4004,5551}, {4018,4980}, {4026,4360}, {4036,8702}, {4054,9612}, {4080,4792}, {4082,4866}, {4086,7253}, {4087,4531}, {4125,4857}, {4160,4761}, {4163,6332}, {4181,4182}, {4208,5249}, {4234,4921}, {4312,5850}, {4314,4512}, {4342,4900}, {4364,4748}, {4373,4862}, {4404,6003}, {4407,4743}, {4421,5217}, {4424,7226}, {4432,4473}, {4439,4527}, {4470,4670}, {4534,6558}, {4542,4582}, {4595,8299}, {4657,4852}, {4658,8025}, {4667,4747}, {4672,4753}, {4675,4688}, {4694,9335}, {4699,4732}, {4729,6002}, {4736,6758}, {4756,9668}, {4767,9669}, {4867,5141}, {4922,9508}, {4999,5432}, {5010,5267}, {5056,8227}, {5059,5493}, {5187,10043}, {5221,5434}, {5284,6767}, {5285,7520}, {5286,9620}, {5429,8258}, {5534,6908}, {5584,7411}, {5590,5604}, {5591,5605}, {5592,6546}, {5597,5600}, {5598,5599}, {5714,9654}, {5791,6857}, {5985,10053}, {6001,6223}, {6062,7068}, {6144,7277}, {6154,9963}, {6174,10031}, {6193,9928}, {6245,6282}, {6260,7971}, {6261,6838}, {6265,6949}, {6292,7977}, {6326,6960}, {6462,8214}, {6463,8215}, {6553,8056}, {6653,6655}, {6691,7231}, {6739,6742}, {6828,7680}, {6835,7686}, {6856,8164}, {6945,7681}, {6995,7713}, {7003,7020}, {7018,7033}, {7028,8242}, {7043,7126}, {7048,8422}, {7090,7133}, {7161,7206}, {7279,9723}, {7373,9342}, {7486,9624}, {7987,9588}, {8092,8125}, {8094,9795}, {8126,8351}, {8162,8167}, {8163,8169}, {8210,8222}, {8211,8223}, {8372,9787}, {8591,9881}, {8972,8983}, {9317,9451}, {9783,9805}, {9859,9943}, {10087,10093}, {10090,10094}
--

X(8) is the {X(69),X(75)}-harmonic conjugate of X(7). For a list of other harmonic conjugates of X(8), click **Tables** at the top of this page.

X(8) = midpoint of X(i) and X(j) for these {i,j}: {1,3632}, {10,3625}, {40,5881}, {145,3621}, {3057,3893}, {3626,4701}, {3679,4677}, {4474,4814}, {4668,4816}, {4900,8275}, {5541,9897}, {5691,7991}, {5903,5904}

X(8) = reflection of X(i) in X(j) for these (i,j):

(1,10), (2,3679), (3,5690), (4,355), (7,2550), (8,8), (10,3626), (11,3036), (20,40), (56,8256), (65,5836), (69,3416), (75,3696), (78,6736), (86,4733), (100,1145), (144,5223), (145,1), (147,9864), (149,80), (192,984), (193,3751), (210,4711), (315,4769), (329,3421), (346,4901), (376,3654), (388,5794), (390,9), (551,4745), (663,4147), (944,3), (950,5795), (960,4662), (962,4), (1019,4807), (1043,3704), (1120,3756), (1125,4691), (1280,4904), (1317,3035), (1320,11), (1361,3042), (1364,3040), (1392,7705), (1482,5), (1483,140), (1697,5837), (2098,1329), (2099,2886), (2102,1313), (2103,1312), (3021,3039), (3022,3041), (3057,960), (3146,5691), (3161,10005), (3189,3913), (3241,2), (3242,141), (3243,142), (3244,1125), (3434,3419), (3476,1376), (3486,958), (3488,9708), (3555,942), (3600,1706), (3616,3617), (3617,4668), (3621,3632), (3623,1698), (3625,4701), (3626,4746), (3632,3625), (3633,3244), (3635,3634), (3679,4669), (3685,3717), (3786,4111), (3868,65), (3869,72), (3872,4847), (3873,3753), (3874,3754), (3875,3755), (3877,210), (3878,3678), (3881,3918), (3883,3686), (3884,4015), (3885,3057), (3886,2321), (3889,3698), (3890,3697), (3892,3968), (3894,3919), (3898,3956), (3899,4134), (3901,4084), (3902,3706), (3952,4738), (4318,1861), (4344,2345), (4360,4026), (4363,4665), (4419,4643), (4430,5902), (4454,4659), (4511,6735), (4560,4041), (4643,4690), (4644,4363), (4673,3714), (4693,4439), (4720,4046), (4861,6734), (4864,3823), (4895,3716), (4922,9508), (5048,5123), (5080,5176), (5180,5080), (5263,594), (5441,3647), (5603,5790), (5697,3878), (5698,5220), (5710,5835), (5716,5793), (5731,5657), (5734,5818), (5882,6684), (5905,1478), (5919,3740), (5984,9860), (6018,3038), (6193,9928), (6224,100), (6327,4680), (6332,4163), (6737,6743), (6740,6741), (6742,6739), (6758,4736), (7192,4761), (7253,4086), (7962,3452), (7970,114), (7971,6260), (7972,214), (7973,2883), (7974,619), (7975,618), (7976,39), (7977,6292), (7978,113), (7979,1209), (7980,642), (7981,641), (7982,946), (7983,115), (7984,125), (8241,2090), (8591,9881), (8596,9875), (8834,6552), (9263,291), (9780,4678), (9785,2551), (9791,1654), (9797,938), (9802,149), (9809,153), (9856,9947), (9884,2482), (9933,1147), (9957,5044), (9963,6154), (9965,2093), (10031,6174)

X(8) = isogonal conjugate of X(56)

X(8) = isotomic conjugate of X(7)

X(8) = cyclocevian conjugate of X(189)

X(8) = complement of X(145)

X(8) = anticomplement of X(1)

X(8) = complementary conjugate of X(2885)

X(8) = anticomplementary conjugate of X(8)

X(8) = circumcircle-inverse of X(17100)

X(8) = X(i)-Ceva conjugate of X(j) for these (i,j):
(2,3161), (4,2899), (7,8055), (69,329), (75,2), (190,3239), (290,3948), (312,346), (314,312), (318,5552), (319,2895), (333,9), (341,7080), (643,7253), (645,3700), (646,650), (664,6332), (668,4391), (765,3952), (1016,644), (1043,78), (1219,3616), (1222,1), (1494,3936), (1909,1655), (2319,7155), (2985,6), (3596,345), (3699,522), (4076,3699), (4102,2321), (4373,6557), (4554,4130), (4555,3904), (4582,1639), (4997,2325), (4998,190), (6063,344), (6064,645), (6079,900), (7017,281), (7033,192), (7155,4903), (7257,4560), (8817,7)

X(8) = X(i)-cross conjugate of X(j) for these (i,j):

(1,280), (4,1034), (9,2), (10,318), (11,522), (40,7080), (55,281), (56,2123), (72,78), (200,346), (210,9), (219,345), (312,7155), (346,6557), (497,7), (521,100), (522,3699), (650,646), (950,29), (960,21), (1145,6735), (1146,4391), (1639,4582), (1837,4), (1857,8805), (1864,282), (2170,4560), (2321,312), (2325,4997), (2968,4397), (3057,1), (3059,200), (3239,190), (3271,650), (3680,6553), (3683,7110), (3686,333), (3687,4451), (3688,55), (3700,645), (3703,3596), (3704,3701), (3706,314), (3717,4518), (3877,2320), (3880,1320), (3885,1392), (3893,3680), (3900,644), (3907,7257), (4012,5423), (4046,2321), (4051,330), (4060,4102), (4081,3239), (4086,3952), (4092,3700), (4111,210), (4124,885), (4130,4554), (4152,2325), (4180,4182), (4531,41), (4534,514), (4542,1639), (4546,6558), (4847,75), (4853,1219), (4863,6601), (4875,274), (4965,7192), (5245,7026), (5246,7043), (5795,1220), (6062,7359), (6068,6745), (6555,3161), (6736,341), (6737,1043), (6741,4086), (7063,3709), (7067,3712), (8058,1897), (8611,4552), (9785,5558)

X(8) = cevapoint of X(i) and X(j) for these (i,j):

{1,40}, {2,144}, {4,3176}, {6,197}, {9,200}, {10,72}, {11,522}, {34,8899}, {42,3588}, {55,219}, {56,2122}, {65,5930}, {175,176}, {210,2321}, {312,4110}, {346,6555}, {497,4012}, {513,3756}, {514,4904}, {519,1145}, {521,2968}, {523,8286}, {650,3271}, {758,6739}, {960,3704}, {966,4859}, {1125,3650}, {1146,3900}, {1639,4542}, {2170,4041}, {2175,4548}, {2325,4152}, {3057,6736}, {3059,4847}, {3161,4859}, {3239,4081}, {3686,4046}, {3688,3703}, {3700,4092}, {3706,4111}, {3709,7063}, {3712,7067}, {3893,4895}, {4136,4531}, {4180,4181}, {4530,4543}, {4534,4546}, {6062,7359}, {6068,6745}, {7358,8058}

X(8) = crosspoint of X(i) and X(j) for these (i,j):

{1,979}, {2,4373}, {7,8051}, {9,2319}, {75,312}, {190,4998}, {314,333}, {643,765}, {645,6064}, {668,1016}, {3596,7017}, {3699,4076}

X(8) = crosssum of X(i) and X(j) for these (i,j): {1,978}, {6,3052}, {25,3209}, {31,604}, {57,1423}, {244,4017}, {649,3271}, {663,7117}, {667,1015}, {1042,1410}, {1284,8850}, {1400,1402}

X(8) = crossdifference of every pair of points on line X(649)X(854)

X(8) = X(1)-aleph conjugate of X(1050)

X(8) = X(i)-beth conjugate of X(j) for these (i,j): (8,1), (341,341), (643,3), (668,8), (1043,8)

X(8) = exsimilicenter of incircle and Spieker circle

X(8) = exsimilicenter of Conway circle and Spieker radical circle

X(8) = trilinear product of vertices of Hutson-intouch triangle

X(8) = trilinear product of vertices of Caelum triangle

X(8) = orthocenter of X(i)X(j)X9k) for these (i,j,k): (1,4,5556), (4,7,885)

X(8) = perspector of ABC and pedal triangle of X(40)

X(8) = perspector of ABC and reflection of medial triangle in X(10)

X(8) = perspector of ABC and reflection of intouch triangle in X(1)

X(8) = pedal antipodal perspector of X(1)

X(8) = pedal antipodal perspector of X(36)

X(8) = X(1498)-of-intouch-triangle

X(8) = X(185)-of-excentral-triangle

X(8) = X(74)-of-Fuhrmann-triangle

X(8) = X(5992)-of-Brocard-triangle

X(8) = perspector of circumconic with center X(3161)

X(8) = center of circumconic that is locus of trilinear poles of lines passing through X(3161)

X(8) = X(2)-Ceva conjugate of X(3161)

X(8) = trilinear pole of line X(522)X(650) (the radical axis of circumcircle and excircles radical circle)

X(8) = pole wrt polar circle of trilinear polar of X(278) (line X(513)X(1835))

X(8) = X(48)-isoconjugate (polar conjugate) of X(278)

X(8) = X(6)-isoconjugate of X(57)

X(8) = X(75)-isoconjugate of X(1397)

X(8) = X(1101)-isoconjugate of X(1365)

X(8) = antigonal image of X(1320)

X(8) = {X(1),X(2)}-harmonic conjugate of X(3616)

X(8) = {X(1),X(10)}-harmonic conjugate of X(2)

X(8) = {X(2),X(10)}-harmonic conjugate of X(667)

X(8) = inverse-in-polar-circle of X(1878)

X(8) = inverse-in-Steiner-circumellipse of X(3912)

X(8) = inverse-in-Mandart-inellipse of X(2325)

X(8) = inverse-in-circumconic-centered-at-X(1) of X(4511)

X(8) = X(4) of 2nd Conway triangle (the extraversion triangle of X(8))

X(8) = trilinear square root of X(341)

X(8) = perspector of 5th extouch triangle and anticevian triangle of X(7)

X(8) = centroid of cross-triangle of Gemini triangles 20 and 28

X(8) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1050), (188,2943), (1222,8)

X(8) = X(i)-beth conjugate of X(j) for these (i,j): (8,1), (99,3160), (200,4517), (333,5222), (341,341), (346,4873), (643,3), (644,3730), (668,8), (1043,8), (2287,4266), (3699,8), (6558,8), (7256,8), (7257,76), (7259,220), (8706,8)

X(8) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1,8), (2,69), (3,20), (5,2888), (6,2), (7,3434), (8,3436), (9,829), (10,1330), (13,621), (14, 622), (15,616), (16,617), (18,634), (19,59015), (20,6225), (21,3869), (22,5596), (24,6193), (25,192), (28,3868), (30,146), (31,192), (32,194), (54,3), (55,144), (56,145), (57,7), (58,1), (59,100), (74,30), (81,75), (98,511), (99,512), and many others

X(8) = X(i)-complementary conjugate of X(j) for these (i,j): (1,2885), (31,3161), (513,5510), (1293,513), (3445,10), (3680,1329), (4373,2887), (8056,141)

X(8) = perspector of ABC and mid-triangle of excentral and 2nd extouch triangles

X(8) = perspector of 5th extouch triangle and cross-triangle of ABC and 5th extouch triangle

X(8) = X(1593)-of-2nd-extouch-triangle

X(8) = excentral-to-2nd-extouch similarity image of X(1697)

X(8) = Cundy-Parry Phi transform of X(104)

X(8) = Cundy-Parry Psi transform of X(517)

X(8) = perspector of ABC and cross-triangle of ABC and Gemini triangle 39

X(8) = barycentric product of vertices of Gemini triangle 39

X(8) = perspector of Gemini triangle 13 and cross-triangle of Gemini triangles 1 and 13

X(8) = excentral-to-ABC barycentric image of X(1)

X(8) = X(4)-of-Yff-contact-triangle

X(8) = Yff-contact-isogonal conjugate of X(5592)

X(8) = isoconjugate of X(i) and X(j) for these {i,j}: {1,56}, {2,604}, {3,34}, {4,603}, {6,57}, {7,31}, {8,1106}, {9,1407}, {10,1408}, {12,849}, {19,222}, {21,1042}, {25,77}, {27,1409}, {28,73}, {29,1410}, {32,85}, {33,7053}, {36,1411}, {37,1412}, {40,1413}, {41,279}, {42,1014}, {48,278}, {54,1393}, {55,269}, {58,65}, {59,244}, {60,1254}, {63,608}, {64,1394}, {69,1395}, {71,1396}, {75,1397}, {78,1398}, {79,1399}, {81,1400}, {82,1401}, {84,221}, {86,1402}, {87,1403}, {88,1404}, {89,1405}, {90,1406}, {101,3669}, {102,1455}, {103,1456}, {104,1457}, {105,1458}, {106,1319}, {108,1459}, {109,513}, {110,4017}, {154,8809}, {158,7335}, {163,7178}, {171,1431}, {172,1432}, {181,757}, {184,273}, {189,2199}, {198,1422}, {200,7023}, {208,1433}, {212,1119}, {213,1434}, {219,1435}, {220,738}, {223,1436}, {225,1437}, {226,1333}, {241,1438}, {255,1118}, {259,7370}, {267,8614}, {270,1425}, {281,7099}, {282,6611}, {283,1426}, {284,1427}, {291,1428}, {292,1429}, {296,1430}, {307,2203}, {326,7337}, {331,9247}, {346,7366}, {347,2208}, {348,1973}, {388,1472}, {393,7125}, {479,1253}, {512,1414}, {514,1415}, {518,1416}, {519,1417}, {552,872}, {560,6063}, {593,2171}, {607,7177}, {614,1037}, {643,7250}, {649,651}, {650,1461}, {657,4617}, {658,3063}, {661,4565}, {662,7180}, {663,934}, {664,667}, {669,4625}, {672,1462}, {692,3676}, {727,1463}, {741,1284}, {756,7341}, {759,1464}, {765,1357}, {798,4573}, {893,7175}, {896,7316}, {904,7176}, {909,1465}, {923,7181}, {937,1466}, {939,1467}, {951,1104}, {959,1468}, {961,1193}, {983,7248}, {985,1469}, {998,1470}, {1002,1471}, {1015,4564}, {1019,4559}, {1020,7252}, {1027,2283}, {1035,3345}, {1036,4320}, {1041,1473}, {1073,3213}, {1086,2149}, {1088,2175}, {1089,7342}, {1096,1804}, {1098,7143}, {1101,1365}, {1110,1358}, {1149,8686}, {1170,1475}, {1174,1418}, {1191,7091}, {1201,1476}, {1214,1474}, {1245,5323}, {1262,2170}, {1279,1477}, {1420,3445}, {1421,3446}, {1423,2162}, {1424,3224}, {1439,2299}, {1440,2187}, {1441,2206}, {1442,6186}, {1443,6187}, {1447,1911}, {1453,2213}, {1576,4077}, {1616,2137}, {1617,2191}, {1631,7213}, {1769,2720}, {1790,1880}, {1795,1875}, {1813,6591}, {1919,4554}, {1922,10030}, {1974,7182}, {1980,4572}, {2003,2160}, {2006,7113}, {2099,2163}, {2114,9500}, {2150,6354}, {2159,6357}, {2176,7153}, {2194,3668}, {2207,7183}, {2210,7233}, {2212,7056}, {2218,4306}, {2221,2285}, {2260,2982}, {2263,3423}, {2275,7132}, {2279,5228}, {2291,6610}, {2306,2307}, {2310,7339}, {2324,6612}, {2334,3361}, {2353,7210}, {2362,6502}, {3121,4620}, {3212,7121}, {3248,4998}, {3271,7045}, {3451,3752}, {3709,4637}, {3733,4551}, {3777,8685}, {3900,6614}, {3911,9456}, {3937,7012}, {3942,7115}, {4252,5665}, {4296,8615}, {4557,7203}, {4626,8641}, {4822,5545}, {5018,8852}, {5546,7216}, {6129,8059}, {6180,9315}, {7011,7129}, {7013,7151}, {7051,7052}, {7054,7147}, {7084,7195}, {7104,7196}, {7117,7128}, {7122,7249}, {9309,9316}, {9363,9435}, {9364,9432}

Trilinears cot(A/2) : cot(B/2) : cot(C/2)

Trilinears a - s : b - s : c - s

Trilinears csc A + cot A : :

Trilinears csc A (1 + cos A) : :

Trilinears tan A' : : , where A'B'C' = excentral triangle

Trilinears d(a,b,c) : : , where d(a,b,c) = distance from A to the Gergonne line

Trilinears square of semi-minor axis of A-Soddy ellipse : :

Barycentrics a(b + c - a) : b(c + a - b) : c(a + b - c)

Barycentrics 1 + cos A : 1 + cos B : 1 + cos C

Tripolars Sqrt[b c (a^4-2 a^2 b^2+b^4-4 b^3 c-2 a^2 c^2+6 b^2 c^2-4 b c^3+c^4)] : :

X(9) = (r + 2R)*X(1) - 6*X(2) -2r*X(3) = 3*X(2) - X(7)

X(9) is the symmedian point of the excentral triangle.

X(9) = perspector of ABC and the medial triangle of the extouch triangle of ABC. (Randy Hutson, 9/23/2011)

Let A' be the orthocorrespondent of the A-excenter, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(9). (Randy Hutson, November 18, 2015)

Let A'B'C' be the orthic triangle. Let La be the antiorthic axis of AB'C', and define Lb, Lc cyclically. Let A" = Lb∩Lc. B" = Lc∩La, C" = La∩Lb. Triangle A"B"C" is inversely similar to ABC, with similitude center X(9). (Randy Hutson, November 18, 2015)

Let E be the locus of the trilinear pole of a line that passes through X(1). The center of E is X(9). Moreover, E passes through the points X(100), X(658), X(662), X(799), X(1821), X(2580), X(2581) and the bicentric pairs PU(34), PU(75), PU(77), PU(79). Also, E is a circumellipse of ABC and an inellipse of the excentral triangle. (Randy Hutson, February 10, 2016)

The locus E also passes through the vertices of Gemini triangle 2. (Randy Hutson, November 30, 2018)

Let A' be the intersection of the tangents to the A-excircle at the intercepts with the circumcircle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(9). (Randy Hutson, December 2, 2017)

Let A' be the perspector of the A-mixtilinear excircle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(9). (Randy Hutson, December 2, 2017)

Let E be the circumellipse of T = ABC with center X(9). Then ABC is a billiard orbit of E (3-periodic). If we fix E in the plane, all its triangular orbits (a set of "rotating" triangles T) have the same X(9). Note that X(9) is the point of concurrence of lines drawn from each excenter to the midpoint of the corresponding side of T. (Dan Reznik, June 30, 2019) See [1] Triangular Orbits in Elliptic Billiards: the Mittenpunkt X(9) is stationary at the origin and [2] Triangular Orbits in Elliptic Billiards: the Mittenpunkt X(9) is stationary at the origin.

Notes from Peter Moses (July 1, 2019) about the circumellipse E with center X(9):

E passes through the vertices of these triangles: ABC; Honsberger (see X(7670); Inner Conway (see X(11677); Gemini 2, Gemini 30.

E passes through the point X(i) for these i:

88, 100, 162, 190, 651, 653, 655, 658, 660, 662, 673, 771, 799,

823, 897, 1156, 1492, 1821, 2349, 2580, 2581, 3257, 4598, 4599

4604, 4606, 4607, 8052, 20332, 23707, 24624, 27834, 32680.

The perspector of E is X(1); the major axis of E passes through X(i) for these i: 9, 2590, 3307, 24646, and the minor axis, for these i: 9, 2591, 3309, 24647. The ellipse E is the isogonal conjugate of the antiorthic axis.

Let g = length of semi-major axis of E; then g = Sqrt[(2 R (r + R + Sqrt[R(R - 2 r)]) s^2)/(r + 4 R)^2].

Let h = length of semi-minor axis of E: then h = Sqrt[(2 R (r + R - Sqrt[R(R - 2 r)]) s^2)/(r + 4 R)^2].

h/g = Sqrt[(OI - R) (OI + 3 R) / ((OI - 3 R) (OI + R))] = Sqrt[(r + R - Sqrt[R (R - 2 r)]) / (r + R + Sqrt[R (R - 2 r)])].

eccentricity of E: 2 Sqrt[OI R / ((3 R - OI) (OI + R))].

If F is a focus of E, then |FX(9)|^2 = 4 s^2 R Sqrt[R (R - 2 r)] / (r + 4 R)^2/.

The axes of E are the asymptotes of the Feuerbach hyperbola.

The area of E is π R S /(r (r + 4 R))^(3/2) = 4 π a b c / ( 2 b c + 2 c a + 2 a b - a^2 - b^2 - c^2 )^(3/2) * area(ABC).

The triangle tangent to the vertices of E is the excentral triangle.

Let A'B'C' be the intouch triangle of the anticomplementary triangle of ABC. The ellipse E passes through A', B', C'. See Three orbits in elliptic billiard. (Dan Reznick, July 1, 2019)

Each of the following cubics passes through the four foci (two real and two imaginary) of the ellipse E: K040, K351, K352, K710, K716, K1060. The two real foci are a pair of isogonal conjugates, and likewise for the two imaginary foci. Moreover, if p : q : r is on the circumcircle, then p/a : q/b : r/c is on E. (Peter Moses, July 2 and 3, 2019)

The ellipse E is the isogonal conjugate of the antiorthic axis, X(44)X(13), with barycentric equation x + y + z = 0, and E is the isotomic conjugate of the X(514)X(661), with barycentric equation a x + b y + c z = 0. This line, called the Gergonne line and denoted by L(55), is the perspectrix of the anticomplementary triangle and the inner Conway triangle (which is the intouch triangle of the anticomplementary triangle). Points lying on X(514)X(661) include X(i) for i = 514, 661, 693, 857, 908, 914, 1577, 1959, 2084, 2582, 2583, 3239, 3250, 3762, 3766, 3835, 3904, 3912, 3936, 3948, 4129, 4358, 4391, 4462, 4468, 4486, 4728, 4766, 4776, 4789, 4791, 4801, 4823, 4978, 5074, 5179, 6332, 6381, 6590, 8045, 14206, 14207, 14208, 14209, 14210, 14281, 14349, 14350, 14963, 18669, 18715, 24018, 30565, 30566, 30804, 30806, 32679. (Peter Moses, July 2 and 3, 2019)

If you have The Geometer's Sketchpad, you can view Mittenpunkt.

If you have GeoGebra, you can view **Mittenpunkt**.

X(9) lies on the these conics: Feuerbach circumhyperbola, Feuerbach circumhyperbola of the medial triangle, Jerabek circumhyperbola of the excentral triangle, Mandart hyperbola

X(9) lies on these curves: K002, K132, K202, K207, K220, K251, K294, K332, K343, K345, K351, K352, K363, K384, K387, K453, K637, K696, K697, K710, K716, K717, K760, K761, K817, K880, K950, K970, K977, K980, K982, K984, K1025, K1038, K1044, K1055, K1059, K1060, K1077, K1079, K1080, K1081, K1082, K1083, K1084, K1090, Q151

X(9) lies on these lines: {1, 6}, {2, 7}, {3, 84}, {4, 10}, {5, 1729}, {8, 346}, {11, 3254}, {12, 5857}, {20, 10429}, {21, 41}, {22, 5314}, {25, 5285}, {30, 3587}, {31, 612}, {32, 987}, {33, 212}, {34, 201}, {35, 90}, {36, 2178}, {38, 614}, {39, 978}, {42, 941}, {43, 256}, {46, 79}, {48, 101}, {51, 3690}, {55, 200}, {56, 1696}, {58, 975}, {65, 4047}, {69, 344}, {75, 190}, {77, 651}, {80, 528}, {81, 5287}, {85, 10509}, {86, 2279}, {87, 292}, {92, 6358}, {100, 1005}, {105, 4712}, {108, 3213}, {109, 2199}, {114, 24469}, {119, 2950}, {123, 20623}, {124, 5513}, {140, 5843}, {141, 4422}, {145, 4029}, {164, 168}, {165, 910}, {171, 1707}, {172, 5019}, {173, 177}, {182, 7193}, {184, 26885}, {192, 239}, {193, 3879}, {194, 16827}, {205, 2359}, {216, 13006}, {222, 17811}, {223, 1073}, {228, 1011}, {241, 269}, {244, 17125}, {255, 25063}, {257, 17743}, {258, 7028}, {259, 15997}, {261, 645}, {264, 1948}, {273, 26003}, {275, 26941}, {291, 17065}, {294, 1253}, {306, 5739}, {312, 314}, {318, 1896}, {319, 17233}, {320, 17234}, {321, 1751}, {326, 6518}, {335, 17000}, {341, 4095}, {342, 653}, {345, 2339}, {348, 738}, {350, 17026}, {354, 4423}, {355, 31789}, {362, 2090}, {363, 5934}, {364, 366}, {374, 517}, {375, 22276}, {377, 9579}, {379, 18655}, {381, 18482}, {386, 10467}, {388, 12527}, {393, 1785}, {394, 2003}, {404, 4652}, {418, 26901}, {427, 21015}, {440, 1211}, {443, 4292}, {474, 3916}, {478, 1038}, {484, 5036}, {497, 4847}, {498, 920}, {499, 25522}, {511, 3781}, {513, 3126}, {514, 23760}, {515, 3427}, {519, 1000}, {521, 4130}, {522, 657}, {524, 4851}, {536, 4361}, {545, 7263}, {551, 18490}, {581, 3682}, {583, 3338}, {584, 15175}, {595, 28594}, {597, 17045}, {599, 17231}, {604, 1420}, {607, 1039}, {608, 1041}, {609, 1333}, {631, 6700}, {644, 1320}, {646, 4110}, {649, 4521}, {650, 15737}, {652, 3239}, {654, 1639}, {660, 3252}, {664, 31169}, {668, 17786}, {726, 16825}, {750, 896}, {758, 2294}, {759, 29127}, {765, 5377}, {813, 2726}, {857, 17052}, {869, 2309}, {899, 4414}, {900, 22108}, {905, 20318}, {912, 3211}, {919, 2751}, {934, 2371}, {937, 2336}, {938, 5129}, {940, 4641}, {942, 3927}, {943, 1802}, {946, 5758}, {948, 3668}, {952, 7966}, {964, 2198}, {970, 15877}, {976, 5037}, {980, 27623}, {981, 1918}, {982, 3290}, {983, 1914}, {986, 1722}, {989, 5255}, {990, 13329}, {991, 1818}, {995, 19246}, {1012, 6282}, {1014, 24557}, {1015, 21826}, {1020, 6356}, {1021, 3700}, {1026, 9319}, {1030, 2932}, {1040, 5452}, {1046, 10381}, {1050, 2275}, {1054, 3038}, {1055, 18450}, {1071, 8726}, {1086, 4859}, {1088, 1223}, {1096, 7076}, {1098, 2150}, {1125, 1732}, {1150, 4358}, {1155, 4413}, {1158, 5811}, {1167, 7129}, {1174, 1621}, {1181, 2910}, {1193, 5105}, {1195, 5552}, {1200, 5281}, {1202, 10578}, {1210, 5084}, {1215, 24511}, {1220, 31359}, {1247, 9560}, {1249, 1712}, {1250, 7126}, {1251, 7127}, {1259, 11344}, {1266, 20073}, {1278, 16816}, {1282, 3041}, {1331, 2000}, {1332, 28978}, {1377, 1703}, {1378, 1702}, {1385, 20818}, {1389, 1953}, {1392, 4861}, {1404, 30318}, {1405, 2171}, {1418, 7271}, {1424, 25918}, {1435, 17917}, {1441, 25001}, {1465, 25939}, {1471, 4327}, {1473, 7484}, {1474, 30733}, {1475, 3616}, {1479, 1752}, {1486, 12329}, {1500, 4263}, {1571, 1574}, {1572, 1573}, {1575, 3551}, {1592, 16028}, {1593, 26935}, {1598, 26938}, {1615, 10178}, {1630, 6261}, {1633, 24309}, {1635, 13266}, {1654, 2893}, {1655, 17033}, {1678, 1705}, {1679, 1704}, {1680, 1701}, {1681, 1700}, {1695, 9565}, {1699, 2886}, {1720, 20224}, {1721, 9441}, {1730, 6708}, {1737, 24005}, {1738, 24248}, {1740, 2235}, {1742, 6184}, {1745, 3330}, {1755, 7413}, {1760, 5224}, {1764, 5737}, {1768, 3035}, {1776, 5218}, {1783, 2331}, {1788, 8582}, {1793, 2341}, {1824, 11323}, {1827, 28044}, {1836, 3925}, {1837, 6598}, {1848, 21062}, {1863, 28120}, {1868, 4185}, {1921, 3403}, {1936, 9817}, {1937, 1945}, {1947, 15466}, {1958, 21511}, {1959, 15988}, {1966, 6376}, {1974, 26924}, {1992, 29574}, {2006, 26611}, {2013, 2018}, {2014, 2017}, {2049, 19859}, {2066, 7133}, {2078, 17615}, {2093, 3753}, {2099, 31165}, {2108, 20603}, {2112, 4579}, {2124, 2125}, {2137, 24150}, {2173, 3647}, {2174, 2278}, {2175, 2330}, {2177, 21805}, {2209, 3728}, {2214, 28615}, {2220, 5301}, {2223, 16688}, {2225, 29828}, {2242, 5042}, {2252, 5553}, {2272, 5658}, {2293, 2340}, {2295, 4274}, {2305, 5277}, {2308, 5311}, {2312, 4220}, {2317, 22356}, {2318, 2335}, {2319, 7081}, {2320, 2364}, {2326, 11107}, {2432, 14298}, {2478, 6734}, {2503, 21381}, {2509, 21189}, {2568, 2573}, {2569, 2572}, {2590, 3307}, {2591, 3308}, {2629, 2634}, {2640, 2645}, {2648, 17963}, {2802, 4752}, {2887, 4703}, {2895, 32858}, {2947, 2954}, {2957, 24250}, {2959, 20666}, {2999, 3666}, {3008, 3663}, {3009, 7032}, {3037, 5539}, {3056, 3271}, {3057, 3680}, {3058, 4863}, {3060, 26911}, {3063, 24307}, {3068, 5393}, {3069, 5405}, {3083, 15890}, {3084, 15889}, {3085, 21075}, {3175, 19723}, {3177, 9312}, {3182, 18641}, {3185, 10434}, {3187, 3995}, {3189, 4314}, {3197, 6001}, {3207, 7987}, {3212, 27288}, {3216, 4261}, {3244, 4098}, {3255, 5432}, {3262, 28974}, {3287, 3709}, {3293, 4277}, {3295, 3991}, {3336, 5356}, {3337, 5043}, {3339, 3812}, {3341, 3344}, {3343, 3352}, {3349, 3351}, {3361, 5022}, {3416, 3932}, {3421, 31397}, {3434, 9580}, {3436, 9578}, {3467, 7301}, {3474, 26040}, {3486, 6737}, {3503, 25994}, {3525, 26877}, {3550, 4386}, {3560, 31837}, {3567, 26915}, {3579, 9709}, {3584, 17699}, {3588, 31330}, {3589, 4364}, {3596, 3975}, {3617, 5175}, {3618, 17023}, {3619, 29596}, {3620, 29579}, {3621, 12630}, {3622, 17474}, {3623, 4982}, {3625, 4072}, {3626, 4058}, {3629, 17390}, {3632, 3943}, {3633, 4898}, {3634, 5714}, {3644, 17160}, {3648, 10123}, {3652, 16005}, {3664, 4644}, {3671, 28629}, {3672, 3946}, {3673, 17681}, {3675, 20275}, {3676, 25924}, {3695, 5814}, {3696, 5695}, {3697, 5687}, {3703, 3966}, {3706, 4042}, {3708, 25095}, {3712, 4023}, {3720, 32912}, {3726, 29820}, {3732, 21232}, {3735, 9620}, {3738, 14427}, {3739, 4363}, {3741, 4011}, {3742, 8167}, {3746, 4006}, {3752, 23511}, {3757, 21101}, {3759, 4360}, {3760, 29433}, {3761, 3770}, {3763, 17237}, {3764, 21035}, {3765, 3963}, {3772, 4415}, {3779, 20683}, {3782, 23681}, {3784, 3819}, {3814, 5535}, {3817, 8166}, {3834, 7232}, {3836, 4655}, {3840, 20785}, {3842, 4672}, {3846, 4438}, {3868, 3951}, {3871, 32635}, {3873, 4666}, {3874, 20116}, {3880, 4900}, {3888, 25279}, {3893, 31509}, {3897, 17440}, {3899, 17443}, {3900, 23351}, {3913, 4515}, {3917, 26892}, {3920, 17127}, {3937, 5650}, {3940, 16418}, {3941, 20990}, {3942, 25097}, {3944, 17064}, {3945, 4667}, {3952, 26227}, {3955, 9306}, {3957, 4661}, {3971, 4362}, {3974, 4082}, {3977, 17740}, {3984, 16865}, {3989, 17017}, {3997, 30116}, {3998, 16368}, {4015, 8715}, {4020, 25079}, {4030, 4126}, {4033, 29712}, {4063, 6008}, {4067, 12559}, {4070, 5233}, {4071, 4388}, {4077, 26017}, {4078, 5847}, {4086, 4529}, {4090, 29670}, {4092, 23902}, {4111, 4433}, {4119, 4514}, {4123, 9447}, {4124, 30224}, {4148, 4768}, {4153, 30172}, {4154, 15628}, {4180, 4182}, {4189, 4855}, {4191, 22060}, {4255, 8951}, {4260, 16850}, {4268, 7113}, {4269, 25516}, {4272, 5312}, {4295, 19855}, {4297, 10864}, {4301, 6766}, {4304, 11111}, {4310, 16020}, {4313, 11106}, {4328, 5228}, {4329, 5813}, {4333, 7700}, {4336, 28125}, {4346, 17067}, {4359, 25734}, {4389, 16706}, {4392, 7292}, {4393, 4704}, {4402, 4452}, {4417, 33116}, {4418, 26037}, {4421, 31508}, {4425, 25453}, {4429, 24723}, {4430, 29817}, {4432, 32941}, {4435, 4526}, {4440, 29628}, {4441, 24592}, {4445, 4690}, {4454, 4488}, {4461, 32087}, {4466, 31261}, {4480, 24199}, {4534, 12641}, {4552, 25243}, {4554, 30988}, {4557, 8053}, {4559, 24806}, {4568, 30108}, {4650, 16570}, {4651, 32929}, {4665, 28634}, {4668, 7300}, {4670, 4698}, {4675, 4888}, {4676, 5263}, {4677, 4908}, {4681, 4852}, {4683, 25957}, {4686, 17119}, {4688, 17118}, {4699, 16815}, {4708, 17327}, {4711, 8168}, {4713, 21264}, {4715, 17313}, {4731, 5183}, {4741, 17232}, {4748, 29604}, {4755, 28639}, {4798, 6707}, {4869, 21296}, {4872, 24694}, {4929, 9053}, {4967, 26998}, {4974, 32921}, {4981, 24552}, {4997, 30608}, {5046, 21029}, {5057, 30311}, {5082, 10624}, {5087, 5536}, {5088, 27472}, {5110, 5529}, {5122, 16417}, {5124, 7280}, {5128, 5177}, {5153, 5313}, {5217, 7285}, {5232, 7291}, {5256, 16579}, {5266, 30435}, {5267, 22054}, {5274, 24386}, {5286, 13161}, {5290, 25466}, {5297, 9330}, {5307, 22001}, {5423, 7172}, {5424, 5426}, {5433, 24954}, {5439, 16842}, {5440, 16370}, {5530, 31402}, {5534, 10267}, {5550, 30340}, {5551, 19862}, {5555, 24982}, {5560, 7297}, {5575, 25891}, {5584, 12565}, {5586, 28646}, {5651, 26884}, {5691, 5794}, {5693, 19350}, {5703, 17558}, {5708, 16853}, {5726, 11236}, {5736, 28627}, {5741, 33113}, {5743, 19542}, {5790, 18499}, {5836, 7991}, {5854, 8275}, {5881, 6936}, {5886, 20330}, {5887, 7971}, {5903, 17098}, {6009, 21385}, {6048, 21857}, {6056, 11429}, {6181, 17601}, {6245, 6865}, {6350, 30675}, {6377, 16576}, {6505, 16585}, {6506, 8068}, {6536, 29647}, {6542, 17242}, {6586, 21173}, {6626, 18784}, {6675, 11374}, {6687, 17235}, {6690, 8255}, {6701, 13159}, {6705, 6926}, {6706, 30494}, {6726, 7014}, {6735, 30513}, {6769, 11496}, {6832, 8227}, {6843, 10175}, {6857, 13411}, {6905, 21165}, {6910, 27385}, {6939, 7682}, {6976, 12703}, {6986, 10884}, {6990, 24045}, {6996, 10444}, {7003, 7008}, {7004, 25096}, {7066, 19366}, {7098, 10588}, {7146, 18726}, {7176, 27340}, {7177, 10004}, {7183, 17095}, {7190, 24554}, {7191, 7226}, {7222, 31211}, {7229, 24590}, {7244, 18068}, {7273, 8898}, {7277, 17392}, {7283, 9534}, {7293, 7485}, {7377, 24702}, {7595, 8231}, {7670, 11691}, {7673, 14923}, {7678, 11680}, {7679, 11681}, {7736, 24239}, {7957, 12651}, {7963, 8572}, {7992, 9943}, {8056, 16602}, {8069, 8573}, {8125, 8388}, {8126, 8389}, {8169, 9814}, {8237, 11687}, {8238, 11688}, {8245, 8424}, {8273, 12680}, {8385, 11685}, {8386, 11686}, {8387, 11690}, {8632, 14408}, {8666, 15179}, {8680, 18698}, {8730, 15348}, {8750, 23050}, {8771, 21508}, {8822, 16054}, {8835, 15856}, {8941, 31459}, {9028, 26130}, {9342, 9352}, {9365, 14936}, {9367, 11512}, {9470, 9499}, {9576, 9640}, {9577, 9639}, {9582, 9679}, {9583, 9678}, {9584, 9689}, {9585, 9688}, {9586, 9702}, {9587, 9701}, {9588, 9711}, {9589, 9710}, {9590, 9713}, {9591, 9712}, {9592, 31449}, {9599, 29676}, {9614, 24390}, {9616, 30354}, {9619, 31456}, {9624, 31458}, {9785, 21627}, {9843, 17559}, {9846, 12125}, {9955, 31493}, {9957, 12629}, {10012, 31269}, {10039, 21074}, {10050, 10058}, {10157, 19541}, {10167, 10857}, {10198, 21077}, {10246, 22147}, {10266, 12639}, {10268, 11500}, {10383, 10391}, {10387, 19589}, {10388, 17658}, {10446, 24705}, {10449, 21071}, {10455, 27164}, {10461, 11110}, {10476, 15825}, {10638, 19551}, {10645, 11791}, {10646, 11790}, {10855, 16411}, {10856, 15509}, {10865, 11678}, {10882, 23361}, {10902, 17857}, {11008, 29601}, {11019, 24477}, {11036, 17554}, {11194, 13462}, {11248, 11434}, {11343, 25083}, {11375, 24953}, {11433, 26872}, {11519, 30337}, {11520, 16859}, {11526, 11682}, {11604, 21014}, {11683, 16609}, {11684, 16133}, {12047, 19854}, {12389, 12399}, {12396, 12397}, {12435, 22299}, {12436, 17582}, {12511, 31871}, {12519, 13089}, {12520, 31803}, {12529, 12706}, {12530, 12718}, {12531, 12730}, {12532, 12755}, {12533, 12846}, {12534, 12847}, {12535, 12850}, {12575, 15998}, {12650, 31786}, {12659, 12693}, {12675, 22153}, {12699, 31419}, {12782, 24478}, {13143, 13144}, {13388, 15891}, {13389, 15892}, {13405, 21060}, {13426, 13427}, {13442, 29181}, {13454, 13456}, {13567, 26942}, {14021, 18650}, {14151, 17439}, {14224, 14400}, {14319, 14321}, {14497, 16200}, {14543, 27039}, {14552, 34255}, {14621, 31323}, {14829, 18743}, {14963, 22073}, {14996, 17021}, {14997, 17012}, {15066, 22128}, {15487, 26034}, {15496, 22080}, {15507, 31394}, {15669, 24315}, {15934, 16857}, {16058, 20760}, {16286, 22458}, {16367, 20769}, {16374, 23206}, {16555, 21366}, {16565, 27688}, {16568, 31144}, {16575, 16592}, {16605, 24440}, {16608, 25964}, {16610, 17595}, {16704, 31035}, {16713, 17183}, {16726, 18186}, {16732, 17885}, {16738, 27261}, {16822, 17760}, {16823, 20459}, {16826, 17120}, {16887, 30110}, {17002, 26247}, {17046, 17671}, {17050, 17753}, {17063, 18193}, {17107, 24797}, {17137, 29966}, {17152, 30036}, {17156, 32864}, {17175, 25508}, {17227, 17273}, {17228, 17271}, {17230, 17268}, {17238, 17252}, {17239, 17251}, {17240, 17295}, {17241, 17297}, {17244, 17300}, {17246, 17301}, {17247, 17302}, {17249, 17305}, {17250, 17307}, {17309, 17372}, {17310, 17373}, {17311, 17374}, {17312, 17375}, {17315, 17377}, {17317, 17378}, {17320, 17380}, {17322, 17381}, {17323, 17382}, {17324, 17383}, {17325, 17384}, {17391, 20090}, {17394, 29597}, {17444, 21398}, {17499, 26110}, {17542, 24473}, {17550, 24995}, {17605, 31245}, {17614, 20991}, {17625, 25893}, {17687, 25500}, {17691, 25242}, {17728, 31249}, {17737, 24892}, {17757, 31434}, {17792, 18788}, {17793, 30546}, {17889, 33099}, {18040, 29396}, {18044, 18133}, {18046, 29561}, {18134, 33066}, {18139, 32859}, {18398, 25542}, {18483, 31418}, {18589, 24316}, {18758, 23863}, {19261, 23169}, {19555, 31090}, {19582, 23640}, {19604, 25731}, {19785, 26723}, {19804, 32939}, {19872, 32632}, {20080, 29583}, {20092, 24184}, {20106, 26934}, {20174, 29773}, {20205, 21370}, {20247, 25261}, {20248, 26653}, {20257, 27304}, {20456, 22172}, {20486, 25613}, {20544, 24329}, {20608, 21250}, {20662, 21320}, {20667, 26069}, {20678, 23868}, {20719, 31785}, {20930, 28980}, {20973, 21858}, {20979, 21389}, {20980, 21348}, {20984, 22174}, {21026, 31134}, {21066, 26793}, {21090, 24298}, {21281, 30030}, {21286, 26581}, {21367, 32782}, {21368, 24611}, {21405, 27390}, {21759, 23566}, {21827, 23543}, {21832, 24121}, {21956, 32865}, {22003, 24435}, {23062, 23618}, {23073, 30389}, {23354, 25292}, {23529, 28118}, {23649, 28352}, {24067, 27368}, {24154, 24156}, {24155, 24157}, {24158, 24242}, {24172, 28090}, {24210, 33137}, {24268, 25252}, {24325, 32935}, {24343, 25107}, {24398, 27918}, {24512, 26102}, {24542, 33122}, {24547, 24612}, {24586, 30758}, {24603, 28827}, {24633, 24993}, {24690, 30822}, {24935, 25669}, {25010, 30312}, {25057, 31171}, {25343, 30779}, {25457, 25682}, {25570, 25571}, {25660, 29456}, {25690, 25693}, {25842, 25856}, {25850, 25858}, {25958, 29873}, {25960, 33119}, {25961, 33067}, {26006, 26668}, {26035, 31339}, {26068, 27020}, {26107, 26959}, {26724, 33146}, {26730, 26731}, {26756, 27073}, {26919, 26940}, {26933, 30739}, {27036, 27102}, {27044, 27136}, {27108, 27514}, {27507, 28789}, {28604, 29576}, {29381, 29536}, {29388, 29504}, {29395, 29423}, {29431, 29514}, {29570, 31313}, {29616, 32099}, {29632, 33065}, {29642, 33064}, {29643, 32843}, {29653, 32946}, {29664, 33107}, {29667, 33166}, {29674, 33082}, {29679, 33083}, {29681, 33153}, {29687, 33080}, {29711, 29716}, {29767, 30939}, {29826, 32944}, {29850, 32776}, {29851, 33069}, {29854, 32949}, {29855, 32775}, {30416, 30429}, {30417, 30430}, {30676, 30701}, {30695, 31994}, {31316, 31343}, {31408, 31533}, {31547, 31565}, {31548, 31566}, {32771, 32938}, {32773, 33118}, {32778, 33164}, {32849, 33077}, {32860, 32936}, {32861, 33092}, {32862, 33075}, {32914, 32925}, {32947, 33117}, {33084, 33158}, {33096, 33111}, {33100, 33131}, {33101, 33130}, {33129, 33151}, {33132, 33154}, {33134, 33139}

X(9) = midpoint of X(i) and X(j) for these (i,j): midpoint of X(i) and X(j) for these {i,j}: {1, 5223}, {2, 6172}, {3, 5779}, {4, 5759}, {7, 144}, {8, 390}, {11, 6068}, {40, 11372}, {63, 8545}, {72, 5728}, {100, 1156}, {190, 673}, {329, 12848}, {346, 5838}, {1001, 5220}, {2262, 21871}, {2294, 3958}, {2550, 5698}, {2951, 3062}, {3059, 14100}, {3587, 18540}, {3621, 12630}, {3681, 7671}, {3869, 7672}, {4361, 17262}, {4915, 9819}, {5817, 21168}, {5839, 17314}, {7670, 11691}, {7673, 14923}, {9846, 12125}, {11495, 16112}, {11684, 16133}, {12389, 12399}, {12526, 12560}, {12527, 12573}, {12528, 12669}, {12529, 12706}, {12530, 12718}, {12531, 12730}, {12532, 12755}, {12533, 12846}, {12534, 12847}, {12535, 12850}, {13359, 13360}, {15254, 15481}, {31547, 31565}, {31548, 31566}

X(9) = reflection of X(i) in X(j) for these {i,j}: {1, 1001}, {3, 31658}, {7, 142}, {8, 24393}, {57, 8257}, {84, 3358}, {100, 6594}, {101, 28345}, {142, 6666}, {1001, 15254}, {2294, 25081}, {2550, 10}, {2951, 11495}, {3062, 16112}, {3174, 6600}, {3243, 1}, {3254, 11}, {3874, 20116}, {4312, 5880}, {4361, 17348}, {4851, 17243}, {5220, 15481}, {5223, 5220}, {5528, 100}, {5542, 1125}, {5732, 3}, {5735, 5805}, {5805, 5}, {5833, 5791}, {5880, 3826}, {6173, 2}, {6601, 24389}, {8255, 6690}, {9623, 9708}, {10427, 3035}, {13159, 6701}, {15185, 5572}, {15298, 15296}, {15299, 15297}, {16593, 4422}, {17151, 4361}, {17314, 3950}, {17668, 15587}, {18443, 6883}, {20195, 18230}, {31657, 140}, {31671, 18482}

X(9) = isogonal conjugate of X(57)

X(9) = isotomic conjugate of X(85)

X(9) = complement of X(7)

X(9) = anticomplement of X(142)

X(9) = polar conjugate of X(273)

X(9) = antigonal image of X(3254)

X(9) = antitomic image of X(14943)

X(9) = symgonal image of X(6594)

X(9) = circumcircle-inverse of X(32625)

X(9) = Spieker-circle-inverse of X(5199)

X(9) = Bevan-circle-inverse of X(5011)

X(9) = Stevanovic-circle inverse of X(15737)

X(9) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 3158}, {2, 1}, {4, 2900}, {6, 3169}, {7, 3174}, {8, 200}, {21, 55}, {27, 3189}, {29, 3190}, {31, 32468}, {41, 7075}, {55, 3208}, {57, 2136}, {63, 40}, {78, 7070}, {81, 3913}, {85, 3870}, {88, 3880}, {92, 3811}, {100, 3900}, {105, 19589}, {144, 2951}, {189, 6765}, {190, 522}, {241, 9451}, {257, 3961}, {294, 3684}, {312, 78}, {318, 33}, {329, 1490}, {333, 8}, {346, 2324}, {348, 8270}, {527, 5528}, {643, 4041}, {644, 650}, {645, 3737}, {650, 4919}, {651, 521}, {653, 8058}, {655, 2804}, {660, 926}, {664, 4105}, {672, 24578}, {673, 5853}, {765, 3939}, {799, 3907}, {894, 1045}, {897, 24394}, {908, 6326}, {1121, 3935}, {1156, 15733}, {1220, 42}, {1223, 142}, {1252, 1018}, {1261, 4513}, {1320, 3689}, {1751, 12625}, {1791, 5285}, {1821, 740}, {2053, 2319}, {2167, 8715}, {2184, 11523}, {2185, 3871}, {2287, 219}, {2297, 1449}, {2319, 4050}, {2322, 281}, {2339, 1697}, {2346, 6600}, {2349, 758}, {2975, 15621}, {3218, 5541}, {3219, 191}, {3257, 3738}, {3305, 3646}, {3596, 4149}, {3699, 663}, {3903, 4477}, {4076, 4069}, {4102, 4420}, {4564, 100}, {4997, 4511}, {5279, 18598}, {5546, 1021}, {6558, 4521}, {6605, 220}, {7123, 3501}, {7131, 57}, {8056, 3680}, {9776, 12658}, {10509, 7674}, {14942, 2340}, {15889, 30557}, {15890, 30556}, {17484, 13146}, {17743, 43}, {21446, 3243}, {23617, 6}, {23618, 7}, {27065, 5506}, {27834, 513}, {28659, 4123}, {30608, 3872}, {30705, 8271}, {30711, 4882}, {31343, 4162}, {31359, 612}, {32008, 2}, {32015, 3957}, {32635, 210}

X(9) = cevapoint of X(i) and X(j) for these (i,j): (1,165), (6,198), (31,205), (37,71), (41,212), (55,220). (2066.5414)

X(9) = X(i)-cross conjugate of X(j) for these (i,j): {1, 19605}, {6, 282}, {8, 2319}, {37, 281}, {41, 33}, {42, 10570}, {55, 1}, {57, 2125}, {71, 219}, {198, 2324}, {200, 3680}, {210, 8}, {212, 78}, {220, 200}, {518, 14943}, {650, 644}, {652, 101}, {654, 5548}, {657, 3939}, {663, 3699}, {672, 2338}, {1146, 1021}, {1212, 2}, {1334, 55}, {1400, 30457}, {1864, 4}, {1903, 8805}, {2082, 57}, {2170, 650}, {2183, 15629}, {2238, 15628}, {2245, 15627}, {2264, 1172}, {2269, 284}, {2310, 522}, {2340, 14942}, {2347, 6}, {2348, 294}, {3059, 6601}, {3119, 3900}, {3271, 3737}, {3287, 645}, {3683, 21}, {3689, 1320}, {3691, 333}, {3693, 4876}, {3700, 1018}, {3709, 4069}, {3711, 4900}, {3715, 4866}, {3900, 100}, {4041, 643}, {4105, 664}, {4162, 31343}, {4266, 2364}, {4326, 10390}, {4477, 3903}, {4517, 7220}, {4936, 3158}, {7008, 3347}, {7069, 318}, {7082, 90}, {7156, 7070}, {8012, 220}, {9404, 5546}, {10382, 5665}, {11429, 3469}, {11436, 3362}, {14100, 7}, {14298, 1783}, {14547, 29}, {15733, 3254}, {15837, 2346}, {18235, 2329}, {20665, 9439}, {21033, 2321}, {21811, 37}, {23544, 893}, {27538, 3208}, {28070, 728}, {30223, 84}, {30456, 27382}, {33299, 312}

X(9) = crosspoint of X(i) and X(j) for these (i,j): {1, 8056}, {2, 8}, {21, 333}, {55, 2053}, {57, 2137}, {63, 271}, {100, 4564}, {188, 7028}, {190, 765}, {236, 24158}, {258, 24242}, {312, 318}, {645, 4076}, {651, 7012}, {1016, 8706}, {1252, 5546}, {1275, 6606}, {2287, 2322}, {3161, 24150}, {6605, 32008}, {24154, 24155}

X(9) = crosssum of X(i) and X(j) for these (i,j): {1, 1743}, {2, 3210}, {3, 3211}, {6, 56}, {7, 3212}, {9, 2136}, {19, 208}, {25, 21058}, {34, 3213}, {37, 3214}, {41, 21059}, {44, 17460}, {48, 3215}, {63, 8897}, {65, 1400}, {173, 8078}, {244, 649}, {269, 17106}, {294, 9453}, {513, 2170}, {514, 24237}, {518, 19593}, {603, 604}, {650, 7004}, {661, 2611}, {798, 4128}, {1015, 6363}, {1086, 7178}, {1357, 7180}, {1407, 6611}, {1418, 1475}, {1575, 20366}, {2082, 28017}, {2446, 2590}, {2447, 2591}, {2488, 14936}, {4000, 28110}, {4466, 21124}, {10490, 18888}

X(9) = crossdifference of every pair of points on line X(513)X(663)

X(9) = X(i)-Hirst inverse of X(j) for these (i,j): {1, 518}, {2, 10025}, {8, 3685}, {43, 8844}, {55, 3684}, {57, 6168}, {192, 239}, {200, 3693}, {294, 28071}, {1282, 8299}, {1575, 16557}, {2170, 4919}, {2348, 3158}, {3307, 24646}, {3308, 24647}, {4876, 7077}, {5239, 5240}, {17792, 18788}

X(9) = X(i)-aleph conjugate of X(j) for these (i,j): {1, 43}, {2, 9}, {4, 1711}, {7, 16572}, {8, 10860}, {9, 170}, {75, 1759}, {76, 21366}, {99, 21383}, {174, 1743}, {188, 165}, {190, 1018}, {259, 32462}, {291, 18787}, {365, 1740}, {366, 1}, {507, 361}, {508, 57}, {509, 978}, {522, 2958}, {556, 1766}, {4146, 169}, {4182, 2951}, {5374, 1745}, {6728, 1053}, {7025, 503}, {14087, 1018}, {14089, 21383}, {18297, 63}, {20034, 1716}

X(9) = X(i)-beth conjugate of X(j) for these (i,j):

(9,6), (190,6), (346,346), (644,9), (645,75)

X(9) = X(1)-line conjugate of X(1279)

X(9) = X(i)-vertex conjugate of X(j) for these (i,j): {1, 3420}, {9, 1436}, {269, 1037}, {3900, 32625}

X(9) = perspector of ABC and extraversion triangle of X(57)

X(9) = X(6)-of-excentral-triangle

X(9) = X(159)-of-intouch-triangle

X(9) = X(6)-of-2nd-extouch-triangle

X(9) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 2886}, {2, 17046}, {6, 142}, {8, 2887}, {9, 141}, {19, 16608}, {21, 3741}, {25, 1210}, {31, 1}, {32, 3752}, {33, 5}, {37, 17052}, {40, 20307}, {41, 2}, {42, 442}, {43, 20338}, {48, 17073}, {55, 10}, {56, 11019}, {57, 21258}, {58, 3742}, {63, 18639}, {71, 18642}, {75, 17047}, {78, 1368}, {81, 17050}, {82, 17049}, {100, 17072}, {101, 4885}, {109, 3900}, {163, 17069}, {184, 17102}, {192, 20547}, {198, 20206}, {200, 1329}, {210, 3454}, {212, 3}, {213, 17056}, {219, 18589}, {220, 3452}, {228, 18641}, {251, 17048}, {281, 20305}, {282, 21239}, {284, 3739}, {294, 20335}, {312, 626}, {318, 21243}, {333, 21240}, {346, 21244}, {512, 8286}, {513, 17059}, {522, 21252}, {560, 17053}, {604, 4000}, {607, 226}, {643, 512}, {644, 3835}, {646, 21262}, {649, 4904}, {650, 116}, {657, 26932}, {662, 17066}, {663, 11}, {667, 3756}, {669, 16613}, {672, 17060}, {692, 522}, {798, 17058}, {893, 17062}, {902, 1145}, {904, 24239}, {923, 17070}, {983, 17792}, {1106, 5573}, {1110, 3035}, {1174, 6706}, {1252, 21232}, {1253, 9}, {1320, 21241}, {1333, 3946}, {1334, 1211}, {1395, 17054}, {1400, 18635}, {1402, 1834}, {1409, 18643}, {1415, 7658}, {1820, 18638}, {1911, 1738}, {1918, 2092}, {1946, 2968}, {1964, 17055}, {1973, 3772}, {1974, 20227}, {1980, 16614}, {2053, 3840}, {2148, 17043}, {2149, 17044}, {2150, 17045}, {2155, 18634}, {2156, 18636}, {2157, 18637}, {2158, 18640}, {2159, 18644}, {2162, 20257}, {2163, 17051}, {2172, 17068}, {2175, 37}, {2176, 20528}, {2177, 17057}, {2187, 7952}, {2192, 946}, {2194, 1125}, {2195, 518}, {2200, 18592}, {2208, 3086}, {2212, 6}, {2258, 25466}, {2268, 10472}, {2287, 21246}, {2289, 6389}, {2299, 942}, {2316, 3834}, {2318, 21530}, {2319, 20255}, {2320, 21242}, {2321, 21245}, {2328, 960}, {2332, 6708}, {2340, 120}, {2342, 517}, {2344, 21264}, {2361, 214}, {3052, 12640}, {3063, 1086}, {3158, 2885}, {3195, 20264}, {3208, 21250}, {3433, 24388}, {3445, 24386}, {3596, 21235}, {3684, 20333}, {3685, 20542}, {3688, 21249}, {3689, 121}, {3693, 20540}, {3699, 21260}, {3700, 21253}, {3709, 8287}, {3711, 21251}, {3724, 6739}, {3900, 124}, {3939, 513}, {4041, 125}, {4069, 31946}, {4105, 5514}, {4162, 5510}, {4166, 20334}, {4182, 20543}, {4548, 16582}, {4814, 15614}, {4845, 5087}, {4876, 20541}, {4895, 3259}, {5546, 4369}, {5547, 4892}, {5548, 4928}, {6059, 24005}, {6065, 24003}, {6066, 24036}, {6187, 1737}, {6602, 6554}, {6603, 31844}, {6614, 17427}, {7037, 6245}, {7054, 21233}, {7069, 1209}, {7070, 2883}, {7071, 20262}, {7072, 21616}, {7073, 25639}, {7074, 6260}, {7077, 3836}, {7084, 1376}, {7104, 28358}, {7110, 21236}, {7118, 57}, {7121, 17063}, {7139, 20268}, {7156, 20207}, {7252, 17761}, {7257, 23301}, {7339, 24009}, {7367, 20205}, {8611, 127}, {8641, 1146}, {8750, 521}, {8851, 20340}, {9439, 3816}, {9447, 39}, {9448, 16584}, {9456, 17067}, {10482, 3740}, {13455, 639}, {14827, 1212}, {14942, 20544}, {15374, 21629}, {18265, 1575}, {18757, 33135}, {18889, 527}, {19624, 6594}, {21059, 6600}, {23990, 16578}, {32652, 8058}, {32666, 676}, {32739, 905}, {33299, 21248}

X(9) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 2890}, {1170, 3434}, {1174, 8}, {2346, 69}, {6605, 3436}, {10482, 329}, {21453, 21285}, {31618, 21280}, {32008, 6327}

X(9) = Thomson-isogonal conjugate of X(3576)

X(9) = medial-isogonal conjugate of X(2886)

X(9) = anticomplementary-isogonal conjugate of X(2890)

X(9) = excentral-isogonal conjugate of X(165)

X(9) = tangential-isogonal conjugate of X(2921)

X(9) = orthic-isogonal conjugate of X(2900)

X(9) = trilinear square root of X(200)

X(9) = trilinear product of extraversions of X(57)

X(9) = trilinear product of PU(112)

X(9) = inverse-in-circumconic-centered-at-X(1) of X(6603)

X(9) = orthocenter of X(1)X(4)X(885)

X(9) = bicentric sum of PU(56)

X(9) = midpoint of PU(56)

X(9) = crossdifference of PU(96)

X(9) = perspector of circumconic centered at X(1)

X(9) = the point in which the extended legs P(6)P(33) and U(6)U(33) of the trapezoid PU(6)PU(33) meet

X(9) = trilinear pole of line X(650)X(663)

X(9) = pole wrt polar circle of trilinear polar of X(273) (line X(514)X(3064))

X(9) = X(48)-isoconjugate (polar conjugate) of X(273)

X(9) = perspector of ABC and unary cofactor triangle of 1st mixtilinear triangle

X(9) = perspector of ABC and unary cofactor triangle of 3rd mixtilinear triangle

X(9) = homothetic center of excentral triangle and 2nd extouch triangle

X(9) = perspector of ABC and medial triangle of extouch triangle

X(9) = perspector of pedal and anticevian triangles of X(1490)

X(9) = SS(A->A') of X(19), where A'B'C' is the excentral triangle

X(9) = homothetic center of medial triangle and tangential triangle of excentral triangle

X(9) = homothetic center of excentral triangle and complement of the intouch triangle

X(9) = Cundy-Parry Phi transform of X(84)

X(9) = Cundy-Parry Psi transform of X(40)

X(9) = trilinear product of circumcircle intercepts of excircles radical circle

X(9) = perspector of Gemini triangle 4 and unary cofactor triangle of Gemini triangle 3

X(9) = eigencenter of Gemini triangle 5

X(9) = perspector of Gemini triangle 35 and cross-triangle of ABC and Gemini triangle 35

X(9) = perspector of ABC and unary cofactor triangle of Gemini triangle 35

X(9) = perspector of ABC and unary cofactor triangle of Gemini triangle 40

X(9) = internal center of similitude of the Bevan circle and Spieker circle; the external center is X(1706)

X(9) = excentral-to-ABC barycentric image of X(9)

X(9) = intouch-to-ABC barycentric image of X(7)

X(9) = intouch-to-excentral similarity image of X(7)

X(9) = ABC-to-excentral barycentric image of X(9)

X(9) = barycentric product X(i)*X(j) for these {i,j}: {1, 8}, {3, 318}, {4, 78}, {6, 312}, {7, 200}, {10, 21}, {11, 765}, {12, 1098}, {19, 345}, {25, 3718}, {27, 3694}, {28, 3710}, {29, 72}, {31, 3596}, {32, 28659}, {33, 69}, {34, 1265}, {37, 333}, {40, 280}, {41, 76}, {42, 314}, {43, 7155}, {44, 4997}, {45, 30608}, {48, 7017}, {55, 75}, {56, 341}, {57, 346}, {58, 3701}, {59, 24026}, {60, 1089}, {63, 281}, {65, 1043}, {66, 4123}, {71, 31623}, {77, 7046}, {79, 4420}, {80, 4511}, {81, 2321}, {82, 3703}, {83, 33299}, {84, 7080}, {85, 220}, {86, 210}, {87, 27538}, {88, 2325}, {89, 4873}, {90, 5552}, {92, 219}, {95, 7069}, {99, 4041}, {100, 522}, {101, 4391}, {104, 6735}, {105, 3717}, {106, 4723}, {109, 4397}, {110, 4086}, {142, 6605}, {144, 19605}, {145, 3680}, {158, 1259}, {171, 4451}, {174, 6731}, {188, 188}, {189, 2324}, {190, 650}, {192, 2319}, {212, 264}, {213, 28660}, {222, 7101}, {225, 1792}, {226, 2287}, {236, 7028}, {238, 4518}, {239, 4876}, {241, 6559}, {244, 4076}, {253, 7070}, {256, 7081}, {257, 2329}, {259, 556}, {261, 756}, {266, 7027}, {269, 5423}, {270, 3695}, {271, 7952}, {273, 1260}, {274, 1334}, {278, 3692}, {279, 728}, {282, 329}, {284, 321}, {285, 21075}, {286, 2318}, {291, 3685}, {292, 3975}, {294, 3912}, {304, 607}, {305, 2212}, {306, 1172}, {307, 4183}, {309, 7074}, {313, 2194}, {319, 7073}, {322, 2192}, {326, 1857}, {330, 3208}, {331, 1802}, {332, 1824}, {335, 3684}, {348, 7079}, {350, 7077}, {366, 4182}, {391, 25430}, {393, 3719}, {480, 1088}, {483, 3082}, {492, 13455}, {512, 7257}, {513, 3699}, {514, 644}, {518, 14942}, {519, 1320}, {521, 1897}, {523, 643}, {561, 2175}, {594, 2185}, {596, 3871}, {612, 30479}, {645, 661}, {646, 649}, {648, 8611}, {651, 3239}, {652, 6335}, {657, 4554}, {658, 4130}, {660, 3716}, {662, 3700}, {663, 668}, {664, 3900}, {673, 3693}, {679, 4152}, {693, 3939}, {726, 8851}, {749, 4387}, {757, 6057}, {758, 6740}, {799, 3709}, {860, 1793}, {870, 4517}, {872, 18021}, {873, 7064}, {885, 1026}, {893, 17787}, {897, 3712}, {898, 14430}, {901, 4768}, {903, 3689}, {932, 4147}, {934, 4163}, {941, 11679}, {943, 6734}, {947, 23528}, {950, 1257}, {958, 31359}, {960, 1220}, {979, 19582}, {983, 3705}, {985, 3790}, {996, 3877}, {997, 30513}, {1000, 3872}, {1002, 3886}, {1014, 4082}, {1016, 2170}, {1018, 4560}, {1019, 30730}, {1021, 4552}, {1022, 30731}, {1025, 28132}, {1034, 1490}, {1036, 4385}, {1096, 1264}, {1100, 4102}, {1111, 6065}, {1120, 3880}, {1121, 6603}, {1125, 32635}, {1126, 3702}, {1146, 4564}, {1156, 6745}, {1174, 1229}, {1212, 32008}, {1214, 2322}, {1219, 1697}, {1222, 3057}, {1231, 2332}, {1240, 20967}, {1252, 4858}, {1253, 6063}, {1255, 3686}, {1261, 3663}, {1267, 13456}, {1268, 3683}, {1275, 3119}, {1280, 5853}, {1318, 4738}, {1332, 3064}, {1333, 30713}, {1390, 3883}, {1392, 3632}, {1407, 30693}, {1420, 6556}, {1434, 4515}, {1435, 30681}, {1441, 2328}, {1476, 6736}, {1492, 4522}, {1502, 9447}, {1577, 5546}, {1635, 4582}, {1639, 3257}, {1743, 6557}, {1751, 27396}, {1783, 6332}, {1785, 1809}, {1807, 5081}, {1812, 1826}, {1896, 3682}, {1903, 27398}, {1911, 4087}, {1928, 9448}, {1937, 7360}, {1959, 15628}, {1978, 3063}, {2052, 2289}, {2053, 6376}, {2057, 10309}, {2082, 30701}, {2125, 30695}, {2136, 6553}, {2137, 6552}, {2149, 23978}, {2150, 28654}, {2161, 32851}, {2162, 4110}, {2171, 7058}, {2176, 27424}, {2184, 27382}, {2195, 3263}, {2269, 30710}, {2279, 28809}, {2297, 18228}, {2298, 3687}, {2299, 20336}, {2310, 4998}, {2311, 3948}, {2316, 4358}, {2320, 3679}, {2323, 18359}, {2326, 26942}, {2330, 7018}, {2334, 4673}, {2335, 5271}, {2339, 2345}, {2340, 2481}, {2341, 3936}, {2342, 3262}, {2344, 3661}, {2346, 4847}, {2347, 32017}, {2349, 7359}, {2361, 20566}, {2363, 3704}, {2364, 4671}, {2643, 6064}, {2968, 7012}, {2985, 17452}, {2997, 3190}, {2998, 7075}, {3056, 7033}, {3059, 21453}, {3061, 17743}, {3083, 13454}, {3084, 13426}, {3112, 3688}, {3158, 4373}, {3161, 8056}, {3198, 5931}, {3219, 7110}, {3241, 4900}, {3247, 30711}, {3254, 3935}, {3271, 7035}, {3287, 27805}, {3452, 23617}, {3467, 27529}, {3478, 4737}, {3495, 26752}, {3615, 3678}, {3616, 4866}, {3623, 31509}, {3667, 31343}, {3669, 6558}, {3676, 4578}, {3691, 32009}, {3714, 5331}, {3715, 30598}, {3737, 3952}, {3762, 5548}, {3869, 10570}, {3870, 6601}, {3903, 3907}, {3996, 13476}, {3998, 8748}, {4007, 25417}, {4017, 7256}, {4024, 4612}, {4033, 7252}, {4034, 27789}, {4036, 4636}, {4069, 7192}, {4079, 4631}, {4081, 7045}, {4092, 24041}, {4105, 4569}, {4124, 5378}, {4140, 4603}, {4146, 6726}, {4149, 7357}, {4166, 18297}, {4171, 4573}, {4319, 8817}, {4433, 18827}, {4435, 4562}, {4494, 30650}, {4512, 5936}, {4513, 9311}, {4516, 4600}, {4521, 27834}, {4524, 4625}, {4526, 4607}, {4530, 5376}, {4534, 5382}, {4543, 4618}, {4551, 7253}, {4555, 4895}, {4557, 18155}, {4561, 18344}, {4567, 21044}, {4571, 7649}, {4572, 8641}, {4587, 17924}, {4597, 4814}, {4604, 4944}, {4606, 4765}, {4614, 4843}, {4624, 4827}, {4645, 7281}, {4791, 5549}, {4811, 8694}, {4845, 30806}, {4853, 7320}, {4861, 5559}, {4882, 5558}, {4919, 6630}, {4936, 27818}, {4985, 8701}, {5205, 9365}, {5239, 7043}, {5240, 7026}, {5380, 14432}, {5391, 13427}, {5430, 24242}, {5547, 14210}, {5665, 20007}, {6358, 7054}, {6554, 7131}, {6555, 19604}, {6606, 6608}, {6615, 8706}, {6737, 17097}, {7004, 15742}, {7020, 7078}, {7068, 24000}, {7071, 7182}, {7072, 20930}, {7090, 30556}, {7097, 27540}, {7105, 7283}, {7162, 10527}, {7178, 7259}, {7180, 7258}, {7220, 24349}, {7285, 27525}, {8012, 31618}, {8058, 13138}, {8707, 17420}, {8806, 13614}, {9368, 9369}, {9404, 15455}, {9436, 28071}, {9442, 28058}, {10025, 14943}, {10482, 20880}, {11609, 17763}, {12644, 12646}, {14121, 30557}, {14206, 15627}, {14534, 21033}, {14624, 17185}, {14827, 20567}, {15416, 32674}, {15891, 30413}, {15892, 30412}, {17780, 23838}, {18265, 18891}, {18750, 30457}, {19607, 21078}, {23705, 23836}, {24150, 24151}, {24152, 24154}, {24153, 24155}, {28070, 30705}

X(9) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 7}, {2, 85}, {3, 77}, {4, 273}, {6, 57}, {7, 1088}, {8, 75}, {10, 1441}, {11, 1111}, {19, 278}, {21, 86}, {22, 7210}, {25, 34}, {29, 286}, {31, 56}, {32, 604}, {33, 4}, {34, 1119}, {35, 1442}, {36, 1443}, {37, 226}, {38, 3665}, {40, 347}, {41, 6}, {42, 65}, {43, 3212}, {44, 3911}, {45, 5219}, {48, 222}, {51, 1393}, {55, 1}, {56, 269}, {57, 279}, {58, 1014}, {59, 7045}, {60, 757}, {63, 348}, {64, 8809}, {65, 3668}, {69, 7182}, {71, 1214}, {72, 307}, {73, 1439}, {75, 6063}, {76, 20567}, {77, 7056}, {78, 69}, {80, 18815}, {81, 1434}, {84, 1440}, {90, 7318}, {92, 331}, {99, 4625}, {100, 664}, {101, 651}, {109, 934}, {110, 1414}, {144, 31627}, {154, 1394}, {163, 4565}, {165, 3160}, {171, 7176}, {172, 7175}, {173, 18886}, {174, 555}, {181, 1254}, {184, 603}, {188, 4146}, {190, 4554}, {192, 30545}, {197, 21147}, {198, 223}, {200, 8}, {201, 6356}, {205, 478}, {210, 10}, {212, 3}, {213, 1400}, {218, 1445}, {219, 63}, {222, 7177}, {223, 14256}, {226, 1446}, {228, 73}, {238, 1447}, {239, 10030}, {244, 1358}, {255, 1804}, {256, 7249}, {258, 21456}, {259, 174}, {261, 873}, {266, 7371}, {269, 479}, {278, 1847}, {279, 23062}, {280, 309}, {281, 92}, {282, 189}, {283, 1444}, {284, 81}, {291, 7233}, {294, 673}, {306, 1231}, {312, 76}, {314, 310}, {318, 264}, {321, 349}, {326, 7055}, {330, 7209}, {333, 274}, {341, 3596}, {344, 21609}, {345, 304}, {346, 312}, {350, 18033}, {354, 10481}, {391, 19804}, {394, 7183}, {461, 5342}, {480, 200}, {497, 3673}, {512, 4017}, {513, 3676}, {514, 24002}, {517, 22464}, {518, 9436}, {521, 4025}, {522, 693}, {523, 4077}, {560, 1397}, {572, 17074}, {573, 17080}, {577, 7125}, {594, 6358}, {603, 7053}, {604, 1407}, {607, 19}, {608, 1435}, {610, 18623}, {612, 388}, {614, 7195}, {643, 99}, {644, 190}, {645, 799}, {646, 1978}, {649, 3669}, {650, 514}, {651, 658}, {652, 905}, {653, 13149}, {654, 3960}, {656, 17094}, {657, 650}, {661, 7178}, {662, 4573}, {663, 513}, {664, 4569}, {668, 4572}, {672, 241}, {678, 1317}, {692, 109}, {728, 346}, {750, 7223}, {756, 12}, {757, 552}, {765, 4998}, {798, 7180}, {846, 17084}, {849, 7341}, {862, 1874}, {869, 1469}, {872, 181}, {884, 1027}, {893, 1432}, {894, 7196}, {896, 7181}, {902, 1319}, {904, 1431}, {906, 1813}, {923, 7316}, {926, 2254}, {934, 4626}, {950, 17863}, {958, 10436}, {960, 4357}, {968, 3485}, {982, 7185}, {984, 7179}, {1018, 4552}, {1019, 17096}, {1021, 4560}, {1026, 883}, {1040, 17170}, {1043, 314}, {1054, 17089}, {1055, 6610}, {1096, 1118}, {1098, 261}, {1100, 553}, {1106, 7023}, {1107, 30097}, {1110, 59}, {1146, 4858}, {1155, 1323}, {1158, 31600}, {1170, 10509}, {1172, 27}, {1174, 1170}, {1193, 24471}, {1201, 1122}, {1212, 142}, {1220, 31643}, {1229, 1233}, {1250, 1082}, {1251, 1081}, {1252, 4564}, {1253, 55}, {1254, 6046}, {1259, 326}, {1260, 78}, {1261, 1222}, {1265, 3718}, {1318, 679}, {1320, 903}, {1331, 6516}, {1333, 1412}, {1334, 37}, {1376, 9312}, {1395, 1398}, {1397, 1106}, {1400, 1427}, {1402, 1042}, {1407, 738}, {1414, 4616}, {1415, 1461}, {1419, 9533}, {1436, 1422}, {1438, 1462}, {1445, 17093}, {1449, 21454}, {1460, 4320}, {1461, 4617}, {1469, 7204}, {1474, 1396}, {1475, 1418}, {1490, 5932}, {1500, 2171}, {1613, 1424}, {1615, 2124}, {1617, 4350}, {1635, 30725}, {1639, 3762}, {1697, 3672}, {1707, 17081}, {1721, 2898}, {1722, 31598}, {1731, 33129}, {1740, 17082}, {1742, 31526}, {1743, 5435}, {1754, 3188}, {1759, 17075}, {1760, 17076}, {1781, 18625}, {1783, 653}, {1792, 332}, {1802, 219}, {1812, 17206}, {1824, 225}, {1837, 17861}, {1857, 158}, {1859, 1838}, {1864, 1210}, {1897, 18026}, {1903, 8808}, {1909, 7205}, {1914, 1429}, {1918, 1402}, {1935, 6359}, {1936, 5088}, {1946, 1459}, {1962, 3649}, {1964, 1401}, {1973, 608}, {1974, 1395}, {2053, 87}, {2066, 13389}, {2082, 4000}, {2098, 4862}, {2115, 9499}, {2124, 17113}, {2136, 4452}, {2149, 1262}, {2150, 593}, {2161, 2006}, {2162, 7153}, {2170, 1086}, {2171, 6354}, {2173, 6357}, {2174, 2003}, {2175, 31}, {2176, 1423}, {2177, 2099}, {2183, 1465}, {2185, 1509}, {2187, 221}, {2188, 1433}, {2192, 84}, {2193, 1790}, {2194, 58}, {2195, 105}, {2199, 6611}, {2200, 1409}, {2204, 1474}, {2206, 1408}, {2208, 1413}, {2209, 1403}, {2210, 1428}, {2212, 25}, {2223, 1458}, {2238, 16609}, {2241, 7225}, {2244, 7214}, {2245, 18593}, {2246, 5723}, {2251, 1404}, {2258, 959}, {2259, 2982}, {2268, 940}, {2269, 3666}, {2276, 7146}, {2280, 5228}, {2284, 1025}, {2285, 7365}, {2287, 333}, {2289, 394}, {2293, 354}, {2295, 4032}, {2299, 28}, {2308, 32636}, {2310, 11}, {2316, 88}, {2318, 72}, {2319, 330}, {2321, 321}, {2322, 31623}, {2323, 3218}, {2324, 329}, {2325, 4358}, {2327, 1812}, {2328, 21}, {2329, 894}, {2330, 171}, {2331, 196}, {2332, 1172}, {2333, 1880}, {2340, 518}, {2341, 24624}, {2342, 104}, {2344, 14621}, {2346, 21453}, {2347, 3752}, {2348, 3008}, {2352, 4306}, {2356, 1876}, {2361, 36}, {2364, 89}, {2427, 24029}, {2632, 1367}, {2638, 1364}, {2640, 17085}, {2643, 1365}, {2646, 3664}, {2900, 12649}, {2911, 1708}, {2939, 18631}, {2951, 31527}, {2968, 17880}, {2997, 15467}, {3009, 1463}, {3022, 2310}, {3024, 7266}, {3052, 1420}, {3056, 982}, {3057, 3663}, {3058, 7264}, {3059, 4847}, {3061, 3662}, {3063, 649}, {3064, 17924}, {3083, 13453}, {3084, 13436}, {3100, 4872}, {3119, 1146}, {3158, 145}, {3161, 18743}, {3169, 3210}, {3172, 3213}, {3185, 10571}, {3190, 3868}, {3195, 208}, {3198, 5930}, {3207, 1419}, {3208, 192}, {3217, 4383}, {3218, 17078}, {3219, 17095}, {3239, 4391}, {3248, 1357}, {3270, 7004}, {3271, 244}, {3287, 4369}, {3295, 7190}, {3303, 4328}, {3304, 7271}, {3306, 17079}, {3309, 31605}, {3445, 19604}, {3452, 26563}, {3496, 17086}, {3596, 561}, {3601, 3945}, {3666, 3674}, {3680, 4373}, {3681, 33298}, {3683, 1125}, {3684, 239}, {3685, 350}, {3686, 4359}, {3687, 20911}, {3688, 38}, {3689, 519}, {3690, 201}, {3691, 3739}, {3692, 345}, {3693, 3912}, {3694, 306}, {3699, 668}, {3700, 1577}, {3701, 313}, {3702, 1269}, {3703, 1930}, {3704, 18697}, {3706, 20888}, {3707, 24589}, {3709, 661}, {3710, 20336}, {3711, 3679}, {3712, 14210}, {3713, 11679}, {3715, 1698}, {3716, 3766}, {3717, 3263}, {3718, 305}, {3719, 3926}, {3720, 4059}, {3721, 16888}, {3723, 3982}, {3724, 1464}, {3731, 5226}, {3733, 7203}, {3737, 7192}, {3738, 4453}, {3745, 4298}, {3746, 7269}, {3747, 1284}, {3786, 30966}, {3870, 6604}, {3871, 4360}, {3876, 5224}, {3877, 4389}, {3880, 1266}, {3883, 26234}, {3885, 4398}, {3886, 4441}, {3900, 522}, {3907, 4374}, {3910, 4509}, {3913, 3875}, {3920, 7247}, {3938, 30617}, {3939, 100}, {3949, 26942}, {3957, 32007}, {3965, 3687}, {3974, 4385}, {3975, 1921}, {3985, 3948}, {3996, 17143}, {4007, 28605}, {4009, 6381}, {4041, 523}, {4042, 32092}, {4046, 4647}, {4050, 1278}, {4055, 22341}, {4060, 4980}, {4069, 3952}, {4073, 3705}, {4076, 7035}, {4081, 24026}, {4082, 3701}, {4086, 850}, {4087, 18891}, {4092, 1109}, {4094, 3027}, {4095, 3963}, {4097, 3896}, {4102, 32018}, {4105, 3900}, {4110, 6382}, {4111, 21020}, {4117, 1356}, {4118, 7217}, {4119, 20432}, {4123, 315}, {4130, 3239}, {4136, 20234}, {4147, 20906}, {4149, 6327}, {4152, 4738}, {4162, 3667}, {4163, 4397}, {4166, 366}, {4167, 21442}, {4171, 3700}, {4178, 20627}, {4182, 18297}, {4183, 29}, {4253, 17092}, {4254, 5256}, {4258, 1449}, {4266, 4850}, {4319, 497}, {4320, 7197}, {4326, 10580}, {4336, 1836}, {4361, 7243}, {4387, 3760}, {4390, 4363}, {4391, 3261}, {4394, 30719}, {4420, 319}, {4421, 25716}, {4433, 740}, {4435, 812}, {4451, 7018}, {4474, 4411}, {4477, 3907}, {4501, 4382}, {4511, 320}, {4512, 3616}, {4513, 3729}, {4515, 2321}, {4516, 3120}, {4517, 984}, {4518, 334}, {4521, 4462}, {4524, 4041}, {4526, 4728}, {4528, 4768}, {4531, 3778}, {4548, 2172}, {4551, 4566}, {4557, 4551}, {4559, 1020}, {4560, 7199}, {4564, 1275}, {4565, 4637}, {4567, 4620}, {4571, 4561}, {4573, 4635}, {4578, 3699}, {4579, 6649}, {4587, 1332}, {4606, 4624}, {4612, 4610}, {4662, 4967}, {4723, 3264}, {4730, 30572}, {4765, 4801}, {4790, 30723}, {4814, 4777}, {4820, 4823}, {4827, 4765}, {4843, 4815}, {4845, 1156}, {4847, 20880}, {4849, 4848}, {4853, 31995}, {4858, 23989}, {4860, 21314}, {4861, 7321}, {4866, 5936}, {4873, 4671}, {4875, 24199}, {4876, 335}, {4877, 5333}, {4882, 32087}, {4895, 900}, {4901, 31130}, {4903, 20943}, {4907, 5274}, {4919, 4440}, {4936, 3161}, {4944, 4791}, {4953, 4939}, {4959, 4926}, {4976, 4978}, {4979, 30724}, {4990, 4985}, {4995, 7278}, {4997, 20568}, {5048, 4887}, {5089, 5236}, {5250, 17321}, {5269, 3600}, {5285, 4296}, {5289, 17274}, {5311, 10404}, {5320, 1451}, {5414, 13388}, {5423, 341}, {5452, 169}, {5532, 1090}, {5546, 662}, {5547, 897}, {5548, 3257}, {5549, 4604}, {5552, 20930}, {5795, 24993}, {5802, 19788}, {5837, 24547}, {6003, 31603}, {6056, 255}, {6057, 1089}, {6058, 1091}, {6059, 1096}, {6060, 1097}, {6061, 1098}, {6062, 1099}, {6064, 24037}, {6065, 765}, {6066, 1110}, {6139, 14413}, {6187, 1411}, {6198, 7282}, {6332, 15413}, {6558, 646}, {6600, 3870}, {6602, 220}, {6603, 527}, {6605, 32008}, {6607, 6608}, {6608, 6362}, {6726, 188}, {6731, 556}, {6735, 3262}, {6736, 20895}, {6740, 14616}, {6741, 17886}, {6745, 30806}, {7004, 1565}, {7007, 7149}, {7014, 558}, {7017, 1969}, {7032, 7248}, {7037, 3345}, {7046, 318}, {7050, 7091}, {7054, 2185}, {7062, 23996}, {7064, 756}, {7067, 24038}, {7068, 17879}, {7069, 5}, {7070, 20}, {7071, 33}, {7072, 90}, {7073, 79}, {7074, 40}, {7075, 194}, {7076, 1940}, {7077, 291}, {7078, 7013}, {7079, 281}, {7080, 322}, {7081, 1909}, {7082, 499}, {7083, 614}, {7084, 1037}, {7085, 1038}, {7087, 7213}, {7101, 7017}, {7110, 30690}, {7115, 7128}, {7117, 3942}, {7118, 1436}, {7123, 7131}, {7124, 7289}, {7131, 30705}, {7133, 1659}, {7154, 7129}, {7155, 6384}, {7156, 1249}, {7177, 30682}, {7180, 7216}, {7252, 1019}, {7253, 18155}, {7256, 7257}, {7257, 670}, {7259, 645}, {7281, 7261}, {7290, 3598}, {7322, 5261}, {7339, 24013}, {7359, 14206}, {7367, 282}, {7368, 2324}, {7675, 14548}, {7707, 234}, {7952, 342}, {7962, 4346}, {8012, 1212}, {8056, 27818}, {8058, 17896}, {8540, 18201}, {8545, 1996}, {8551, 8012}, {8580, 31994}, {8606, 7100}, {8611, 525}, {8641, 663}, {8647, 1279}, {8653, 4822}, {8676, 23800}, {8750, 108}, {8835, 15913}, {8851, 3226}, {9310, 6180}, {9404, 14838}, {9439, 9309}, {9440, 9446}, {9441, 14189}, {9447, 32}, {9448, 560}, {9629, 3583}, {10382, 938}, {10387, 3677}, {10393, 5738}, {10482, 2346}, {10501, 10491}, {10502, 10489}, {10570, 2995}, {10581, 21127}, {10582, 32086}, {10638, 559}, {11075, 26743}, {11124, 21105}, {11193, 21201}, {11429, 3075}, {11934, 21185}, {11997, 24210}, {11998, 24237}, {12329, 8270}, {13427, 1336}, {13455, 485}, {13456, 1123}, {14077, 30181}, {14100, 11019}, {14298, 14837}, {14308, 17898}, {14392, 6366}, {14427, 1639}, {14547, 942}, {14827, 41}, {14936, 2170}, {14942, 2481}, {15627, 2349}, {15628, 1821}, {15733, 26015}, {15837, 13405}, {16011, 2091}, {16012, 177}, {16283, 9310}, {16502, 28017}, {16545, 18626}, {16546, 18627}, {16552, 17077}, {16556, 17083}, {16566, 17087}, {16568, 17088}, {16569, 17090}, {16571, 17091}, {16572, 8732}, {16588, 17451}, {16601, 21617}, {16666, 4031}, {16686, 1421}, {16713, 16708}, {16721, 18176}, {16777, 4654}, {16780, 28079}, {16885, 31231}, {17185, 16705}, {17194, 17169}, {17197, 16727}, {17412, 13401}, {17420, 3004}, {17452, 3782}, {17453, 7251}, {17469, 7198}, {17742, 28739}, {17744, 28780}, {17787, 1920}, {17798, 5018}, {18098, 18097}, {18163, 18600}, {18191, 17205}, {18265, 1911}, {18344, 7649}, {18594, 18624}, {18595, 18628}, {18596, 18629}, {18597, 18630}, {18598, 18632}, {18599, 18633}, {18887, 21624}, {18888, 14596}, {18889, 2291}, {19605, 10405}, {19624, 2078}, {20229, 1475}, {20359, 24215}, {20663, 8850}, {20665, 2275}, {20672, 2114}, {20684, 3721}, {20753, 3784}, {20967, 1193}, {21010, 4334}, {21033, 1211}, {21039, 3925}, {21044, 16732}, {21059, 1617}, {21104, 23599}, {21127, 21104}, {21333, 4920}, {21334, 24214}, {21677, 18698}, {21789, 3737}, {21795, 21808}, {21803, 7211}, {21809, 4415}, {21811, 17056}, {21832, 7212}, {21840, 5244}, {21859, 4605}, {21879, 27691}, {22074, 22097}, {22079, 22053}, {23207, 4303}, {23344, 23703}, {23544, 28358}, {23638, 24443}, {23838, 6548}, {23990, 2149}, {24010, 4081}, {24012, 3022}, {24027, 7339}, {24041, 7340}, {24151, 27828}, {24394, 4442}, {24430, 17181}, {25082, 17234}, {25128, 23807}, {25268, 21580}, {26885, 1935}, {27382, 18750}, {27396, 18134}, {27424, 6383}, {27508, 20921}, {27523, 20923}, {27538, 6376}, {27540, 20914}, {27549, 30758}, {27958, 8033}, {28043, 2550}, {28070, 6554}, {28071, 14942}, {28125, 5880}, {28659, 1502}, {28660, 6385}, {28808, 20925}, {28809, 21615}, {30223, 3086}, {30457, 2184}, {30568, 18135}, {30608, 20569}, {30618, 17353}, {30706, 2082}, {30713, 27801}, {30730, 4033}, {30731, 24004}, {32008, 31618}, {32462, 31604}, {32635, 1268}, {32652, 8059}, {32666, 32735}, {32674, 32714}, {32739, 1415}, {32851, 20924}, {33299, 141}

X(9) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6, 1449}, {1, 37, 3247}, {1, 44, 16670}, {1, 45, 16676}, {1, 72, 11523}, {1, 238, 7290}, {1, 405, 5436}, {1, 960, 15829}, {1, 984, 7174}, {1, 1723, 2257}, {1, 1724, 1453}, {1, 1728, 10396}, {1, 1743, 6}, {1, 1757, 3751}, {1, 3731, 37}, {1, 3973, 1743}, {1, 5234, 958}, {1, 10398, 5728}, {1, 16468, 16475}, {1, 16469, 1386}, {1, 16552, 21384}, {1, 16667, 1100}, {1, 16673, 16777}, {1, 17744, 17742}, {1, 30330, 5572}, {2, 7, 142}, {2, 57, 5437}, {2, 63, 57}, {2, 142, 20195}, {2, 144, 7}, {2, 226, 25525}, {2, 307, 18634}, {2, 329, 226}, {2, 672, 17754}, {2, 894, 10436}, {2, 908, 5219}, {2, 3218, 3306}, {2, 3219, 63}, {2, 3305, 7308}, {2, 3452, 30827}, {2, 3662, 17282}, {2, 3911, 31190}, {2, 3929, 3928}, {2, 4357, 17306}, {2, 5273, 5745}, {2, 5282, 3509}, {2, 5296, 5257}, {2, 5435, 6692}, {2, 5744, 3911}, {2, 5749, 5750}, {2, 5905, 5249}, {2, 6646, 3662}, {2, 9965, 9776}, {2, 17236, 17291}, {2, 17257, 4357}, {2, 17333, 17274}, {2, 17350, 894}, {2, 17483, 27186}, {2, 17484, 31019}, {2, 17781, 4654}, {2, 18228, 3452}, {2, 18230, 6666}, {2, 20347, 30949}, {2, 20348, 30097}, {2, 26125, 25521}, {2, 26685, 17353}, {2, 26792, 31053}, {2, 26806, 27147}, {2, 26836, 26997}, {2, 27065, 3305}, {2, 27131, 30852}, {2, 27184, 25527}, {2, 27420, 27384}, {2, 28287, 27626}, {2, 29696, 29740}, {2, 30414, 5242}, {2, 30415, 5243}, {2, 30946, 20335}, {2, 31018, 908}, {2, 31053, 31266}, {2, 31300, 26806}, {3, 84, 9841}, {3, 936, 5438}, {3, 5044, 936}, {3, 5777, 1490}, {3, 7330, 84}, {3, 12684, 31805}, {3, 24320, 3220}, {3, 31445, 31424}, {3, 31658, 21153}, {4, 21168, 5759}, {5, 5791, 5705}, {5, 5812, 5715}, {5, 26921, 5709}, {6, 37, 1}, {6, 44, 1743}, {6, 45, 37}, {6, 219, 2323}, {6, 220, 219}, {6, 1001, 16503}, {6, 1100, 16667}, {6, 1743, 16670}, {6, 2176, 2300}, {6, 3731, 3247}, {6, 8557, 2257}, {6, 8609, 3554}, {6, 15492, 3973}, {6, 16672, 16884}, {6, 16675, 16777}, {6, 16677, 3723}, {6, 16777, 1100}, {6, 16814, 3731}, {6, 16884, 16666}, {6, 16885, 44}, {6, 16969, 21785}, {6, 16970, 7290}, {6, 16972, 16475}, {6, 21769, 20228}, {7, 142, 6173}, {7, 1445, 57}, {7, 6172, 144}, {7, 6666, 20195}, {7, 8232, 226}, {7, 18230, 2}, {7, 29007, 8545}, {8, 346, 2321}, {8, 391, 3686}, {8, 452, 950}, {8, 950, 12625}, {8, 1334, 3208}, {8, 1697, 2136}, {8, 2269, 3169}, {8, 2321, 4007}, {8, 2325, 4873}, {8, 3161, 346}, {8, 3208, 4050}, {8, 3685, 3886}, {8, 3686, 4034}, {8, 3717, 4901}, {8, 5250, 1697}, {8, 5686, 24393}, {8, 27549, 3717}, {10, 40, 1706}, {10, 3730, 3501}, {10, 6554, 23058}, {10, 12514, 40}, {10, 12572, 4}, {10, 12618, 1861}, {10, 17355, 2345}, {10, 18250, 2551}, {10, 31594, 14121}, {10, 31595, 7090}, {19, 71, 40}, {19, 169, 16547}, {19, 1766, 16548}, {19, 2183, 2270}, {19, 7079, 281}, {21, 78, 3601}, {21, 2287, 284}, {21, 3876, 78}, {25, 7085, 5285}, {25, 26867, 7085}, {31, 612, 5269}, {31, 756, 612}, {33, 212, 7070}, {37, 44, 6}, {37, 45, 3731}, {37, 72, 22021}, {37, 220, 2324}, {37, 1100, 16777}, {37, 1743, 1449}, {37, 2911, 3553}, {37, 3723, 16672}, {37, 3731, 16676}, {37, 3973, 16670}, {37, 4426, 5336}, {37, 15479, 15829}, {37, 15492, 44}, {37, 16517, 7174}, {37, 16666, 3723}, {37, 16669, 1100}, {37, 16671, 16884}, {37, 16777, 16673}, {37, 16814, 45}, {37, 16885, 1743}, {37, 21873, 4053}, {37, 21879, 21810}, {38, 614, 3677}, {38, 748, 614}, {39, 21796, 2277}, {39, 31442, 31429}, {41, 2268, 284}, {41, 33299, 78}, {43, 846, 17594}, {43, 21369, 21387}, {44, 45, 1}, {44, 1100, 16669}, {44, 3723, 16671}, {44, 3731, 1449}, {44, 8557, 16572}, {44, 15492, 16885}, {44, 16675, 16667}, {44, 16814, 37}, {44, 16885, 3973}, {45, 1743, 3247}, {45, 3973, 1449}, {45, 15492, 1743}, {45, 16669, 16673}, {45, 16777, 16675}, {45, 16884, 16677}, {45, 16885, 6}, {48, 2265, 2261}, {48, 2267, 572}, {51, 3690, 26893}, {55, 200, 3158}, {55, 210, 200}, {55, 480, 6600}, {55, 1864, 10382}, {55, 2264, 380}, {55, 3059, 3174}, {55, 3683, 4512}, {55, 3711, 3689}, {55, 3715, 210}, {55, 7082, 30223}, {55, 14100, 4326}, {56, 8581, 4321}, {56, 25917, 8583}, {57, 63, 3928}, {57, 3929, 63}, {57, 7308, 2}, {63, 3219, 3929}, {63, 3305, 2}, {63, 3306, 3218}, {63, 7308, 5437}, {63, 21371, 16574}, {63, 25880, 28017}, {63, 25894, 28039}, {63, 27065, 7308}, {69, 344, 3912}, {69, 3912, 17296}, {71, 2183, 573}, {71, 2345, 3501}, {72, 405, 1}, {72, 10396, 6762}, {72, 14054, 5904}, {75, 190, 3729}, {75, 3729, 4659}, {75, 17277, 4384}, {75, 17335, 17277}, {75, 17336, 190}, {75, 20927, 20236}, {77, 651, 1419}, {85, 32024, 30625}, {85, 32088, 32008}, {85, 32100, 32024}, {86, 4687, 16831}, {86, 18206, 18164}, {101, 572, 48}, {101, 2265, 16554}, {101, 12034, 2265}, {105, 4712, 9451}, {141, 4422, 17279}, {141, 4643, 17272}, {141, 17279, 17284}, {141, 17332, 4643}, {142, 6666, 2}, {144, 6666, 6173}, {144, 18230, 142}, {165, 1709, 10860}, {165, 1750, 7580}, {165, 2951, 11495}, {165, 3062, 2951}, {165, 8580, 1376}, {165, 30326, 1750}, {165, 30393, 8580}, {169, 573, 2270}, {169, 1766, 19}, {169, 3730, 40}, {169, 4456, 7713}, {169, 7079, 23058}, {169, 12514, 3496}, {171, 7262, 1707}, {173, 8078, 18888}, {188, 236, 16016}, {190, 4384, 4659}, {190, 17277, 75}, {190, 17335, 4384}, {190, 17336, 25728}, {191, 1698, 46}, {191, 1781, 1761}, {192, 239, 3875}, {192, 17349, 239}, {193, 17316, 3879}, {198, 1436, 1604}, {198, 1903, 1490}, {198, 2182, 610}, {200, 4326, 3174}, {200, 4512, 55}, {200, 10382, 2900}, {210, 3683, 55}, {210, 3689, 3711}, {210, 4512, 3158}, {210, 13615, 2900}, {210, 14100, 3059}, {210, 15837, 480}, {212, 7069, 33}, {213, 5283, 1}, {218, 16601, 1}, {220, 958, 2329}, {220, 1212, 1}, {220, 8557, 15286}, {226, 329, 28609}, {226, 1708, 57}, {238, 984, 1}, {238, 16517, 1449}, {239, 17261, 192}, {241, 6180, 269}, {261, 645, 27958}, {281, 20262, 23058}, {284, 4877, 21}, {312, 333, 11679}, {319, 17233, 17294}, {319, 17264, 17233}, {320, 17234, 17298}, {320, 17263, 17234}, {321, 5278, 5271}, {329, 5749, 5746}, {333, 17185, 18163}, {344, 4416, 17296}, {345, 14555, 3687}, {346, 391, 8}, {346, 2269, 3208}, {346, 2321, 4873}, {346, 2347, 3169}, {346, 3161, 2325}, {346, 3686, 4007}, {346, 3692, 728}, {346, 3707, 4034}, {354, 4423, 10582}, {374, 21871, 2262}, {380, 3694, 3158}, {381, 31671, 18482}, {390, 5686, 8}, {390, 5809, 950}, {390, 5825, 10392}, {391, 452, 5802}, {391, 1334, 3169}, {391, 2321, 4034}, {391, 2325, 4007}, {391, 3161, 2321}, {392, 956, 1}, {405, 954, 1001}, {405, 15650, 72}, {474, 3916, 15803}, {480, 3059, 200}, {480, 4326, 3158}, {480, 14100, 3174}, {497, 4847, 24392}, {573, 1766, 40}, {573, 3730, 71}, {573, 17355, 3501}, {579, 5750, 17754}, {579, 8558, 1741}, {594, 4370, 17340}, {594, 17275, 3679}, {594, 17330, 17275}, {594, 17340, 17281}, {599, 17267, 17231}, {612, 756, 7322}, {644, 2170, 4919}, {672, 1400, 579}, {672, 5282, 63}, {728, 1697, 3208}, {728, 2082, 2136}, {894, 17260, 2}, {894, 21371, 57}, {894, 27420, 1944}, {908, 31018, 31142}, {936, 31424, 3}, {950, 10392, 5809}, {954, 5223, 11523}, {954, 5728, 1}, {954, 5729, 5728}, {958, 960, 1}, {958, 5302, 5234}, {958, 15479, 1449}, {960, 1212, 3061}, {960, 5302, 958}, {960, 30618, 220}, {965, 5782, 5783}, {966, 2345, 10}, {966, 6554, 20262}, {982, 5272, 5573}, {982, 17123, 5272}, {984, 16970, 3247}, {992, 2277, 978}, {993, 997, 3576}, {993, 10176, 997}, {1001, 5223, 3243}, {1001, 15481, 5223}, {1006, 18446, 3576}, {1038, 1935, 1394}, {1086, 17276, 4862}, {1086, 17278, 4859}, {1086, 17334, 17276}, {1086, 17337, 17278}, {1100, 16667, 1449}, {1100, 16669, 6}, {1100, 16675, 16673}, {1100, 16777, 1}, {1107, 2176, 1}, {1108, 2256, 1}, {1124, 8965, 1}, {1156, 6594, 5528}, {1212, 30618, 2329}, {1213, 1901, 442}, {1213, 7359, 7110}, {1213, 17303, 1698}, {1213, 17369, 17303}, {1253, 2310, 4319}, {1253, 21039, 28043}, {1260, 1864, 2900}, {1260, 10382, 3158}, {1260, 13615, 55}, {1276, 1277, 40}, {1276, 6192, 1277}, {1277, 6191, 1276}, {1278, 16816, 17117}, {1279, 3242, 1}, {1329, 18253, 26066}, {1329, 26066, 1698}, {1334, 2082, 1697}, {1334, 2347, 2269}, {1334, 3691, 8}, {1376, 3740, 8580}, {1376, 4640, 165}, {1376, 5574, 19605}, {1376, 16112, 17668}, {1376, 30624, 5574}, {1400, 2285, 57}, {1400, 5279, 3509}, {1400, 5749, 17754}, {1436, 1604, 32625}, {1445, 3305, 6666}, {1445, 8545, 7}, {1449, 3247, 1}, {1449, 16670, 6}, {1449, 16676, 3247}, {1621, 3681, 3870}, {1621, 3870, 10389}, {1621, 6605, 6602}, {1652, 1653, 57}, {1654, 3661, 17270}, {1654, 4473, 17280}, {1654, 17280, 3661}, {1654, 17339, 17286}, {1697, 10384, 390}, {1698, 9612, 442}, {1707, 5268, 171}, {1723, 1728, 1713}, {1723, 1743, 16572}, {1726, 21361, 1763}, {1728, 15298, 5728}, {1731, 4266, 2082}, {1743, 2324, 2323}, {1743, 3731, 1}, {1743, 3973, 44}, {1743, 16673, 16667}, {1743, 16814, 16676}, {1743, 17744, 5227}, {1743, 21061, 21384}, {1750, 30326, 5927}, {1757, 20372, 24727}, {1759, 16549, 46}, {1778, 2303, 58}, {1826, 26063, 5587}, {1864, 3683, 13615}, {1903, 2182, 5776}, {1903, 7367, 282}, {2082, 3692, 3169}, {2161, 4370, 16561}, {2170, 4390, 3872}, {2170, 21809, 17452}, {2220, 5301, 7031}, {2223, 20992, 16688}, {2238, 2276, 43}, {2245, 17369, 16549}, {2246, 14439, 100}, {2257, 5227, 6762}, {2264, 3965, 3684}, {2268, 3217, 41}, {2268, 21033, 78}, {2269, 2347, 4266}, {2269, 3691, 3686}, {2270, 2345, 1706}, {2278, 3204, 2174}, {2280, 3930, 3870}, {2284, 5701, 1}, {2285, 28070, 27382}, {2287, 27396, 78}, {2310, 4319, 4907}, {2318, 14547, 3190}, {2321, 2325, 346}, {2321, 3169, 4050}, {2321, 3686, 8}, {2321, 3707, 3686}, {2321, 4266, 3169}, {2321, 5802, 12625}, {2324, 9119, 11523}, {2325, 3686, 2321}, {2325, 3707, 8}, {2325, 4266, 3208}, {2329, 3061, 1}, {2345, 5819, 2550}, {2345, 6554, 281}, {2347, 3161, 3208}, {2347, 3691, 391}, {2348, 3693, 3684}, {2478, 6734, 9581}, {2886, 24703, 1699}, {2949, 5715, 5709}, {2975, 19861, 1420}, {3008, 3663, 4000}, {3009, 22343, 7032}, {3056, 4517, 3688}, {3057, 4853, 3680}, {3057, 4875, 4051}, {3059, 15837, 6600}, {3062, 8580, 15587}, {3068, 6351, 5393}, {3069, 6352, 5405}, {3161, 3686, 4873}, {3161, 3707, 4007}, {3174, 6600, 3158}, {3218, 3306, 57}, {3219, 3305, 57}, {3219, 7308, 3928}, {3219, 17260, 16574}, {3219, 27065, 2}, {3230, 16975, 1}, {3230, 20228, 21769}, {3247, 16670, 1449}, {3247, 16676, 37}, {3271, 3688, 3056}, {3271, 7064, 3688}, {3287, 3709, 3737}, {3294, 16552, 1}, {3294, 21061, 37}, {3299, 3301, 16473}, {3305, 3929, 5437}, {3333, 3646, 1125}, {3419, 11113, 3586}, {3436, 24987, 9578}, {3452, 5325, 5745}, {3452, 5745, 2}, {3452, 20258, 21246}, {3487, 16845, 1125}, {3496, 3501, 40}, {3509, 17754, 57}, {3586, 3679, 3419}, {3589, 4364, 4657}, {3589, 4657, 29598}, {3596, 17787, 4494}, {3618, 17321, 17023}, {3624, 6763, 3338}, {3633, 4898, 17388}, {3661, 17280, 17286}, {3661, 17331, 1654}, {3661, 17339, 17280}, {3662, 6646, 17274}, {3662, 17333, 6646}, {3662, 17338, 2}, {3664, 25072, 29571}, {3664, 29571, 4648}, {3666, 4383, 2999}, {3672, 5222, 3946}, {3678, 5248, 3811}, {3683, 3715, 200}, {3686, 3707, 391}, {3688, 7064, 4517}, {3689, 3711, 200}, {3693, 3965, 3694}, {3700, 9404, 1021}, {3713, 3965, 200}, {3717, 3883, 8}, {3723, 16666, 16884}, {3723, 16671, 16666}, {3723, 16884, 1}, {3729, 4384, 75}, {3729, 25728, 190}, {3731, 3973, 6}, {3731, 16667, 16673}, {3731, 16885, 16670}, {3739, 4363, 25590}, {3739, 17259, 16832}, {3739, 17351, 4363}, {3740, 4640, 1376}, {3740, 18227, 18236}, {3758, 4687, 86}, {3759, 4360, 16834}, {3759, 4664, 4360}, {3763, 17253, 17237}, {3782, 24789, 23681}, {3834, 17345, 7232}, {3869, 5260, 19860}, {3869, 19860, 3340}, {3872, 3877, 7962}, {3873, 5284, 4666}, {3876, 25082, 33299}, {3877, 4390, 4919}, {3883, 27549, 4901}, {3911, 5316, 2}, {3912, 4416, 69}, {3912, 25101, 344}, {3913, 4662, 4882}, {3927, 11108, 942}, {3928, 5437, 57}, {3929, 7308, 57}, {3940, 16418, 24929}, {3943, 17362, 17299}, {3944, 33138, 17064}, {3951, 5047, 11518}, {3961, 8616, 3749}, {3970, 16783, 1}, {3970, 17746, 5904}, {3973, 16673, 16669}, {3973, 16814, 3247}, {3975, 17787, 3596}, {3995, 19742, 3187}, {4000, 4419, 3663}, {4007, 4034, 8}, {4007, 4873, 2321}, {4029, 4700, 145}, {4030, 4126, 30615}, {4034, 4873, 4007}, {4042, 4387, 3706}, {4053, 21873, 24048}, {4067, 30143, 12559}, {4271, 17340, 1018}, {4313, 20007, 12437}, {4314, 6743, 3189}, {4357, 17353, 2}, {4361, 17348, 16833}, {4363, 17259, 3739}, {4370, 17330, 17281}, {4384, 25728, 3729}, {4386, 17735, 3550}, {4389, 16706, 17304}, {4389, 17352, 16706}, {4393, 4704, 17319}, {4416, 25101, 3912}, {4422, 4643, 17284}, {4422, 17332, 141}, {4445, 17269, 17229}, {4452, 24599, 4402}, {4473, 17280, 17339}, {4473, 17331, 17286}, {4488, 31995, 4454}, {4513, 4875, 4853}, {4520, 4875, 3057}, {4557, 8053, 15624}, {4640, 15587, 11495}, {4643, 17279, 141}, {4644, 4648, 3664}, {4670, 4698, 15668}, {4675, 17365, 4888}, {4681, 4852, 17318}, {4690, 17229, 4445}, {4708, 17385, 17327}, {4741, 17232, 17288}, {4851, 17243, 29573}, {4853, 4936, 4513}, {4859, 4862, 1086}, {4859, 31183, 17278}, {4862, 31183, 4859}, {4866, 4882, 4662}, {4999, 25681, 3624}, {5044, 5779, 5785}, {5044, 31424, 5438}, {5044, 31445, 3}, {5219, 31142, 908}, {5220, 15254, 1}, {5224, 17289, 17308}, {5224, 17354, 17289}, {5239, 5240, 1}, {5242, 5243, 2}, {5245, 5246, 8}, {5249, 5905, 4654}, {5249, 17781, 5905}, {5251, 5526, 16788}, {5251, 5692, 1}, {5257, 5746, 25525}, {5257, 5750, 2}, {5259, 5904, 1}, {5259, 17745, 16783}, {5269, 7322, 612}, {5273, 18228, 2}, {5279, 5749, 2285}, {5296, 5749, 2}, {5316, 5744, 31190}, {5325, 5745, 5273}, {5436, 11523, 1}, {5440, 16370, 30282}, {5506, 6763, 3624}, {5584, 12688, 12565}, {5686, 5838, 3686}, {5692, 18397, 72}, {5705, 31446, 5791}, {5728, 5729, 10398}, {5729, 15650, 5220}, {5732, 5785, 5784}, {5732, 21153, 3}, {5735, 5833, 5832}, {5739, 17776, 306}, {5742, 5830, 5831}, {5745, 18228, 30827}, {5758, 6846, 946}, {5759, 5817, 4}, {5766, 5809, 390}, {5766, 5825, 5809}, {5779, 31658, 5732}, {5785, 31424, 5732}, {5785, 31658, 5438}, {5795, 5837, 8}, {5809, 5838, 5802}, {5811, 6908, 6260}, {5904, 10399, 14054}, {5927, 7580, 1750}, {6172, 18230, 7}, {6173, 20195, 142}, {6191, 6192, 40}, {6203, 6204, 57}, {6210, 6211, 40}, {6211, 7609, 6210}, {6212, 6213, 40}, {6260, 6684, 6908}, {6601, 24389, 24392}, {6646, 17338, 17282}, {6687, 17235, 17356}, {6986, 12528, 10884}, {7001, 7010, 188}, {7026, 7043, 3679}, {7081, 20665, 2319}, {7090, 14121, 10}, {7174, 7290, 1}, {7174, 15601, 7290}, {7232, 17265, 3834}, {7290, 15601, 238}, {8165, 18231, 9780}, {8232, 12848, 7}, {8257, 8545, 6173}, {8580, 30393, 3740}, {9330, 17126, 5297}, {9776, 9965, 553}, {10177, 15185, 5572}, {10396, 31435, 5436}, {10398, 15299, 10396}, {10436, 16574, 57}, {10456, 18229, 10472}, {10857, 30304, 10167}, {11106, 20007, 4313}, {11679, 30568, 312}, {11683, 26671, 16609}, {11752, 11789, 3576}, {12572, 17355, 8804}, {13405, 21060, 25568}, {14100, 15837, 55}, {14829, 18743, 30567}, {15296, 15297, 1001}, {15298, 15299, 1}, {15492, 16814, 6}, {16482, 23343, 1}, {16517, 16970, 1}, {16519, 16974, 1}, {16547, 16548, 19}, {16566, 20602, 1760}, {16572, 17742, 6762}, {16666, 16671, 6}, {16666, 16672, 1}, {16667, 16673, 1}, {16667, 16675, 3247}, {16668, 16674, 1}, {16669, 16673, 1449}, {16669, 16675, 1}, {16669, 16777, 16667}, {16669, 16814, 16675}, {16670, 16676, 1}, {16671, 16677, 1}, {16672, 16677, 37}, {16672, 16884, 3723}, {16673, 16777, 3247}, {16675, 16777, 37}, {16675, 16885, 16669}, {16677, 16884, 16672}, {16706, 17258, 4389}, {16713, 17183, 17197}, {16779, 16973, 1449}, {16788, 21078, 3553}, {16814, 16885, 1}, {16815, 17116, 4699}, {16816, 25269, 1278}, {16826, 17120, 17379}, {16832, 25590, 3739}, {16833, 17151, 4361}, {16969, 17448, 1}, {17121, 17319, 4393}, {17227, 17329, 17273}, {17227, 17341, 17283}, {17228, 17328, 17271}, {17228, 17342, 17285}, {17230, 17343, 17287}, {17231, 17344, 599}, {17233, 17346, 319}, {17234, 17347, 320}, {17235, 17356, 17290}, {17237, 17357, 3763}, {17238, 17358, 17292}, {17239, 17359, 17293}, {17240, 17360, 17295}, {17241, 17361, 17297}, {17242, 17363, 6542}, {17244, 17364, 17300}, {17245, 17365, 4675}, {17246, 17366, 17301}, {17247, 17367, 17302}, {17248, 17368, 2}, {17249, 17370, 17305}, {17250, 17371, 17307}, {17251, 17293, 17239}, {17252, 17292, 17238}, {17254, 17291, 17236}, {17255, 17290, 17235}, {17256, 17289, 5224}, {17256, 17354, 17308}, {17257, 17353, 17306}, {17257, 26685, 2}, {17257, 28287, 1423}, {17257, 29497, 29740}, {17258, 17352, 17304}, {17259, 17351, 25590}, {17260, 17350, 10436}, {17261, 17349, 3875}, {17262, 17348, 17151}, {17263, 17347, 17298}, {17264, 17346, 17294}, {17266, 17288, 17232}, {17268, 17287, 17230}, {17270, 17286, 3661}, {17271, 17285, 17228}, {17272, 17284, 141}, {17273, 17283, 17227}, {17274, 17282, 3662}, {17275, 17281, 594}, {17276, 17278, 1086}, {17276, 17337, 4859}, {17277, 17336, 3729}, {17277, 25728, 4659}, {17278, 17334, 4862}, {17278, 17337, 31183}, {17279, 17332, 17272}, {17280, 17331, 17270}, {17281, 17330, 3679}, {17299, 17362, 3632}, {17300, 20072, 17364}, {17315, 17377, 29605}, {17322, 17381, 29603}, {17324, 29630, 17383}, {17328, 17342, 17228}, {17329, 17341, 17227}, {17330, 17340, 594}, {17331, 17339, 3661}, {17333, 17338, 3662}, {17334, 17337, 1086}, {17335, 17336, 75}, {17353, 29497, 1423}, {17375, 29572, 17312}, {17379, 27268, 16826}, {17484, 31019, 31164}, {18228, 26059, 21246}, {18249, 18250, 10}, {20090, 29569, 17391}, {20196, 31231, 2}, {20683, 21746, 3779}, {21296, 29627, 4869}, {21616, 26363, 8227}, {21811, 33299, 27396}, {24152, 24153, 200}, {24313, 24314, 8}, {24477, 26105, 11019}, {24697, 33159, 32784}, {24909, 24952, 2}, {25019, 28739, 18634}, {25447, 25651, 2}, {25525, 28609, 226}, {25760, 33115, 29857}, {25842, 25856, 25860}, {26059, 27420, 10436}, {26885, 26890, 184}, {27254, 29967, 5219}, {27509, 28731, 63}, {27819, 27834, 19604}, {28606, 32911, 5256}, {29382, 29492, 3662}, {29395, 29423, 29511}, {29492, 29698, 17274}, {29509, 29541, 29396}, {30324, 30325, 226}, {30327, 30328, 226}, {30412, 30413, 2}, {30414, 30415, 2}, {30556, 30557, 1}, {30557, 31438, 1449}, {30557, 31453, 18991}, {31561, 31562, 4}, {31594, 31595, 10}, {32008, 32024, 85}, {32008, 32100, 30625}, {32088, 32100, 85}, {32555, 32556, 3}, {32784, 33159, 1698}, {32864, 32915, 17156}, {32917, 32931, 29828}, {33076, 33165, 3679}

Trilinears 1/(r cos A - s sin A) : :

Trilinears csc(A - U) : :, U as at X(572) and X(573)

Trilinears (cos B + cos C)/(1 - cos A) : :

Trilinears 1 + 2 csc A/2 sin B/2 sin C/2 : :

Trilinears |AP(1)| + |AU(1)| : :

Trilinears (r/R) - 2 sin B sin C : :

Barycentrics b + c : c + a : a + b

Barycentrics semi-major axis of A-Soddy ellipse : :

Tripolars Sqrt[a^3-2 a b^2-b^3+a b c-2 a c^2-c^3] : :

Let A' be the intersection of these three lines: the perpendicular from midpoint of CA to line BX(1), the perpendicular from midpoint of AB to line CX(1), the perpendicular from midpoint of AX(1) to line BC, and define B' and C' cyclically. The orthocenter of A'B'C' is X(10), and X(10) is also the perspector of A'B'C' and the medial triangle. Note that A'B'C' is the complement of the excentral triangle, and the extraversion triangle of X(10). (Randy Hutson, December 2, 2017)

The Spieker circle is the incircle of the medial triangle; its center, X(10), is the centroid of the perimeter of ABC. If you have The Geometer's Sketchpad, you can view Spieker center.

If you have GeoGebra, you can view **Spieker center**.

A construction of X(10) is given at 24163. (Antreas Hatzipolakis, August 29, 2016)

Let A'B'C' be the excentral triangle. X(10) is the radical center of the polar circles of triangles A'BC, B'CA, C'AB. (Randy Hutson, July 31 2018)

Let A_{20}B_{20}C_{20} be the Gemini triangle 20. Let L_{A} be the line through A_{20} parallel to BC, and define L_{B} and L_{C} cyclically. Let A'_{20} = L_{B}∩L_{C}, and define B'_{20} and C'_{20} cyclically. Triangle A'_{20}B'_{20}C'_{20} is homothetic to ABC at X(10). (Randy Hutson, November 30, 2018)

X(10) lies on the Kiepert hyperbola and these lines:

1,2 3,197 4,9 5,517
6,1377 11,121 12,65
20,165 21,35 28,1891
29,1794 31,964 33,406
34,475 36,404 37,594
38,596 39,730 44,752
46,63 55,405 56,474
57,388 58,171 69,969
75,76 81,1224 82,83
86,319 87,979 92,1838
98,101 106,1222 116,120
117,123 119,124 140,214
141,142 150,1282 153,1768
158,318 182,1678 190,671
191,267 201,225 219,965
227,1214 235,1902 255,1771
257,1581 261,1326 274,291
307,1254 321,756 348,1323
391,1743 407,1867 427,1829
429,1824 480,954 485,1686
486,1685 497,1697 514,764
535,1155 537,1086 626,760
631,944 632,1483 750,1150
774,1736 775,801 846,1247
894,1046 908,994 962,1695
1018,1334 1074,1735 1146,1212
1482,1656 1587,1703 1588,1702
1762,1782 1828,1883 1900,1904

X(10) is the {X(1),X(2)}-harmonic conjugate of X(1125). For a list
of other harmonic conjugates of X(10), click **Tables** at the top
of this page. X(10) is the internal center of similitude of the
Apollonius and nine-points circles.

Let A'B'C' be the 2nd extouch triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(10). Also, let A''B''C'' be the 1st circumperp triangle. The Simson lines of A'', B'', C'' concur in X(10). (Randy Hutson, November 18, 2015)

X(10) = midpoint of X(i) and X(j) for these (i,j): (1,8), (3,355), (4,40), (6,3416), (10,3421), (55,3419), (65,72), (80,100), (2948,3448)

X(10) = reflection of X(i) in X(j) for these (i,j): (1,1125), (551,2), (946,5), (1385,140)

X(10) = isogonal conjugate of X(58)

X(10) = isotomic conjugate of X(86)

X(10) = circumcircle-inverse of X(1324)

X(10) = nine-point-circle-inverse of X(3814)

X(10) = complement of X(1)

X(10) = anticomplement of X(1125)

X(10) = complementary conjugate of X(10)

X(10) = anticomplementary conjugate of X(2891)

X(10) = X(15319)-complementary conjugate of X(32767)

X(10) = radical center of the excircles.

X(10) = radical center of extraversions of Conway circle

X(10) = radical center of the polar circles of triangles BCI, CAI, ABI

X(10) = X(20)-of-3rd-Euler-triangle

X(10) = X(4)-of-4th-Euler-triangle

X(10) = perspector of ABC and the tangential triangle of the Feuerbach triangle

X(10) = X(2)-Hirst inverse of X(6542)

X(10) = inverse-in-Steiner-circumellipse of X(6542)

X(10) = SS(a->a') of X(5), where A'B'C' is the excentral triangle (barycentric substitution)

X(10) = orthocenter of X(2)X(4)X(4049)

X(10) = midpoint of PU(10)

X(10) = bicentric sum of PU(i) for these i: 10, 66

X(10) = PU(66)-harmonic conjugate of X(351)

X(10) = crosssum of X(i) and X(j) for these (i,j): (6,31), (56,603)

X(10) = crossdifference of every pair of points on line X(649)X(834)

X(10) = X(i)-beth conjugate of X(j) for these (i,j): (8,10), (10,65), (100,73), (318,225), (643,35), (668,349)

X(10) = radical trace of Bevan circle and anticomplementary circle

X(10) = insimilicenter of Bevan circle and anticomplementary circle

X(10) = insimilicenter of nine-point circle and Apollonius circle

X(10) = X(i)-Ceva conjugate of X(j) for these (i,j):

(2,37), (8,72), (75,321), (80,519), (100,522), (313,306)

X(10) = cevapoint of X(i) and X(j) for these (i,j): (1,191), (6,199), (12,201), (37,210), (42,71), (65,227)

X(10) = X(i)-cross conjugate of X(j) for these (i,j): (37,226), (71,306), (191,502), (201,72)

X(10) = crosspoint of X(i) and X(j) for these (i,j): (2,75), (8,318)

X(10) = centroid of ABCX(8)

X(10) = Kosnita(X(8),X(2)) point

X(10) = X(578)-of-2nd-extouch-triangle

X(10) = X(389)-of-excentral triangle

X(10) = X(125)-of-Fuhrmann triangle

X(10) = perspector of ABC and triangle formed from orthocenters of JaBC, JbCA, JcAB, where Ja, Jb, Jc are excenters

X(10) = perspector of circumconic centered at X(37)

X(10) = center of circumconic that is locus of trilinear poles of lines passing through X(37)

X(10) = trilinear pole of line X(523)X(661) (the polar of X(27) wrt polar circle)

X(10) = pole wrt polar circle of trilinear polar of X(27) (line X(242)X(514))

X(10) = X(48)-isoconjugate (polar conjugate)-of-X(27)

X(10) = X(6)-isoconjugate of X(81)

X(10) = X(75)-isoconjugate of X(2206)

X(10) = X(1101)-isoconjugate of X(3120)

X(10) = X(1)-of-X(1)-Brocard triangle

X(10) = perspector of medial triangle and Ayme triangle

X(10) = homothetic center of Ayme triangle and anticevian triangle of X(37)

X(10) = perspector of Ayme triangle and Danneels-Bevan triangle

X(10) = X(1)-of-Danneels-Bevan-triangle

X(10) = homothetic center of medial triangle and Danneels-Bevan triangle

X(10) = homothetic center of ABC and anticomplementary triangle of Danneels-Bevan triangle

X(10) = {X(2),X(8)}-harmonic conjugate of X(1)

X(10) = inverse-in-polar-circle of X(242)

X(10) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5205)

X(10) = inverse-in-Steiner-inellipse of X(3912)

X(10) = inverse-in-Feuerbach-hyperbola of X(3057)

X(10) = perspector of Feuerbach and Apollonius triangles

X(10) = perspector of symmedial triangles of Feuerbach and Apollonius triangles

X(10) = perspector of circumsymmedial triangles of Feuerbach and Apollonius triangles

X(10) = perspector of tangential triangles of Feuerbach and Apollonius triangles

X(10) = X(214)-of-inner-Garcia-triangle

X(10) = Cundy-Parry Phi transform of X(13478)

X(10) = Cundy-Parry Psi transform of X(573)

X(10) = perspector of Ayme and 4th Euler triangles

X(10) = barycentric product X(101)*X(850)

X(10) = perspector of Gemini triangle 12 and cross-triangle of ABC and Gemini triangle 12

X(10) = perspector of ABC and cross-triangle of ABC and Gemini triangle 15

X(10) = trilinear product of vertices of Gemini triangle 15

X(10) = homothetic center of Ayme triangle and Gemini triangle 16

X(10) = center of the {ABC, Gemini 18}-circumconic

X(10) = Gemini-triangle-19-to-ABC parallelogic center

X(10) = centroid of Gemini triangle 20

X(10) = perspector of ABC and cross-triangle of ABC and Gemini triangle 25

X(10) = perspector of ABC and Gemini triangle 26

X(10) = perspector of Gemini triangle 39 and cross-triangle of ABC and Gemini triangle 39

X(10) = excentral-to-ABC barycentric image of X(3)

X(10) = incentral-to-ABC barycentric image of X(1)

Trilinears sin

Trilinears bc(b + c - a)(b - c)

Trilinears 1 - cos A - 2 cos B cos C : :

Trilinears 1 + cos A - 2 sin B sin C : :

Barycentrics a(1 - cos(B - C)) : b(1 - cos(C - A)) : c(1 - cos(A -B))

Barycentrics (b + c - a)(b - c)

Tripolars Sqrt[(a-b-c) (a^4-2 a^2 b^2+b^4+a^2 b c+a b^2 c-2 b^3 c-2 a^2 c^2+a b c^2+2 b^2 c^2-2 b c^3+c^4)] : :

X(11) = R*X(1) - 3rX(2) + r*X(3)

X(11) is the point of tangency of the nine-point circle and the incircle. The nine-point circle is the circumcenter of the medial triangle, as well as the orthic triangle. Feuerbach's famous theorem states that the nine-point circle is tangent to the incircle and the three excircles.

Let L_{A} be the line through A parallel to X(1)X(3), and define L_{B} and L_{C} cyclically. Let M_{A} be the reflection of BC in L_{A}, and define M_{B} and M_{C} cyclically. Let A' = M_{B}∩M_{C}, and define cyclically B' and C' cyclically. The triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in line X(1)X(3). The triangle A"B"C" is homothetic to ABC, with center of homothety X(11); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

The circumcircle of the incentral triangle intersects the incircle at 2 points, X(11) and X(3024), and the nine-point circle at 2 points, X(11) and X(115). (Randy Hutson, April 9, 2016)

X(11) lies on the bicevian conic of X(1) and X(2), which is also QA-Co1 (Nine-point Conic) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/other-quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/76-qa-co1.html) (Randy Hutson, April 9, 2016)

Let Na = X(5) of BCX(1), Nb = X(5) of CAX(1), Nc = X(5) of ABX(1). Then X(11) = X(186) of NaNbNc. (Randy Hutson, April 9, 2016)

Let JaJbJc be the excentral triangle and FaFbFc be the Feuerbach triangle. Let Fa' = {X(5),Ja}-harmonic conjugate of Fa, and define Fb', Fc' cyclically. The lines AFa', BFb', CFc' concur in X(11).

Let A'B'C' be the orthic triangle. Let La be the antiorthic axis of AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B'' and C'' cyclically. The triangle A"B"C" is inversely similar to ABC, with similitude center X(9), and X(11) = X(55)-of-A"B"C". (Randy Hutson, December 10, 2016)

Let A'B'C' be the orthic triangle. Let Na be the Nagel line of AB'C', and define Nb and Nc cyclically. Let A" = Nb∩Nc, and define B'' and C'' cyclically. The triangle A"B"C" is inversely similar to ABC, and X(11) = X(36)-of-A"B"C". (Randy Hutson, December 10, 2016)

Let A'B'C' be the orthic triangle. The lines IO of AB'C', BC'A', CA'B' concur in X(11). (Randy Hutson, December 10, 2016)

Let A'B'C' be the orthic triangle. The lines IO of AB'C', BC'A', CA'B' concur in X(11). (Randy Hutson, June 27, 2018)

If you have The Geometer's Sketchpad, you can view Feuerbach point.

If you have GeoGebra, you can view **Feuerbach point**.

X(11) is the {X(1),X(5)}-harmonic conjugate of X(12) and also the {X(5),X(12)}-harmonic conjugate of X(3614) . For a list of other harmonic conjugates of X(11), click **Tables** at the top of this page.

X(11) lies on the incentral circle, Mandart circle, cevian circle of every point on the Feuerbach hyperbola, and these lines:

1,5 2,55 3,499 4,56
7,658 8,1320 9,3254 10,121
13,202 14,203 28,1852
30,36 33,427 34,235
35,140 57,1360 65,117
68,1069 110,215 113,942
115,1015 118,226 124,1364
182,1848 133,1838 153,388
212,748 214,442 244,867
278,1857 325,350 381,999
403,1870 429,1104 485,1124
486,1335 498,1656 515,1319
516,1155 517,1737 518,908
523,1090 613,1352 650,1566
774,1393 944,1388 962,1788
971,1538 1012,1470 1040,1368
1111,1358 1146,1639 1193,1834
1427,1856 1428,1503 1455,1877
1500,1506 1697,1698

X(11) = midpoint of X(i) and X(j) for these (i,j): (1,80), (4,104), (5,1484), (9,3254), (100,149)

X(11) = reflection of X(i) in X(j) for these (i,j): (1,1387), (119,5), (214,1125), (1145,10), (1317,1), (1537,946)

X(11) = isogonal conjugate of X(59)

X(11) = isotomic conjugate of X(4998)

X(11) = inverse-in-Fuhrmann-circle of X(1837)

X(11) = complement of X(100)

X(11) = anticomplement of X(3035)

X(11) = complementary conjugate of X(513)

X(11) = circumcircle-inverse of X(14667)

X(11) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,523), (4,513), (7,514), (8,522), (262,1491)

X(11) = crosspoint of X(i) and X(j) for these (i,j): (7,514), (8,522)

X(11) = crosssum of X(i) and X(j) for these (i,j): (6,692), (55,101), (56,109), (1381,1382), (1397,1415)

X(11) = crossdifference of every pair of points on line X(101)X(109)

X(11) = X(i)-beth conjugate of X(j) for these (i,j): (11,244), (522,11), (693,11)

X(11) = orthopole of line X(1)X(3)

X(11) = anticenter of cyclic quadrilateral ABCX(104)

X(11) = perspector of ABC and extraversion triangle of X(12)

X(11) = homothetic center of intouch and 3rd Euler triangles

X(11) = trilinear square root of X(6728)

X(11) = perspector of Feuerbach triangle and Schroeter triangle

X(11) = X(110)-of-intouch-triangle

X(11) = X(403) of Fuhrmann triangle

X(11) = perspector of circumconic centered at X(650)

X(11) = center of circumconic that is locus of trilinear poles of lines passing through X(650)

X(11) = X(2)-Ceva conjugate of X(650)

X(11) = trilinear pole wrt intouch triangle of Soddy line

X(11) = trilinear pole wrt extouch triangle of line X(8)X(9)

X(11) = midpoint of PU(i) for these i: 121, 123

X(11) = bicentric sum of PU(i) for these i: 121, 123

X(11) = inverse-in-polar-circle of X(108)

X(11) = inverse-in-{circumcircle, nine-point circle}-inverter of X(105)

X(11) = inverse-in-Fuhrmann-circle of X(1837)

X(11) = inverse-in-excircles-radical-circle of X(3030)

X(11) = homothetic center of medial triangle and Mandart-incircle triangle

X(11) = X(100) of Mandart-incircle triangle

X(11) = X(3659) of orthic triangle if ABC is acute

X(11) = homothetic center of intangents triangle and reflection of extangents triangle in X(100)

X(11) = homothetic center of 3rd Euler triangle and intouch triangle

X(11) = QA-P2 (Euler-Poncelet Point) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/quadrangle-objects/10-mathematics/quadrangle-objects/12-qa-p2.html)

X(11) = intersection of tangents to Steiner inellipse at X(1086) and X(1146)

X(11) = crosspoint wrt medial triangle of X(1086) and X(1146)

X(11) = perspector of orthic triangle and tangential triangle of hyperbola {{A,B,C,X(1),X(2)}}

X(11) = homothetic center of cyclic quadrilateral ABCX(104) and congruent quadrilateral formed by orthocenters of vertices taken 3 at a time

X(11) = perspector of ABC and cross-triangle of ABC and Feuerbach triangle

X(11) = homothetic center of medial triangle and cross-triangle of ABC and inner Johnson triangle

X(11) = homothetic center of Euler triangle and cross-triangle of ABC and 1st Johnson-Yff triangle

X(11) = homothetic center of medial triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle

X(11) = X(6)-isoconjugate of X(4564)

X(11) = orthic-isogonal conjugate of X(513)

X(11) = homothetic center of cyclic quadrilateral ABCX(104) and congruent quadrilateral formed by orthocenters of vertices taken 3 at a time

X(11) = homothetic center of Ursa-minor and Ursa-major triangles

X(11) = homothetic center of ABC and inner Johnson triangle

X(11) = trilinear product X(57)*X(1146)

X(11) = barycentric product X(7)*X(1146)

X(11) = homothetic center of Garcia reflection triangle (aka Gemini triangle 8) and 2nd Schiffler triangle

X(11) = excentral-to-ABC functional image of X(3659)

X(11) = excentral-to-ABC barycentric image of X(100)

X(11) = inverse-in-Stevanovic-circle of X(1566)

Trilinears ; cos

Trilinears ; 1 + cos A + 2 cos B cos C : :

Trilinears ; 1 - cos A + 2 sin B sin C : :

Trilinears ; bc(b + c)

Barycentrics a(1 + cos(B - C)) : :

Barycentrics (b + c)

Tripolars (a-b-c) Sqrt[(a^3+a^2 b-a b^2-b^3+a^2 c+3 a b c+b^2 c-a c^2+b c^2-c^3) (a^4-2 a^2 b^2+b^4+a^2 b c-a b^2 c-2 b^3 c-2 a^2 c^2-a b c^2+2 b^2 c^2-2 b c^3+c^4)] : :

Let A'B'C' be the touch points of the nine-point circle with the A-, B-, C- excircles, respectively. The lines AA', BB', CC' concur in X(12).

X(12) lies on these lines:

1,5 2,56 3,498 4,55 7,1268 10,65 17,203
18,202 30,35 33,235
34,427 36,140 37,225
38,1393 40,1836 42,1834
54,215 57,1224 63,1454
71,1901 79,484 85,120
108,451 115,1500 116,1362
117,1364 121,1357 123,1359
124,1361 125,1425 141,1469
171,1399 172,230 201,756
208,1360 221,1853 228,407
281,1118 313,349 354,1210
377,1259 381,1479 431,1824
443,1466 474,1470 485,1124
486,1124 499,999 603,750
611,1352 908,960 942,1209
946,1532 968,1904 1015,1506
1038,1368 1091,1109 1125,1319
1213,1400 1452,1892 1594,1870
1697,1699 1861,1887 1877,1883

X(12) is the {X(1),X(5)}-harmonic conjugate of X(11). For a list of other harmonic conjugates of X(12), click **Tables** at the top of this page.

If you have The Geometer's Sketchpad, you can view X(12).

If you have GeoGebra, you can view **X(12)**.

X(12) = isogonal conjugate of X(60)

X(12) = isotomic conjugate of X(261)

X(12) = complement of X(2975)

X(12) = circumcircle-inverse of X(32626)

X(12) = X(10)-Ceva conjugate of X(201)

X(12) = crosssum of X(58) and X(1437)

X(12) = X(i)-beth conjugate of X(j) for these (i,j): (10,12), (1089,1089)

X(12) = insimilicenter of incircle and nine-point circle

X(12) = X(1594)-of-Fuhrmann triangle

X(12) = homothetic center of Euler and Mandart-incircle triangles

X(12) = homothetic center of intouch and 4th Euler triangles

X(12) = X(6)-isoconjugate of X(2185)

X(12) = trilinear pole of line X(2610)X(4024)

X(12) = trilinear square of X(6724)

X(12) = homothetic center of medial triangle and cross-triangle of ABC and outer Johnson triangle

X(12) = homothetic center of medial triangle and cross-triangle of ABC and 1st Johnson-Yff triangle

X(12) = homothetic center of Euler triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle

X(12) = homothetic center of ABC and triangular hull of circumcircles of BCX(4), CAX(4), and ABX(4); i.e., the outer Johnson triangle

X(12) = centroid of curvatures of nine-point circle and excircles

Trilinears sec(A - π/6) : sec(B - π/6) : sec(C - π/6)

Barycentrics a

Barycentrics (SA + Sqrt[3] S) (SB + SC) + 4 SB SC : :

Tripolars 1/(a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4+2 Sqrt[3] a^2 S) : :

X(13) = 3

(Peter Moses, April 2, 2013)

Construct the equilateral triangle BA'C having base BC and vertex A' on the negative side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA', BB', CC' concur in X(13). If each of the angles A, B, C is < 2π/3, then X(13) is the only center X for which the angles BXC, CXA, AXB are equal, and X(13) minimizes the sum |AX| + |BX| + |CX|. The antipedal triangle of X(13) is equilateral.

If, however, A> 2π/3, then the Fermat point, defined geometrically as the minimizer of |AX| + |BX| + |CX|, is not the 1st isogonic center (which is defined by the above trilinears). Trilinears for the Fermat point when A> 2π/3 are simply 1:0:0. To represent the Fermat point in the form f(a,b,c) : f(b,c,a) : f(c,a,b), one must use Boolean variables, as shown at Fermat point.

If you have The Geometer's Sketchpad, you can view these sketches:

Fermat Dynamic

1st isogonic center

Kiepert Hyperbola, showing X(13) and X(14) on the hyperbola, with midpoint X(115)

**Evans Conic**, passing through X(13), X(14), X(15), X(16), X(17), X(18), X(3070), X(3071).

**X(3054)**, center of the Evans Conic and 19 other triangle centers.

If you have GeoGebra, you can view **1st isogonic center**.

The Evans conic is introduced in **Evans, Lawrence S.,** "A Conic Through Six Triangle Centers," **Forum Geometricorum** 2 (2002) 89-92.

Let NaNbNc, Na'Nb'Nc' be the outer and inner Napoleon triangles, respectively. Let A' be the isogonal conjugate of Na', wrt NaNbNc, and define B' and C' cyclically. The lines NaA', NbB', NcC' concur in X(13). (Randy Hutson, January 29, 2015)

Let P be a point inside triangle ABC such that the line AP bisects angle BPC, and NBP bisets CPA, and CP bisects APB. Then P = X(13). The locus of P such that AP bisects BPC is the circumcubic given by the barycentric equation
c^{2}xy^{2} - b^{2}xz^{2} + (a^{2} - b^{2} + c^{2})y^{2}z - (a^{2} + b^{2} - c^{2})yz^{2} = 0,
and the other two cubics are given cyclically. Bernard Gibert discusses these cubics as K053A, K053B, K053C; see
**Apollonian strophoids.** (Paul Hanna and Peter Moses, August 6, 2017)

The line X(13)X(15) is parallel to the Euler line, and the distance between the two lines is (SB - SC)(SC - SA)(SA - SB)/|(cot(ω) + 3^{1/2})|*((E - 8F)S^{2})^{1/2}. (Kiminari Shinagawa, February 20, 2018)

Let A'B'C' be the outer Napoleon triangle and A"B"C" the inner Napoleon triangle. Let A* be the isogonal conjugate, wrt A'B'C', of A", and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(13). (Randy Hutson, December 2, 2017)

Let F be X(13) or X(14). Let L and L' be lines through F such that the angle between them is π/3; if you have GeoGebra, see Figure 13A. Let L_{BC} = L∩BC, and define L_{CA} and L_{AB} cyclically. Let L'_{BC} = L'∩BC, and define L'_{CA} and L'_{AB} cyclically. The lines L_{BC}L'_{CA}, L_{CA}L'_{AB}, L_{AB}L'_{BC} concur. (Dao Thanh Oai, 2014)

Let F be X(13) or X(14). Let A_{0}, B_{0}, C_{0} be points on BC, CA, AB, respectively, such that the directed angles FA_{0}-to-FC_{0} = π/3 and FC_{0}-to-FB_{0} = π/3; if you have GeoGebra, see Figure 13B. The points A_{0}, B_{0}, C_{0} are collinear. (Dao Thanh Oai, 2014)

X(13) lies on the Neuberg cubic and these lines:

2,16 3,17 4,61 5,18
6,14 11,202 15,30
76,299 80,1251 98,1080
99,303 148,617 203,1478
226,1081 262,383 275,472
298,532 484,1277 531,671
533,621 634,635

X(13) is the {X(6),X(381)}-harmonic conjugate of X(14). For a list
of other harmonic conjugates of X(13), click **Tables** at the top
of this page.

X(13) = reflection of X(i) in X(j) for these (i,j): (14,115), (15,396), (99,619), (298,623), (616,618)

X(13) = isogonal conjugate of X(15)

X(13) = isotomic conjugate of X(298)

X(13) = circumcircle-inverse of X(6104)

X(13) = orthocentroidal-circle inverse of X(14)

X(13) = complement of X(616)

X(13) = anticomplement of X(618)

X(13) = cevapoint of X(15) and X(62)

X(13) = X(i)-cross conjugate of X(j) for these (i,j): (15,18), (30,14), (396,2)

X(13) = trilinear pole of line X(395)X(523) (polar of X(470) wrt polar circle)

X(13) = pole wrt polar circle of trilinear polar of X(470)

X(13) = X(48)-isoconjugate (polar conjugate) of X(470)

X(13) = antigonal image of X(14)

X(13) = reflection of X(14) in line X(115)X(125)

X(13) = X(15)-of-4th-Brocard-triangle

X(13) = X(15)-of-orthocentroidal-triangle

X(13) = orthocorrespondent of X(13)

X(13) = homothetic center of outer Napoleon triangle and antipedal triangle of X(13)

X(13) = inner-Napoleon-to-outer-Napoleon similarity image of X(15)

X(13) = outer-Napoleon-isogonal conjugate of X(3)

X(13) = outer-Napoleon-to-inner-Napoleon similarity image of X(14)

X(13) = orthocenter of X(14)X(98)X(2394)

X(13) = X(15)-of-pedal-triangle of X(13)

X(13) = {X(265),X(1989)}-harmonic Conjugate of X(14)

X(13) = homothetic center of (equilateral) antipedal triangle of X(13) and triangle formed by circumcenters of BCX(13), CAX(13), ABX(13)

X(13) = homothetic center of triangle formed by circumcenters of BCX(14), CAX(14), ABX(14) and triangle formed by nine-point centers of BCX(13), CAX(13), ABX(13)

X(13) = Cundy-Parry Phi transform of X(17)

X(13) = Cundy-Parry Psi transform of X(61)

X(13) = Kosnita(X(13),X(1)) point

X(13) = Kosnita(X(13),X(13)) point

X(13) = Thomson-isogonal conjugate of X(34317)

= sec(A + π/6) : sec(B + π/6) : sec(C + π/6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where

f(a,b,c) = a^{4} - 2(b^{2} - c^{2})^{2} + a^{2}(b^{2} + c^{2} - 4*sqrt(3)*Area(ABC))

Barycentrics (SA - Sqrt[3] S) (SB + SC) + 4 SB SC : :

Tripolars 1/(a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4-2 Sqrt[3] a^2 S : :

X(14) = 3^{1/2}(r^{2} + 2rR + s^{2})*X(1) - 6r(3^{1/2}R + 2s)*X(2) + 2r(3^{1/2}r - 3s)*X(3) (Peter Moses, April 2, 2013)

Construct the equilateral triangle BA'C having base BC and vertex A' on the positive side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA',BB',CC' concur in X(14). The antipedal triangle of X(14) is equilateral.

Let NaNbNc, Na'Nb'Nc' be the outer and inner Napoleon triangles, resp. Let A' be the isogonal conjugate of Na, wrt Na'Nb'Nc', and define B' and C' cyclically. The lines Na'A', Nb'B', Nc'C' concur in X(14). (Randy Hutson, January 29, 2015)

Let A'B'C' be the outer Napoleon triangle and A"B"C" the inner Napoleon triangle. Let A* be the isogonal conjugate, wrt A"B"C", of A', and define B* and C* cyclically. The lines A"A*, B"B*, C"C* concur in X(14). (Randy Hutson, December 2, 2017)

The line X(14)X(16) is parallel to the Euler line, and the distance between the two lines is (SB - SC)(SC - SA)(SA - SB)/|(cot(ω) - 3^{1/2})|((E - 8F)S^{2})^{1/2}. (Kiminari Shinagawa, February 20, 2018)

If you have The Geometer's Sketchpad, you can view 2nd isogonic center

If you have GeoGebra, you can view **2nd isogonic center**.

X(14) lies on the Neuberg cubic and these lines:

2,15 3,18 4,62 5,17 6,13 11,203 16,30 76,298 98,383 99,302 148,616 202,1478 226,554 262,1080 275,473 299,533 397,546 484,1276 530,671 532,622 633,636

X(14) is the {X(6),X(381)}-harmonic conjugate of X(13). For a list of other harmonic conjugates of X(14), click **Tables** at the top of this page.

X(14) = reflection of X(i) in X(j) for these (i,j): (13,115), (16,395), (99,618), (299,624), (617,619)

X(14) = isogonal conjugate of X(16)

X(14) = isotomic conjugate of X(299)

X(14) = complement of X(617)

X(14) = anticomplement of X(619)

X(14) = circumcircle-inverse of X(6105)

X(14) = orthocentroidal-circle-inverse of X(13)

X(14) = cevapoint of X(16) and X(61)

X(14) = X(i)-cross conjugate of X(j) for these (i,j): (16,17), (30,13), (395,2)

X(14) = trilinear pole of line X(396)X(523) (polar of X(471) wrt polar circle)

X(14) = pole wrt polar circle of trilinear polar of X(471)

X(14) = X(48)-isoconjugate (polar conjugate) of X(471)

X(14) = antigonal image of X(13)

X(14) = reflection of X(13) in line X(115)X(125)

X(14) = X(16)-of-4th-Brocard triangle

X(14) = X(16)-of-orthocentroidal-triangle

X(14) = orthocorrespondent of X(14)

X(14) = homothetic center of inner Napoleon triangle and antipedal triangle of X(14)

X(14) = inner-Napoleon-isogonal conjugate of X(3)

X(14) = outer-Napoleon-to-inner-Napoleon similarity image of X(16)

X(14) = inner-Napoleon-to-outer-Napoleon similarity image of X(13)

X(14) = orthocenter of X(13)X(98)X(2394)

X(14) = X(16)-of-pedal-triangle of X(14)

X(14) = {X(265),X(1989)}-harmonic Conjugate of X(13)

X(14) = homothetic center of (equilateral) antipedal triangle of X(14) and triangle formed by circumcenters of BCX(14), CAX(14), ABX(14)

X(14) = homothetic center of triangle formed by circumcenters of BCX(13), CAX(13), ABX(13) and triangle formed by nine-point centers of BCX(14), CAX(14), ABX(14)

X(14) = Cundy-Parry Phi transform of X(18)

X(14) = Cundy-Parry Psi transform of X(62)

X(14) = Kosnita(X(14),X(1)) point

X(14) = Kosnita(X(14),X(14)) point

X(14) = Thomson-isogonal conjugate of X(34318)

Trilinears cos(A - π/6) : cos(B - π/6) : cos(C - π/6)

Trilinears 3 cos A + sqrt(3) sin A : :

Barycentrics a sin(A + π/3) : b sin(B + π/3) : c sin(C + π/3)

Barycentrics (SB + SC) ((SA + Sqrt[3] S) (SB + SC) + 4 SB SC) : :

Tripolars b c : :

Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through one vertex and both isodynamic points. (This configuration, based on the cevians of X(1), generalizes to arbitrary cevians; see TCCT, p. 98, problem 8.)

Let A'B'C' be the 4th Brocard triangle and A"B"C" be the 4th anti-Brocard triangle. The circumcircles of AA'A", BB'B", CC'C" concur in two points, X(15) and X(16). (Randy Hutson, July 20, 2016)

The line X(13)X(15) is parallel to the Euler line, and the distance between the two lines is (SB - SC)(SC - SA)(SA - SB)/|(cot(ω) + 3^{1/2})|*((E - 8F)S^{2})^{1/2}. (Kiminari Shinagawa, February 20, 2018)

The pedal triangle of X(15) is equilateral. If you have The Geometer's Sketchpad, you can view
1st isodynamic point and X(15)&X(16), with Brocard axis and Lemoine axis.

If you have GeoGebra, you can view **1st isodynamic point**.

X(15) lies on the Parry circle, Neuberg cubic, and these lines:

1,1251 2,14 3,6 4,17
13,30 18,140 35,1250
36,202 55,203 298,533
303,316 395,549 397,550
532,616 628,636 1337,2981

X(15) is the {X(3),X(6)}-harmonic conjugate of X(16). For a list of other harmonic conjugates of X(15), click **Tables** at the top of this page.

X(15) = reflection of X(i) in X(j) for these (i,j): (13,396), (16,187), (298,618), (316,624), (621,623)

X(15) = isogonal conjugate of X(13)

X(15) = isotomic conjugate of X(300)

X(15) = complement of X(621)

X(15) = anticomplement of X(623)

X(15) = circumcircle-inverse of X(16)

X(15) = Brocard-circle-inverse of X(16)

X(15) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,202), (13,62), (74,16)

X(15) = crosspoint of X(i) and X(j) for these (i,j): (13,18), (298,470)

X(15) = crosssum of X(i) and X(j) for these (i,j): (15,62), (532,619)

X(15) = crossdifference of every pair of points on line X(395)X(523)

X(15) = X(6)-Hirst inverse of X(16)

X(15) = X(15)-of-2nd-Brocard-triangle

X(15) = X(15)-of-circumsymmedial-triangle

X(15) = {X(371),X(372)}-harmonic conjugate of X(61)

X(15) = X(75)-isoconjugate of X(3457)

X(15) = X(1577)-isoconjugate of X(5995)

X(15) = outer-Napoleon-to-inner-Napoleon similarity image of X(13)

X(15) = orthocentroidal-to-ABC similarity image of X(13)

X(15) = 4th-Brocard-to-circumsymmedial similarity image of X(13)

X(15) = X(2378)-of-2nd-Parry triangle

X(15) = radical center of Lucas(2/sqrt(3)) circles

X(15) = homothetic center of (equilateral) 1st isogonal triangle of X(13) and pedal triangle of X(15)

X(15) = homothetic center of (equilateral) 1st isogonal triangle of X(14) and triangle formed by circumcenters of BCX(14), CAX(14), ABX(14)

X(15) = eigencenter of inner Napoleon triangle

X(15) = X(13)-of-4th-anti-Brocard-triangle

X(15) = X(15)-of-X(3)PU(1)

X(15) = Thomson-isogonal conjugate of X(5463)

X(15) = X(10657)-of-orthocentroidal-triangle

X(15) = {X(16),X(61)}-harmonic conjugate of X(6)

X(15) = Cundy-Parry Phi transform of X(61)

X(15) = Cundy-Parry Psi transform of X(17)

Trilinears cos(A + π/6) : cos(B + π/6) : cos(C + π/6)

Trilinears 3 cos A - sqrt(3) sin A : :

Barycentrics a sin(A - π/3) : b sin(B - π/3) : c sin(C- π/3)

Barycentrics (SB + SC) ((SA - Sqrt[3] S) (SB + SC) + 4 SB SC) : :

Tripolars b c : :

Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through a vertex and both isodynamic points. The pedal triangle of X(16) is equilateral. If you have The Geometer's Sketchpad, you can view 2nd isodynamic point.

If you have GeoGebra, you can view **2nd isodynamic point**.

Let A'B'C' be the 4th Brocard triangle and A"B"C" be the 4th anti-Brocard triangle. The circumcircles of AA'A", BB'B", CC'C" concur in two points, X(15) and X(16). (Randy Hutson, July 20, 2016)

The line X(14)X(16) is parallel to the Euler line, and the distance between the two lines is (SB - SC)(SC - SA)(SA - SB)/|(cot(ω) - 3^{1/2})|((E - 8F)S^{2})^{1/2}. (Kiminari Shinagawa, February 20, 2018)

X(16) lies on the Parry circle, Neuberg cubic, and these lines:

1,1250 2,13 3,6 4,18
14,30 17,140 36,203
55,202 299,532 302,316
358,1135 396,549 398,550
533,617 627,635 1338,3458

X(16) is the {X(3),X(6)}-harmonic conjugate of X(15). For a list of other harmonic conjugates of X(16), click **Tables** at the top of this page.

X(16) = reflection of X(i) in X(j) for these (i,j): (14,395), (15,187), (299,619), (316,623), (622,624)

X(16) = isogonal conjugate of X(14)

X(16) = isotomic conjugate of X(301)

X(16) = complement of X(622)

X(16) = anticomplement of X(624)

X(16) = circumcircle-inverse of X(15)

X(16) = Brocard-circle-inverse of X(15)

X(16) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,203), (14,61), (74,15)

X(16) = crosspoint of X(i) and X(j) for these (i,j): (14,17), (299,471)

X(16) = crosssum of X(i) and X(j) for these (i,j): (16,61), (533,618)

X(16) = crossdifference of every pair of points on line X(396)X(523)

X(16) = X(6)-Hirst inverse of X(15)

X(16) = X(16) of 2nd Brocard triangle

X(16) = X(16)-of-circumsymmedial-triangle

X(16) = {X(371),X(372)}-harmonic conjugate of X(62)

X(16) = X(75)-isoconjugate of X(3458)

X(16) = X(1577)-isoconjugate of X(5994)

X(16) = inner-Napoleon-to-outer-Napoleon similarity image of X(14)

X(16) = orthocentroidal-to-ABC similarity image of X(14)

X(16) = 4th-Brocard-to-circumsymmedial similarity image of X(14)

X(16) = X(2379)-of-2nd-Parry-triangle

X(16) = homothetic center of (equilateral) 1st isogonal triangle of X(14) and pedal triangle of X(16)

X(16) = homothetic center of (equilateral) 1st isogonal triangle of X(13) and triangle formed by circumcenters of BCX(13), CAX(13), ABX(13)

X(16) = radical center of Lucas(-2/sqrt(3)) circles

X(16) = eigencenter of outer Napoleon triangle

X(16) = X(14) of 4th anti-Brocard triangle

X(16) = X(16)-of-X(3)PU(1)

X(16) = Thomson-isogonal conjugate of X(5464)

X(16) = X(10658)-of-orthocentroidal-triangle

X(16) = {X(15),X(62)}-harmonic conjugate of X(6)

X(16) = Cundy-Parry Phi transform of X(62)

X(16) = Cundy-Parry Psi transform of X(18)

Trilinears sec(A - π/3) : sec(B - π/3) : sec(C - π/3)

Barycentrics a csc(A + π/6) : b csc(B + π/6) : c csc(C + π/6)

Tripolars (a^2-b^2-c^2-2 Sqrt[3] S) Sqrt[(a^2-3 b^2-3 c^2-2 Sqrt[3] S)] : :

Let X,Y,Z be the centers of the equilateral triangles in the construction of X(13). The lines AX, BY, CZ concur in X(17).

**John Rigby,** "Napoleon revisited," *Journal of Geometry,*33 (1988) 126-146.

If you have The Geometer's Sketchpad, you can view 1st Napoleon point.

If you have GeoGebra, you can view **1st Napoleon point**.

X(17) lies on the Napoleon cubic and these lines:

2,62 3,13 4,15 5,14 6,18 12,203 16,140 76,303 83,624 202,499 275,471 299,635 623,633

X(17) is the {X(231),X(1209)}-harmonic conjugate of X(18). For a list of other harmonic conjugates of X(17), click **Tables** at the top of this page.

X(17) = reflection of X(627) in X(629)

X(17) = isogonal conjugate of X(61)

X(17) = isotomic conjugate of X(302)

X(17) = complement of X(627)

X(17) = anticomplement of X(629)

X(17) = circumcircle-inverse of X(32627)

X(17) = X(i)-cross conjugate of X(j) for these (i,j): (16,14), (140,18), (397,4)

X(17) = polar conjugate of X(473)

X(17) = trilinear product of vertices of outer Napoleon triangle

X(17) = Kosnita(X(13),X(3)) point

X(17) = Kosnita(X(17),X(17)) point

X(17) = Cundy-Parry Phi transform of X(13)

X(17) = Cundy-Parry Psi transform of X(15)

Trilinears sec(A + π/3) : sec(B + π/3) : sec(C + π/3)

Barycentrics a csc(A - π/6) : b csc(B - π/6) : c csc(C - π/6)

Tripolars (a^2-b^2-c^2+2 Sqrt[3] S) Sqrt[(a^2-3 b^2-3 c^2+2 Sqrt[3] S)] : :

Let X,Y,Z be the centers of the equilateral triangles in the construction of X(14). The lines AX, BY, CZ concur in X(18).

If you have The Geometer's Sketchpad, you can view 2nd Napoleon point.

If you have GeoGebra, you can view **2nd Napoleon point**.

X(18) lies on the Napoleon cubic and these lines: 2,61 3,14 4,16 5,13 6,17 12,202 15,140 76,302 83,623 203,499 275,470 298,636 624,634

X(18) is the {X(231),X(1209)}-harmonic conjugate of X(17). For a list of other harmonic conjugates of X(18), click **Tables** at the top of this page.

X(18) = reflection of X(628) in X(630)

X(18) = isogonal conjugate of X(62)

X(18) = isotomic conjugate of X(303)

X(18) = complement of X(628)

X(18) = anticomplement of X(630)

X(18) = circumcircle-inverse of X(32628)

X(18) = X(i)-cross conjugate of X(j) for these (i,j): (15,13), (140,17), (398,4)

X(18) = polar conjugate of X(472)

X(18) = trilinear product of vertices of inner Napoleon triangle

X(18) = Kosnita(X(14),X(3)) point

X(18) = Cundy-Parry Phi transform of X(14)

X(18) = Cundy-Parry Psi transform of X(16)

Trilinears sin 2B + sin 2C - sin 2A : :

Trilinears 1/(b

Trilinears a

Trilinears SB*SC : :

Barycentrics a tan A : b tan B : c tan C

Tripolars (-a^2+b^2+c^2) Sqrt[b c (a^6-a^4 b^2-a^2 b^4+b^6-2 a^4 b c-2 b^5 c-a^4 c^2+2 a^2 b^2 c^2-b^4 c^2+4 b^3 c^3-a^2 c^4-b^2 c^4-2 b c^5+c^6)] : :

X(19) = (r + 2R - s)(r + 2R + s)*X(1) - 6R(r + 2R)*X(2) - 2(r

X(19) is the homothetic center of the orthic and extangents triangles. The Ayme triangle, constructed at X(3610), is perspective to ABC with perspector X(19).

If you have The Geometer's Sketchpad, you can view Clawson point.

If you have GeoGebra, you can view **Clawson point**.

Further information is available from

Paul Yiu's Website.

Although John Clawson studied this point in 1925, it was studied earlier by Lemoine:

**Emile Lemoine,** "Quelques questions se rapportant à l'étude des antiparallèles des côtes d'un triangle", *Bulletin de la S. M.
F.,* tome 14 (1886), p. 107-128, specifically, on page 114. This article is available online at Numdam.

Let A'B'C' be the 4th Brocard triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(19). (Randy Hutson, November 18, 2015)

Let A'B'C' be the orthic triangle. Let A" be the trilinear product of the (real or imaginary) circumcircle intercepts of line B'C'. Define B" and C" cyclically. The lines AA", BB", CC" concur in X(19). (Randy Hutson, December 26, 2015)

Let A'B'C' be the excentral triangle. Let A" be the pole, wrt the polar circle, of line B'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(19). (Randy Hutson, December 2, 2017)

Let A'B'C' be the hexyl triangle. Let La be the trilinear polar of A', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(19). (Randy Hutson, December 2, 2017)

Let A'B'C' be the hexyl triangle. Let Ab = BC∩C'A', Ac = BC∩A'B', and define Bc, Ba, Ca, Cb cyclically. Then Ab, Ac, Bc, Ba, Ca, Cb lie on an ellipse. Let A" be the intersection of the tangents to the ellipse at Ba and Ca, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(19). (Randy Hutson, December 2, 2017)

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is homothetic to the anti-Ara triangle at X(19). (Randy Hutson, December 2, 2017)

Let La be the A-extraversion of line X(650)X(663) (the trilinear polar of X(9)), and define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(19). (Randy Hutson, December 2, 2017)

Let La be the A-extraversion of line X(661)X(663) (the trilinear polar of X(19)), and define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(19). (Randy Hutson, December 2, 2017)

X(19) lies on these lines:

1,28 2,534 3,1871 4,9
6,34 8,1891 25,33 27,63
31,204 41,1825 44,1828
45,1900 46,579 47,921
53,1846 56,207 57,196
64,1903 81,969 91,920
101,913 102,282 112,759
158,1712 162,897 163,563
208,225 219,517 220,1902
226,1763 232,444 273,653
294,1041 318,1840 379,1441
407,1865 429,1213 560,1910
604,909 672,1851 960,965
1158,1715 1212,1593 1405,1866
1449,1870 1581,1740 1598,1872
1633,1721 1707,1719 1708,1713
1743,1783 1836,1901 1837,1852

X(19) is the {X(607),X(608)}-harmonic conjugate of X(6). For a list
of other harmonic conjugates of X(19), click **Tables** at the top
of this page.

X(19) = isogonal conjugate of X(63)

X(19) = isotomic conjugate of X(304)

X(19) = circumcircle-inverse of X(32756)

X(19) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,204), (4,33), (27,4), (28,25), (57,208), (92,1), (196,207), (278,34)

X(19) = X(i)-cross conjugate of X(j) for these (i,j): (25,34), (31,1)

X(19) = crosspoint of X(i) and X(j) for these (i,j): (4,278), (27,28), (57,84), (92,158)

X(19) = crosssum of X(i) and X(j) for these (i,j): (1,610), (3,219), (9,40), (48,255), (71,72)

X(19) = crossdifference of every pair of points on line X(521)X(656)

X(19) = X(i)-Hirst inverse of X(j) for these (i,j): (1,240), (4,242)

X(19) = X(i)-aleph conjugate of X(j) for these (i,j): (2,610), (92,19), (508,223), (648,163)

X(19) = X(i)-beth conjugate of X(j) for these (i,j): (9,198), (19,608), (112,604), (281,281), (648,273), (653,19)

X(19) = inverse-in-polar-circle of X(5179)

X(19) = inverse-in-circumconic-centered-at-X(9) of X(1861)

X(19) = Zosma transform of X(9

X(19) = perspector of ABC and extraversion triangle of X(19) (which is also the anticevian triangle of X(19))

X(19) = intersection of tangents at X(9) and X(57) to Thomson cubic K002

X(19) = intersection of tangents at X(40) and X(84) to Darboux cubic K004

X(19) = trilinear product of PU(i) for these i: 4, 23, 157

X(19) = barycentric product of PU(15)

X(19) = vertex conjugate of PU(19)

X(19) = bicentric sum of PU(127)

X(19) = PU(127)-harmonic conjugate of X(656)

X(19) = perspector of ABC and unary cofactor triangle of hexyl triangle

X(19) = perspector of unary cofactor triangles of 2nd and 4th extouch triangles

X(19) = perspector of ABC and unary cofactor triangle of extraversion triangle of X(9)

X(19) = complement of X(4329)

X(19) = {X(48),X(1953)}-harmonic conjugate of X(1)

X(19) = {X(92),X(1748)}-harmonic conjugate of X(63)

X(19) = trilinear product X(2)*X(25)

X(19) = trilinear pole of line X(661)X(663) (the polar of X(75) wrt polar circle)

X(19) = pole wrt polar circle of trilinear polar of X(75) (line X(514)X(661))

X(19) = polar conjugate of X(75)

X(19) = X(i)-isoconjugate of X(j) for these {i,j}: {1,63}, {6,69}, {31,304}, {48,75}, {67, 22151}, {92,255}

X(19) = X(571)-of-excentral-triangle

X(19) = perspector, wrt excentral triangle, of polar circle

2- 5, 140, 186, 199, 235, 237, 297, 376- 379, 381- 384,

401- 475, 546- 550, 631, 632 (and others) lie on the Euler line.

Trilinears sec A - sec B sec C : sec B - sec C sec A : sec C - sec A sec B

Trilinears 2 cos A - sin B sin C : 2 cos B - sin C sin A : 2 cos C - sin A sin B

Trilinears (csc A)(tan B + tan C - tan A) : :

Barycentrics tan B + tan C - tan A : tan C + tan A - tan B : tan A + tan B - tan C

Barycentrics [-3a

Barycentrics S^2 - 2 SB SC : :

Tripolars Sqrt[a^6-3 a^2 b^4+2 b^6+6 a^2 b^2 c^2-2 b^4 c^2-3 a^2 c^4-2 b^2 c^4+2 c^6] : :

X(20) = (1 - J) X(1113) + (1 + J) X(1114)

X(20) = 9 X(2) - 8 X(5) = 3 X(4) - 4 X(5) = 3 X(3) - 2 X(5) = 15 X(2) - 16 X(140) = 5 X(4) - 8 X(140) = 5 X(5) - 6 X(140) = 5 X(3) - 4 X(140) = 2 X(10) - 3 X(165) = 8 X(140) - 15 X(376) = 4 X(5) - 9 X(376) = 2 X(3) - 3 X(376) = X(4) - 3 X(376) = 10 X(5) - 9 X(381) = 5 X(4) - 6 X(381) = 5 X(2) - 4 X(381) = 5 X(3) - 3 X(381) = 4 X(140) - 3 X(381) = 5 X(376) - 2 X(381) = 12 X(140) - 5 X(382) = 9 X(381) - 5 X(382) = 9 X(2) - 4 X(382) = 3 X(4) - 2 X(382) = 9 X(376) - 2 X(382) = 3 X(3) - X(382) = 7 X(382) - 12 X(546)

As a point on the Euler line, X(20) has Shinagawa coefficients (1, -2).

Let La be the polar of X(4) wrt the circle centered at A and passing through X(3), and define Lb and Lc cyclically. (Note: X(4) is the perspector of any circle centered at a vertex of ABC.) Let A" = Lb∩\Lc, and define B" and C" cyclically. Triangle A"B"C" is homothetic to the anticomplementary triangle, and the center of homothety is X(20), which is also the orthocenter of A"B"C". Also, let La be the line through the intersections of the B- and C-Soddy ellipses, and define Lb and Lc cyclically. Then La,Lb,Lc concur in X(20). Also, let A'B'C' be the cevian triangle of X(253). Let A" be the orthocenter of AB'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(20). (Randy Hutson, November 18, 2015)

Let L be the Brocard axis of the intouch triangle. Let La be the Brocard axis of the A-extouch triangle, and define Lb and Lc cyclically. The lines L, La, Lb, Lc concur in X(20). (Randy Hutson, September 14, 2016)

Let A' be the reflection in BC of the A-vertex of the anticevian triangle of X(4), and define B' and C' cyclically. The circumcircles of A'BC, B'CA, and C'AB concur at X(20). (Randy Hutson, December 10, 2016)

Let A'B'C' be the reflection of ABC in X(3) (i.e., the circumcevian triangle of X(3)). Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", and CC" concur in X(20). (Randy Hutson, December 10, 2016)

Let A'B'C' be the hexyl triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(20).

Let A'B'C' be the half-altitude triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(20).

Let A'B'C' be the hexyl triangle and A"B"C" be the side-triangle of ABC and hexyl triangle. Let A* be the {B',C'}-harmonic conjugate of A", and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(20). (Randy Hutson, June 27, 2018)

If you have The Geometer's Sketchpad, you can view De Longchamps point.

If you have GeoGebra, you can view **De Longchamps point**.

X(20) lies on the the following curves: Q046, Q063, Q070, Q073, Q115, K004, K007, K032, K041, K047, K071, K077, K080, K096, K099, K122, K169, K182, K236, K268, K270, K313, K329, K344, K364, K401, K425, K426, K443, K449, K462, K499, K522, K566, K609, K617, K648, K649, K650, K651, K652, K706, K753, K763, K778, K809, K814, K824, K825, K827, K850, K894. Euler-Gergonne-Soddy circle, GEOS circle, Steiner/Wallace rectangular hyperbola, anticomplement of Kiepert hyperbola, anticomplement of Feuerbach hyperbola, anticomplement of Jerabek hyperbola, and these lines:

{1,7}, {2,3}, {6,6459}, {8,40}, {9,10429}, {10,165}, {11,5204}, {12,5217}, {13,5238}, {14,5237}, {15,3412}, {16,3411}, {17,5352}, {18,5351}, {32,2549}, {33,1038}, {34,1040}, {35,1478}, {36,1479}, {39,7737}, {46,10572}, {51,9729}, {52,5890}, {54,4846}, {55,388}, {56,497}, {57,938}, {58,387}, {61,10653}, {62,10654}, {64,69}, {65,3474}, {68,74}, {72,144}, {76,3424}, {78,329}, {81,5706}, {97,1217}, {98,148}, {99,147}, {100,153}, {101,152}, {103,150}, {104,149}, {107,3184}, {109,151}, {110,146}, {112,10316}, {113,10721}, {114,7912}, {115,5206}, {116,10725}, {117,10726}, {118,10727}, {119,10728}, {120,10729}, {121,10730}, {122,10152}, {123,10731}, {124,10732}, {125,10733}, {126,10734}, {127,10735}, {142,5436}, {145,517}, {154,2883}, {155,323}, {159,2139}, {172,9598}, {182,7787}, {184,9545}, {185,193}, {187,3767}, {190,1265}, {192,9962}, {200,5815}, {212,1935}, {216,3087}, {220,5781}, {222,3562}, {224,4511}, {226,3601}, {227,9371}, {230,5023}, {243,1118}, {254,1300}, {262,5395}, {265,11270}, {284,5746}, {298,5868}, {299,5869}, {316,7763}, {325,6337}, {333,5786}, {343,6247}, {345,7270}, {346,1766}, {348,4872}, {355,3579}, {371,1587}, {372,1588}, {385,6392}, {386,9535}, {389,3060}, {391,573}, {392,9856}, {393,577}, {394,1032}, {395,5339}, {396,5340}, {399,6188}, {476,2693}, {477,10420}, {484,10573}, {485,1131}, {486,1132}, {487,638}, {488,637}, {495,9655}, {496,9668}, {498,3585}, {499,3583}, {518,3189}, {519,5493}, {524,11148}, {527,11523}, {529,3913}, {535,5537}, {541,9143}, {542,8591}, {543,7751}, {551,11522}, {553,11518}, {568,10263}, {574,2548}, {576,5032}, {578,5012}, {579,5802}, {590,6409}, {597,10541}, {601,3072}, {602,3073}, {603,1936}, {610,8804}, {615,6410}, {616,633}, {617,634}, {620,7825}, {621,627}, {622,628}, {648,9530}, {650,8142}, {651,7078}, {653,3176}, {664,7973}, {671,11623}, {691,2697}, {754,7758}, {901,2734}, {908,4855}, {910,6554}, {936,1750}, {942,3488}, {946,3576}, {952,3621}, {956,5082}, {958,2550}, {960,5698}, {986,11031}, {999,1058}, {1001,8273}, {1007,7773}, {1056,3295}, {1060,6198}, {1062,1870}, {1074,1838}, {1075,5667}, {1076,1785}, {1078,7616}, {1104,4000}, {1124,9660}, {1125,1699}, {1141,11671}, {1147,1614}, {1151,3068}, {1152,3069}, {1154,11271}, {1155,1788}, {1160,10784}, {1161,10783}, {1176,10548}, {1181,1993}, {1204,1899}, {1210,3586}, {1212,5819}, {1216,4549}, {1249,3172}, {1290,2694}, {1293,2370}, {1296,2373}, {1320,10305}, {1327,3590}, {1328,3591}, {1330,3430}, {1335,9647}, {1340,2542}, {1341,2543}, {1342,2546}, {1343,2547}, {1351,7839}, {1352,2896}, {1376,2551}, {1384,5305}, {1385,3622}, {1394,5930}, {1420,9580}, {1440,1804}, {1445,5809}, {1453,5222}, {1482,3623}, {1483,8148}, {1499,6563}, {1511,7728}, {1519,4881}, {1568,11202}, {1578,3092}, {1579,3093}, {1610,1633}, {1619,9914}, {1621,11496}, {1632,2892}, {1689,2545}, {1690,2544}, {1697,10106}, {1698,10164}, {1706,5795}, {1729,5011}, {1743,10443}, {1764,10449}, {1768,9803}, {1834,4252}, {1836,2646}, {1853,6696}, {1857,1940}, {1891,10319}, {1902,7718}, {1914,9597}, {1992,8550}, {1994,7592}, {2077,5080}, {2128,3685}, {2130,2131}, {2287,5776}, {2420,6794}, {2456,10131}, {2482,7888}, {2781,6293}, {2782,5984}, {2797,9409}, {2800,6224}, {2801,5904}, {2822,2939}, {2823,4552}, {2888,3357}, {2893,10432}, {2894,2975}, {2899,5205}, {2917,2935}, {2944,3923}, {2947,3682}, {2979,5562}, {3047,5504}, {3053,5254}, {3054,5585}, {3057,3476}, {3058,3304}, {3062,5785}, {3095,7709}, {3180,5865}, {3181,5864}, {3182,3347}, {3183,3348}, {3218,5709}, {3219,3587}, {3241,5882}, {3244,11531}, {3278,3608}, {3303,5434}, {3311,7581}, {3312,7582}, {3313,5596}, {3316,6451}, {3317,6452}, {3333,10580}, {3334,3609}, {3339,6738}, {3353,3354}, {3355,3637}, {3359,5554}, {3361,11019}, {3398,10788}, {3419,3916}, {3421,5687}, {3431,3521}, {3452,5438}, {3472,3473}, {3475,10404}, {3564,7893}, {3567,5446}, {3598,3673}, {3618,5085}, {3619,10516}, {3624,3817}, {3634,7989}, {3635,11224}, {3648,5693}, {3655,10222}, {3666,5716}, {3667,5592}, {3697,9947}, {3734,7800}, {3788,7842}, {3796,11425}, {3812,10178}, {3813,11194}, {3849,7759}, {3869,6001}, {3870,6769}, {3871,10306}, {3872,9874}, {3876,5777}, {3911,5704}, {3917,5907}, {3933,10513}, {3935,5534}, {3972,7803}, {4257,5292}, {4385,7172}, {4640,5794}, {4652,5175}, {4678,5690}, {4848,5128}, {4857,10072}, {5007,7739}, {5013,7736}, {5044,5927}, {5126,11373}, {5174,6350}, {5208,10441}, {5223,6743}, {5226,9612}, {5247,9441}, {5250,9800}, {5270,10056}, {5303,11680}, {5316,9842}, {5318,11480}, {5321,11481}, {5328,6700}, {5418,6564}, {5420,6565}, {5422,10982}, {5432,10588}, {5433,10589}, {5439,5806}, {5440,5658}, {5441,5902}, {5447,5891}, {5450,10527}, {5462,9781}, {5550,8227}, {5587,6684}, {5601,9834}, {5602,9835}, {5640,10110}, {5654,7712}, {5663,6101}, {5714,11374}, {5720,5811}, {5730,10609}, {5749,10445}, {5758,5905}, {5766,8545}, {5841,10528}, {5853,6762}, {5876,10627}, {5893,10192}, {6102,6243}, {6146,6515}, {6193,6241}, {6197,9536}, {6214,10518}, {6215,10517}, {6221,7583}, {6249,9751}, {6264,9802}, {6326,9809}, {6390,7776}, {6398,7584}, {6449,8981}, {6455,8976}, {6462,6465}, {6463,6466}, {6526,11589}, {6680,7872}, {6737,7992}, {6744,10980}, {6765,7994}, {6766,9797}, {6767,10386}, {7074,9370}, {7596,10885}, {7618,7775}, {7620,8182}, {7694,7752}, {7730,11802}, {7731,11562}, {7749,8588}, {7755,11648}, {7761,7795}, {7768,11057}, {7774,7783}, {7784,7789}, {7785,9737}, {7797,9753}, {7799,7860}, {7801,7873}, {7818,7863}, {7820,7935}, {7832,7910}, {7835,7911}, {7836,7898}, {7864,9748}, {7885,7891}, {7921,10983}, {7971,11682}, {7998,11439}, {8069,10629}, {8081,9793}, {8082,9795}, {8111,9783}, {8112,9787}, {8117,8118}, {8119,8124}, {8120,8123}, {8164,9654}, {8234,9789}, {8235,9791}, {8726,9776}, {8861,9474}, {8983,9615}, {9529,9979}, {9786,11433}, {9927,11468}, {9957,11035}, {9993,10583}, {9996,10357}, {10246,10595}, {10267,10532}, {10269,10531}, {10282,11449}, {10359,10796}, {10453,10476}, {10470,10478}, {10525,10785}, {10526,10786}, {10543,11246}, {10584,10893}, {10585,10894}, {10601,11745}, {10679,10805}, {10680,10806}, {11180,11645}, {11451,11695}, {11470,11511}, {11472,11487}, {11473,11513}, {11474,11514}, {11475,11515}, {11476,11516}

X(20) = midpoint of X(i) and X(j) for these {i,j}: {3, 1657}, {4, 3529}, {376, 11001}, {944, 6361}, {1498, 5925}, {3146, 5059}, {3869, 9961}, {6241, 11412}, {10575, 10625}

X(20) = reflection of X(i) in X(j) for these (i,j): (1, 4297), (2, 376), (3, 550), (4, 3), (5, 548), (7, 5732), (8, 40), (23, 10295), (64, 5894), (65, 9943), (68, 7689), (69, 1350), (76, 5188), (107, 3184), (144, 5759), (145, 944), (146, 110), (147, 99), (148, 98), (149, 104), (150, 103), (151, 109), (152, 101), (153, 100), (176, 8984), (193, 6776), (194, 11257), (329, 6282), (355, 3579), (376, 3534), (381, 8703), (382, 5), (616, 5473), (617, 5474), (650, 8142), (938, 9841), (962, 1), (1330, 3430), (1352, 3098), (1375, 8153), (2475, 3651), (2550, 11495), (2888, 7691), (3091, 3522), (3146, 4), (3153, 2071), (3421, 6244), (3434, 3428), (3436, 10310), (3448, 74), (3529, 1657), (3543, 2), (3627, 140), (3830, 549), (3832, 3528), (3839, 10304), (3853, 3530), (3868, 1071), (4846, 8717), (5059, 3529), (5073, 3627), (5080, 2077), (5189, 7464), (5691, 10), (5768, 7171), (5876, 10627), (5878, 6759), (5881, 11362), (5889, 185), (5895, 2883), (5921, 69), (5984, 9862), (6223, 1490), (6225, 1498), (6241, 10575), (6243, 6102), (6256, 6796), (6515, 10605), (6655, 7470), (6764, 6762), (6839, 7411), (6840, 6909), (6895, 6906), (6925, 7580), (7379, 4229), (7391, 378), (7408, 3537), (7620, 8182), (7710, 8719), (7728, 1511), (7731, 11562), (7758, 7781), (7982, 5882), (7991, 5493), (8148, 1483), (9589, 4301), (9797, 9845), (9799, 84), (9802, 6264), (9803, 1768), (9809, 6326), (9812, 3576), (9863, 7750), (9965, 2096), (10152, 122), (10296, 858), (10431, 1012), (10446, 991), (10721, 113), (10722, 114), (10723, 115), (10724, 11), (10725, 116), (10726, 117), (10727, 118), (10728, 119), (10729, 120), (10730, 121), (10731, 123), (10732, 124), (10733, 125), (10734, 126), (10735, 127), (10736, 1313), (10737, 1312), (11185, 8722), (11381, 5907), (11412, 10625), (11455, 5891), (11477, 8550), (11531, 3244), (11541, 5073), (11671, 1141)

X(20) = isogonal conjugate of X(64)

X(20) = isotomic conjugate of X(253)

X(20) = cyclocevian conjugate of X(1032)

X(20) = complement of X(3146)

X(20) = anticomplement of X(4)

X(20) = anticomplementary conjugate of X(4)

X(20) = X(i)-Ceva conjugate of X(j) for these (i,j): (69,2), (489,487), (490,488), (801,6), (1043,1), (1350,6194), (1503,147), (1975,194), (5921,9742), (7750,2896), (8822,63)

X(20) = X(i)-cross conjugate of X(j) for these (i,j): (64,2131), (122,8057), (154,1249), (1249,2), (1498,6616), (3183,2060), (3198,610), (5895,4), (5930,1895), (6525,3344)

X(20) = crosspoint of X(1) and X(7038)

X(20) = crosssum of X(i) and X(j) for these (i,j): {1,1044}, {512,3269}, {649,3270}

X(20) = crossdifference of every pair of points on line X(647)X(657)

X(20) = trilinear pole of X(6587)X(8057)

X(20) = circumcircle-inverse of X(2071)

X(20) = orthocentroidal-circle-inverse of X(3091)

X(20) = Steiner-circle-inverse of X(858)

X(20) = polar-circle-inverse of X(10151)

X(20) = orthoptic-circle-of-Steiner-inellipse-inverse of X(5159)

X(20) = orthoptic-circle-of-Steiner-circumelipse-inverse of X(858)

X(20) = anticomplement-of-circumcircle-inverse of X(3153)

X(20) = X(i)-aleph conjugate of X(j) for these (i,j): (8,191), (9,1045), (21,3216), (29,1714), (188,1046), (333,2), (556,1762), (645,3882), (1043,20), (3699,4427), (4182,846), (6731,2938)

X(20) = X(i)-beth conjugate of X(j) for these (i,j): (8,5691), (20,1394), (21,4306), (643,1259), (664,20), (1043,280)

X(20) = X(i)-gimel conjugate of X(j) for these (i,j): (21,6848), (1792,20), (3900,20), (4397,20), (7253,20)

X(20) = X(i)-he conjugate of X(j) for these (i,j): (645,20), (799,20), (7256,20), (7258,20)

X(20) = X(i)-zayin conjugate of X(j) for these (i,j): (1,64), (200,7580), (1043,20), (2287,573), (4397,3667), (6737,40)

X(20) = antigonal conjugate of X(10152)

X(20) = syngonal conjugate of X(3184)

X(20) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1,4), (3,8), (4,5906), (6,5905), (19,6515), (31,193), (47,6193), (48,2), (55,5942), (58,3868), (63,69), (69,6327), (71,2895), (72,1330), (73,2475), (75,11442), (77,3434), (78,3436), (82,3060), (91,68), (92,317), (101,4391), (102,5081), (109,521), (110,7253), (162,520), (163,525), (184,192), (212,144), (219,329), (222,7), (228,1654), (255,20), (268,189), (283,3869), (284,92), (293,511), (295,4645), (304,315), (326,1370), (394,4329), (577,6360), (603,145), (656,3448), (662,850), (810,148), (905,150), (906,514), (921,11411), (922,7665), (947,318), (951,5174), (1069,11415), (1101,110), (1110,3732), (1214,2893), (1262,4566), (1331,513), (1333,3187), (1433,962), (1437,1), (1459,149), (1496,11469), (1790,75), (1794,72), (1795,517), (1796,319), (1797,320), (1803,85), (1807,5080), (1813,693), (1815,4872), (1822,2574), (1823,2575), (1923,10340), (1964,8878), (1973,6392), (2148,1993), (2149,651), (2159,3580), (2164,2994), (2167,264), (2168,5392), (2169,3), (2190,5889), (2193,63), (2196,6542), (2200,1655), (2216,52), (2349,340), (2359,321), (2360,1895), (2576,2592), (2577,2593), (2964,11271), (3916,2891), (3990,3151), (4020,2896), (4303,2894), (4558,7192), (4575,523), (4587,4462), (4592,512), (6507,6527), (7011,5932), (7015,4388), (7078,6223), (7099,4452), (7116,6646), (7125,347), (7177,6604), (9247,194), (9255,1899)

X(20) = X(3532)-complementary conjugate of X(10)

X(20) = X(i)-vertex conjugate of X(j) for these (i,j): {3,3346}, {4,5879}, {523,2071}

X(20) = X(4)-of-anticomplementary triangle

X(20) = X(52)-of-hexyl-triangle

X(20) = reflection of X(10296) in the De Longchamps line

X(20) = perspector of anticomplementary triangle and polar triangle of de Longchamps circle

X(20) = isogonal conjugate of X(4) wrt anticevian triangle of X(4) (or 'anticevian-isogonal conjugate of X(4)')

X(20) = perspector of ABC and pedal triangle of X(1498)

X(20) = exsimilicenter of circumcircle and 1st Steiner circle (the insimilicenter is X(631))

X(20) = X(4)-of-circumcevian-triangle-of-X(30)

X(20) = anticomplementary isotomic conjugate of X(193)

X(20) = excentral isogonal conjugate of X(1046)

X(20) = excentral isotomic conjugate of X(1045)

X(20) = cevapoint of X(i) and X(j) for these {i,j}: {1,3182}, {3,1498}, {4,3183}, {6,1661}, {30,3184}, {40,1490}, {64,2130}, {84,3353}, {122,8057}, {577,1660}, {610,7070}, {1249,3079}, {3198,8804}, {3345,3472}, {3346,3355}

X(20) = radical center of power circles

X(20) = radical center of circles centered at the vertices of ABC with radius equal to opposite side

X(20) = intersection of tangents to conic {X(4),X(13),X(14),X(15),X(16)} at X(15) and X(16)

X(20) = trilinear pole wrt anticomplementary triangle of de Longchamps line

X(20) = trilinear pole of polar of X(459) wrt polar circle, which is also the perspectrix of ABC and the half-altitude triangle

X(20) = pole wrt polar circle of trilinear polar of X(459)

X(20) = isoconjugate of X(j) and X(j) for these (i,j): {1,64}, {2,2155}, {6,2184}, {19,1073}, {31,253}, {48,459}, {55,8809}, {255,6526}, {656,1301}, {1036,10375}, {1402,5931}, {2190,8798}

X(20) = circumcevian isogonal conjugate of X(3)

X(20) = perspector of ABC and circumcircle antipode of circumanticevian triangle of X(4)

X(20) = X(98)-of-6th-Brocard-triangle

X(20) = perspector of hexyl triangle and cross-triangle of ABC and hexyl triangle

X(20) = Thomson isogonal conjugate of X(3167)

X(20) = Lucas isogonal conjugate of X(2)

X(20) = inner-Conway-to-Conway similarity image of X(8)

X(20) = cyclocevian conjugate of X(2) wrt anticevian triangle of X(2)

X(20) = trilinear product of vertices of X(4)-anti-altimedial triangle

X(20) = homothetic center of X(20)-altimedial and X(2)-anti-altimedial triangles

X(20) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(5159)

X(20) = inverse-in-circumconic-centered-at-X(4) of X(1559)

X(20) = anticevian isogonal conjugate of X(4)

X(20) = X(5562)-of-excentral-triangle

X(20) = X(74)-of-X(3)-Fuhrmann-triangle

X(20) = Ehrmann-mid-to-Johnson similarity image of X(3)

X(20) = perspector of hexyl triangle and anticevian triangle of X(63)

X(20) = perspector of excentral triangle and tangential triangle wrt hexyl triangle of the excentral-hexyl ellipse

X(20) = perspector of excentral triangle and extraversion triangle of X(7)

X(20) = homothetic center of ABC and the reflection in X(3) of the pedal triangle of X(3) (medial triangle)

X(20) = homothetic center of ABC and the reflection in X(4) of the antipedal triangle of X(4) (anticomplementary triangle)

X(20) = orthic-isogonal conjugate of X(32605)

X(20) = QA-P5 (Isotomic Center) of the incenter-excenters quadrangle (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/26-qa-p5.html)

X(20) = barycentric product X(i)*X(j) for these {i,j}: {63,1895}, {69,1249}, {75,610}, {76,154}, {85,7070}, {86,8804}, {99,6587}, {204,304}, {274,3198}, {305,3172}, {312,1394}, {333,5930}, {648,8057}, {801,2883}, {1032,6616}, {1097,2184}, {3213,3718}, {3344,6527}, {3926,6525}, {7156,7182}, {10152,11064}

X(20) = barycentric quotient X(i)/X(j) for these (i,j): (1,2184), (2,253), (3,1073), (4,459), (6,64), (31,2155), (57,8809), (112,1301), (154,6), (204,19), (216,8798), (333,5931), (393,6526), (610,1), (1249,4), (1394,57), (1498,3343), (1562,125), (1895,92), (2285,10375), (3079,1249), (3172,25), (3198,37), (3213,34), (3284,11589), (3344,3346), (5930,226), (6525,393), (6587,523), (7070,9), (7156,33), (8057,525), (8804,10)

X(20) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7,11036), (1,1044,1042), (1,1742,4300), (1,1770,4295), (1,3100,9538), (1,4292,7), (1,4293,3600), (1,4294,390), (1,4297,5731), (1,4298,11037), (1,4299,4293), (1,4301,5734), (1,4302,4294), (1,4304,4313), (1,4311,4308), (1,4312,3671), (1,4316,4299), (1,4324,4302), (1,4325,4317), (1,4330,4309), (1,4333,1770), (1,4340,3945), (1,4355,5542), (1,5732,10884), (1,9589,4301), (1,10624,9785), (2,3,3523), (2,4,3091), (2,5,7486), (2,22,10565), (2,23,4232), (2,376,10304), (2,377,4208), (2,452,5129), (2,1370,7396), (2,2475,5177), (2,3091,5056), (2,3146,4), (2,3522,3), (2,3523,10303), (2,3543,3839), (2,3832,5), (2,4190,6904), (2,5046,6919), (2,5059,3146), (2,5068,3090), (2,6837,6884), (2,6838,6960), (2,6839,6993), (2,6847,6888), (2,6848,6979), (2,6872,452), (2,6890,6972), (2,6987,6992), (2,6995,7398), (2,7391,7378), (2,7408,6997), (2,7409,5133), (2,7500,6995), (2,11106,405), (3,4,2), (3,5,631), (3,22,7488), (3,26,186), (3,140,3524), (3,376,3522), (3,381,140), (3,382,5), (3,405,6986), (3,546,3525), (3,548,3528), (3,549,10299), (3,550,376), (3,1012,21), (3,1532,6921), (3,1593,7503), (3,1597,7395), (3,1656,549), (3,1885,6816), (3,2937,1658), (3,3079,2060), (3,3091,10303), (3,3146,3091), (3,3149,404), (3,3522,10304), (3,3526,3530), (3,3529,3146), (3,3534,550), (3,3543,5056), (3,3560,1006), (3,3575,6815), (3,3627,3090), (3,3830,1656), (3,3843,3526), (3,3845,3533), (3,3851,5054), (3,3853,5067), (3,5059,3543), (3,5073,381), (3,5076,3628), (3,6638,417), (3,6756,6803), (3,6823,7494), (3,6827,6926), (3,6831,6910), (3,6836,6890), (3,6840,6972), (3,6842,6954), (3,6850,6908), (3,6851,6847), (3,6868,6987), (3,6872,6992), (3,6882,6961), (3,6895,6888), (3,6905,4188), (3,6906,4189), (3,6907,6988), (3,6911,6940), (3,6914,6875), (3,6917,6889), (3,6923,6825), (3,6925,6838), (3,6928,6891), (3,6929,6967), (3,6934,4190), (3,6938,6872), (3,6985,6905), (3,7387,24), (3,7395,7485), (3,7488,10298), (3,7491,6827), (3,7517,6644), (3,7553,7401), (3,7580,411), (3,8727,6857), (3,9122,1817), (3,9818,7509), (3,9909,3515), (3,10323,6636), (3,10431,6837), (3,11001,5059), (3,11413,2071), (3,11414,22), (3,11479,7484), (3,11676,3552), (4,5,3832), (4,21,6837), (4,24,3089), (4,140,5068), (4,186,3542), (4,376,3), (4,377,6839), (4,378,3088), (4,404,6953), (4,411,6838), (4,443,6835), (4,550,3522), (4,631,5), (4,1006,6846), (4,1595,7409), (4,1656,3854), (4,1657,5059), (4,3079,6616), (4,3088,7378), (4,3090,381), (4,3091,3839), (4,3146,3543), (4,3147,403), (4,3520,3541), (4,3522,3523), (4,3523,5056), (4,3524,3090), (4,3525,3545), (4,3528,631), (4,3533,3851), (4,3537,6803), (4,3538,6804), (4,3542,6623), (4,3545,546), (4,3651,6908), (4,3855,3843), (4,4188,6979), (4,4189,6888), (4,5067,3855), (4,5084,6957), (4,6353,235), (4,6616,1559), (4,6756,7408), (4,6803,6997), (4,6815,7544), (4,6824,6870), (4,6825,6871), (4,6826,6894), (4,6827,5046), (4,6833,6844), (4,6836,6840), (4,6850,2475), (4,6851,6895), (4,6852,6866), (4,6853,6867), (4,6854,6849), (4,6856,7548), (4,6857,6828), (4,6865,2478), (4,6868,6872), (4,6875,6824), (4,6876,6825), (4,6878,6990), (4,6880,6941), (4,6889,6843), (4,6891,5187), (4,6897,6826), (4,6899,6827), (4,6902,6929), (4,6903,6928), (4,6905,6848), (4,6906,6847), (4,6908,5177), (4,6909,6890), (4,6916,377), (4,6926,6919), (4,6927,1532), (4,6935,6831), (4,6940,6964), (4,6942,6834), (4,6947,6893), (4,6948,4190), (4,6949,6968), (4,6950,6833), (4,6951,6917), (4,6954,5141), (4,6961,5154), (4,6977,6830), (4,6986,6886), (4,6987,452), (4,6988,2476), (4,7386,6816), (4,7401,7394), (4,7412,4194), (4,7487,6995), (4,7512,3547), (4,7714,5198), (4,8889,7507), (4,10299,1656), (4,10304,10303), (4,10323,7400), (4,10996,6815), (4,11001,3529), (4,11111,6912), (5,140,5070), (5,382,4), (5,548,3), (5,550,548), (5,631,2), (5,3526,5067), (5,3530,3526), (5,3627,3861), (5,3832,3091), (5,3843,3855), (5,3845,3859), (5,3853,3843), (5,3859,3851), (5,3861,381), (5,5070,3090), (5,7486,5056), (5,9715,7493), (7,3188,279), (7,4313,1), (8,9778,40), (8,10430,9799), (8,10538,280), (11,5204,7288), (12,5217,5218), (21,377,2), (21,6839,6884), (21,7411,3), (22,858,7493), (22,1370,2), (22,2071,10298), (22,11413,3), (24,3089,4232), (24,6643,2), (24,7387,23), (25,1885,4), (25,7386,2), (25,7667,7386), (27,464,2), (32,2549,5286), (32,5286,5304), (32,7756,2549), (32,7765,5319), (35,1478,3085), (35,3085,5281), (35,10483,1478), (36,1479,3086), (36,3086,5265), (40,84,63), (40,3101,9537), (40,5881,11362), (55,7354,388), (56,6284,497), (57,950,938), (65,5918,9943), (68,11457,3448), (69,6527,253), (76,5188,6194), (98,5171,7793), (99,315,3926), (99,7802,315), (100,3436,7080), (140,381,3090), (140,546,10109), (140,3090,2), (140,3627,381), (140,3861,5), (140,5073,4), (140,8703,3), (145,9965,3868), (154,5895,2883), (165,5691,10), (175,176,347), (187,7748,3767), (226,3601,5703), (235,3515,6353), (315,8721,147), (316,7782,7763), (355,3579,5657), (355,5657,3617), (371,1587,7585), (371,6560,1587), (372,1588,7586), (372,6561,1588), (376,631,3528), (376,1370,2071), (376,1657,3146), (376,3146,3523), (376,3524,8703), (376,3528,548), (376,3529,4), (376,5059,3091), (376,6240,7400), (376,6851,4189), (376,6869,4190), (376,6916,7411), (376,6938,6987), (376,11541,140), (377,1012,6837), (377,6837,6993), (377,10431,4), (378,6240,4), (378,10323,3), (381,3090,5068), (381,3524,2), (381,3627,4), (381,5068,3091), (381,5070,5), (381,5073,3627), (381,8703,3524), (381,10109,3545), (382,548,631), (382,550,3528), (382,631,3832), (382,3526,3843), (382,3528,2), (382,3530,3855), (382,3843,3853), (382,5070,3861), (384,7791,2), (384,7833,7791), (390,3600,1), (394,1498,11441), (404,2478,2), (404,11114,2478), (405,443,2), (405,6835,6886), (405,11111,11106), (405,11112,443), (411,6836,2), (411,6840,6960), (411,6909,3), (411,6943,6962), (427,7494,2), (428,7484,7392), (428,11403,4), (440,7490,2), (442,6857,2), (442,8727,6828), (443,6912,6886), (443,11111,405), (452,6904,2), (474,5084,2), (474,11113,5084), (485,6200,9540), (485,9540,8972), (487,638,1271), (488,637,1270), (489,490,69), (498,3585,10590), (499,3583,10591), (546,549,1656), (546,1656,3545), (546,3545,3854), (546,3830,4), (546,10299,2), (547,3858,5072), (548,3528,3522), (548,3853,3530), (549,1656,3525), (549,3545,2), (549,3830,3545), (550,1657,4), (550,3146,10304), (550,3529,2), (550,3627,8703), (550,5059,3523), (550,6240,6636), (550,11001,3146), (574,7747,2548), (578,10984,5012), (631,3528,3), (631,3832,7486), (631,3855,5067), (631,5067,3526), (632,3850,5055), (858,7493,2), (858,10296,3153), (944,2096,1071), (946,3576,3616), (958,11495,5584), (962,5731,1), (962,5734,4301), (962,10884,11036), (991,3332,3945), (1006,6826,2), (1011,6817,2), (1012,6916,2), (1042,3000,1044), (1092,6759,110), (1113,1114,2071), (1131,8972,485), (1147,1614,9544), (1151,3070,3068), (1152,3071,3069), (1155,1837,1788), (1352,3098,10519), (1352,10519,3620), (1368,6353,2), (1368,9909,6353), (1370,7493,858), (1385,5603,3622), (1478,3085,5261), (1479,3086,5274), (1482,7967,3623), (1490,6282,78), (1532,6922,4193), (1583,6805,2), (1584,6806,2), (1585,1589,2), (1586,1590,2), (1587,9541,371), (1593,3575,4), (1593,7503,7527), (1593,10996,2), (1595,7399,5133), (1597,3537,2), (1597,6756,4), (1598,3538,2), (1610,1633,3556), (1656,3525,2), (1656,3830,546), (1656,10109,3090), (1657,3534,3), (1657,8703,11541), (1658,2937,7556), (1699,7987,1125), (1764,10454,10449), (1764,10463,10461), (1836,2646,3485), (1853,8567,6696), (1975,7750,69), (2041,2042,4), (2043,2044,376), (2045,2046,3533), (2060,3146,1559), (2071,7488,3), (2071,10296,858), (2077,6256,5552), (2475,4189,2), (2475,6847,3091), (2475,6895,4), (2475,6906,6888), (2476,6910,2), (2478,3149,6953), (2479,2480,441), (2549,5319,7765), (3053,5254,7735), (3060,10574,389), (3088,6636,3523), (3088,7400,2), (3090,3524,140), (3090,3529,11541), (3090,11541,3627), (3091,3523,2), (3091,3543,4), (3091,4208,6993), (3091,6992,5129), (3091,7486,5), (3091,10304,3523), (3098,9873,2896), (3100,4296,1), (3146,3522,2), (3146,3523,3839), (3146,3528,7486), (3146,3854,3830), (3146,7411,4208), (3146,10304,5056), (3146,11413,7396), (3147,11585,2), (3149,6865,2), (3151,7560,2), (3152,7538,2), (3153,10298,2), (3474,3486,65), (3520,7512,3), (3522,3529,3543), (3522,3543,10303), (3522,3854,10299), (3522,5059,4), (3523,3543,3091), (3523,10304,3), (3524,3529,5073), (3524,3627,5068), (3524,11541,4), (3525,3545,1656), (3525,3830,3854), (3525,10299,549), (3526,3530,631), (3526,3843,5), (3526,3853,3855), (3526,5067,2), (3528,3529,382), (3528,3832,3523), (3528,3855,3530), (3529,3534,3522), (3529,11001,1657), (3530,3843,5067), (3530,3853,5), (3530,3855,2), (3533,3628,2), (3533,5071,3628), (3534,11001,2), (3541,3547,2), (3542,3546,2), (3543,7396,3153), (3543,10304,2), (3545,3854,3091), (3545,10299,3525), (3548,7505,2), (3552,6655,2), (3560,6826,6846), (3560,6897,2), (3567,5446,11002), (3575,7503,7544), (3583,7280,499), (3585,5010,498), (3601,9579,226), (3616,9812,946), (3627,8703,140), (3627,11541,3146), (3628,3845,3851), (3628,3851,5071), (3628,3859,5), (3628,5054,3533), (3651,6851,2), (3651,6906,3), (3734,7830,7800), (3839,5056,3091), (3839,10303,5056), (3843,3853,4), (3843,3855,3832), (3845,5054,5071), (3845,5076,4), (3851,5054,3628), (3851,5076,3845), (3853,5067,3832), (3855,5067,5), (3868,11220,1071), (3911,9581,5704), (3917,5907,11444), (3917,11381,5907), (3972,7847,7803), (4188,5046,2), (4188,6926,3523), (4189,6895,6847), (4189,6908,3523), (4190,6868,6992), (4190,6872,2), (4190,6987,3523), (4191,6818,2), (4193,6921,2), (4195,4201,2), (4197,10883,5), (4208,6837,5056), (4292,4297,10884), (4292,4304,1), (4292,4313,11036), (4293,4294,1), (4293,4302,390), (4294,4299,3600), (4295,4305,1), (4298,4314,1), (4299,4302,1), (4299,4309,4317), (4299,4317,4325), (4299,4324,4294), (4301,9589,962), (4302,4309,4330), (4302,4316,4293), (4302,4317,4309), (4304,5732,5731), (4308,9785,1), (4309,4317,1), (4309,4330,4294), (4311,10624,1), (4316,4324,1), (4316,4330,4325), (4317,4325,4293), (4319,4320,1), (4324,4325,4330), (4325,4330,1), (4345,6049,1), (4348,7221,1), (4351,4354,1), (4652,6734,5744), (5004,5005,25), (5013,7745,7736), (5046,6848,3091), (5046,6905,6979), (5054,5071,2), (5054,5076,3851), (5056,10303,2), (5059,5068,5073), (5068,11541,3543), (5073,8703,3090), (5077,7866,8357), (5085,5480,3618), (5128,5727,4848), (5175,5744,6734), (5177,6888,5056), (5189,7492,2), (5218,5229,12), (5225,7288,11), (5261,5281,3085), (5265,5274,3086), (5318,11480,11488), (5319,7765,5286), (5321,11481,11489), (5432,10895,10588), (5433,10896,10589), (5446,9730,3567), (5447,5891,7999), (5550,9779,8227), (5550,10248,9779), (5584,6253,2550), (5587,6684,9780), (5806,11227,5439), (5878,6759,5656), (5881,11362,8), (5882,7982,3241), (6143,6639,2), (6143,7552,6639), (6225,11206,1498), (6459,6460,6), (6560,9541,7585), (6636,7391,2), (6643,7387,3089), (6644,7517,3518), (6676,8889,2), (6756,7395,6997), (6781,7756,32), (6803,7395,2), (6815,7503,2), (6824,6889,2), (6824,6917,6843), (6825,6833,2), (6827,6848,6919), (6827,6905,2), (6827,6970,6963), (6827,6985,6848), (6830,6842,5141), (6830,6954,2), (6831,6907,2476), (6831,6988,2), (6832,6989,2), (6833,6923,6871), (6834,6891,2), (6834,6928,5187), (6835,6986,2), (6836,6925,4), (6836,6962,6943), (6836,7580,6838), (6837,6839,3091), (6838,6840,3091), (6838,6890,2), (6842,6977,2), (6843,6870,3091), (6844,6871,3091), (6845,6937,5), (6846,6894,3091), (6847,6850,5177), (6847,6908,2), (6848,6926,2), (6850,6851,4), (6850,6892,6937), (6850,6906,2), (6853,6862,2), (6854,6883,2), (6856,7483,2), (6863,6952,2), (6868,6869,4), (6868,6885,6936), (6868,6934,2), (6868,6948,3), (6869,6938,3146), (6869,6948,6934), (6872,6904,5129), (6875,6917,2), (6875,6951,6889), (6876,6950,3), (6878,6881,2), (6880,6882,2), (6882,6941,5154), (6884,6993,5056), (6885,6930,5), (6885,6936,2), (6892,6937,2), (6893,6911,6964), (6893,6940,2), (6895,6908,3091), (6899,6905,6926), (6899,6985,2), (6902,6924,2), (6903,6942,6891), (6904,6992,10303), (6905,6963,6970), (6905,7491,5046), (6906,6937,6892), (6907,6935,2), (6909,6925,2), (6909,6932,6966), (6911,6947,2), (6912,6986,405), (6914,6917,6824), (6914,6951,2), (6916,10431,6839), (6919,6979,5056), (6922,6927,2), (6923,6950,2), (6924,6929,6944), (6925,6966,6932), (6928,6942,2), (6930,6948,6955), (6930,6955,2), (6932,6943,5), (6932,6966,2), (6934,6936,6885), (6934,6938,4), (6934,6987,6904), (6935,6988,6910), (6936,6955,631), (6938,6948,2), (6938,6955,6930), (6941,6961,2), (6943,6962,2), (6944,6967,2), (6949,6958,2), (6960,6972,2), (6962,6966,631), (6963,6970,2), (6985,7491,4), (6997,7485,2), (7381,11340,2), (7383,7404,2), (7388,11291,2), (7389,11292,2), (7390,7406,3091), (7391,7400,3091), (7392,7484,2), (7395,7408,3091), (7396,7488,10303), (7396,10565,2), (7399,7409,3091), (7401,7509,2), (7411,10431,2), (7464,7512,3520), (7470,11676,3), (7484,11403,11479), (7502,11250,3), (7509,7553,7394), (7509,7576,7401), (7511,7549,7557), (7538,7560,7520), (7540,7550,7533), (7553,9818,4), (7555,7574,7552), (7576,9818,7394), (7689,11750,11457), (7714,10691,2), (7761,7816,7795), (7783,7823,7774), (8226,8728,6991), (8227,10165,5550), (8357,8369,7866), (8550,11477,1992), (8703,11541,5068), (9778,10430,63), (9825,11479,7392), (10267,10532,10587), (10269,10531,10586), (10304,10565,10298), (10310,11500,100), (11015,11220,944), (11111,11112,2), (11291,11292,7819), (11293,11294,2), (11413,11414,7488)

= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b + c)

Barycentrics a/(cos B + cos C) : b/(cos C + cos A) : c/(cos A + cos B)

X(21) = 3R*X(2) + 2r*X(3)

As a point on the Euler line, X(21) has Shinagawa coefficients ($aS_{A}$, abc - $aS_{A}$).

Let A'B'C' be the incentral triangle of ABC, and let L_{A}
be the reflection of line B'C' in line BC; define L_{B} and
L_{C} cyclically. The triangle formed by the lines
L_{A}, L_{B}, L_{C} is perspective to ABC, and
the perspector is X(21). (Randy Hutson, 9/23/2011)

Write I for the incenter; the Euler lines of the four triangles IBC, ICA, IAB, and ABC concur in X(21). This configuration extends to Kirikami-Schiffler points and generalizations found by Peter Moses, as introduced just before X(3648).

Let A'B'C' be the 2nd circumperp triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", and CC" concur in X(21). (Randy Hutson, April 9, 2016)

Let A'B'C' be the Feuerbach triangle. Let La be the tangent to the nine-point circle at A', and define Lb and Lc cyclically. Let A" be the isogonal conjugate of the trilinear pole of La, and define B" and C" cyclically. Let A*B*C* be the circumcevian triangle, wrt A"B"C", of X(1). The lines AA*, BB*, CC* concur in X(21). (Randy Hutson, April 9, 2016)

Let A'B'C' be the 2nd circumperp triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. Then A", B", C" are collinear on line X(36)X(238) (the trilinear polar of X(81)). The lines A'A", B'B", C'C" concur in X(21). (Randy Hutson, April 9, 2016)

Let Oa be the reflection of the A-excircle in the perpendicular bisector of BC, and define Ob, Oc cyclically. Then X(21) is the radical center of Oa, Ob, Oc. (Randy Hutson, April 9, 2016)

Let Ab, Ac, Bc, Ba, Ca, Cb be as in the construction of the Conway circle (see http://mathworld.wolfram.com/ConwayCircle.html). Let Oa be the circumcircle of AAbAc, and define Ob, Oc cyclically. X(21) is the radical center of Oa, Ob, Oc; see also X(8) and X(274). (Randy Hutson, April 9, 2016)

Let A'B'C' be the excentral triangle. X(21) is the radical center of the circles O(3,4) of triangles A'BC, B'CA, C'AB. (Randy Hutson, July 31 2018)

If you have The Geometer's Sketchpad, you can view Schiffler point.

If you have GeoGebra, you can view **Schiffler point**.

**Lev Emelyanov and Tatiana Emelyanova,**
A note on the Schiffler point, *Forum Geometricorum* 3 (2003) pages 113-116.

The name of this point honors Kurt Schiffler.

X(21) lies on these lines:

{1, 31}, {2, 3}, {6, 941}, {7, 56}, {8, 55}, {9, 41}, {10, 35}, {11, 4996}, {12, 5080}, {15, 5362}, {16, 5367}, {19, 4288}, {32, 981}, {36, 79}, {37, 172}, {40, 3577}, {42, 4281}, {44, 4273}, {45, 3285}, {51, 970}, {57, 4652}, {60, 960}, {65, 4640}, {71, 4269}, {72, 943}, {73, 651}, {75, 272}, {77, 1394}, {84, 285}, {85, 3188}, {90, 224}, {99, 105}, {101, 3294}, {104, 110}, {107, 1295}, {141, 4265}, {144, 954}, {145, 956}, {149, 2894}, {162, 3194}, {187, 5277}, {198, 5296}, {200, 4866}, {210, 4420}, {214, 501}, {219, 2335}, {238, 256}, {243, 1896}, {261, 314}, {268, 280}, {270, 1172}, {286, 1441}, {294, 1212}, {307, 2062}, {323, 5453}, {329, 5703}, {332, 1036}, {385, 1655}, {386, 1724}, {390, 6601}, {391, 4254}, {476, 2687}, {484, 3754}, {517, 1389}, {518, 2346}, {519, 3746}, {535, 5270}, {551, 5557}, {572, 1765}, {593, 6051}, {612, 989}, {614, 988}, {643, 1320}, {644, 1334}, {662, 1156}, {691, 2752}, {741, 932}, {748, 978}, {756, 5293}, {884, 885}, {902, 5255}, {915, 925}, {936, 3305}, {938, 5744}, {940, 4252}, {942, 3218}, {950, 5745}, {961, 1402}, {962, 3428}, {976, 983}, {986, 3924}, {987, 2206}, {992, 5110}, {999, 3296}, {1030, 1213}, {1038, 1041}, {1039, 1040}, {1060, 1063}, {1061, 1062}, {1064, 3073}, {1083, 3110}, {1104, 3666}, {1107, 1914}, {1155, 3812}, {1214, 1396}, {1251, 5240}, {1254, 1758}, {1261, 4723}, {1304, 2694}, {1319, 1408}, {1329, 5432}, {1330, 3936}, {1376, 5217}, {1392, 2098}, {1412, 1420}, {1453, 5256}, {1466, 5435}, {1470, 5555}, {1500, 5291}, {1610, 2217}, {1617, 3600}, {1682, 3271}, {1697, 3680}, {1698, 5010}, {1761, 2294}, {1936, 2654}, {1946, 4391}, {2077, 6684}, {2096, 5553}, {2276, 4426}, {2310, 2648}, {2320, 5289}, {2341, 5549}, {2344, 3061}, {2551, 5218}, {2782, 5985}, {2886, 6284}, {3006, 5015}, {3052, 5710}, {3053, 5275}, {3058, 3813}, {3060, 5752}, {3062, 5732}, {3085, 3436}, {3207, 5781}, {3208, 4390}, {3216, 4256}, {3220, 4357}, {3241, 3303}, {3244, 5288}, {3256, 4848}, {3304, 5558}, {3315, 3953}, {3333, 4666}, {3336, 5883}, {3337, 4973}, {3419, 5791}, {3427, 5731}, {3434, 4294}, {3487, 5905}, {3496, 5060}, {3555, 3957}, {3579, 3753}, {3585, 3822}, {3586, 5705}, {3589, 5096}, {3614, 6668}, {3617, 5687}, {3624, 5561}, {3681, 3811}, {3684, 3691}, {3689, 4662}, {3737, 6615}, {3757, 4968}, {3816, 5433}, {3833, 5131}, {3841, 4324}, {3886, 4483}, {3895, 4853}, {3920, 5266}, {3929, 3951}, {4084, 5425}, {4101, 4416}, {4255, 4383}, {4314, 4847}, {4423, 5204}, {4516, 4612}, {4520, 6603}, {4567, 5377}, {4646, 4689}, {4668, 4803}, {4867, 5424}, {5044, 5440}, {5294, 5314}, {5554, 5657}, {5686, 6600}

X(21) is the {X(2),X(3)}-harmonic conjugate of X(404). For a list of other harmonic conjugates of X(21), click **Tables** at the top of this page.

X(21) = midpoint of X(1) and X(191)

X(21) = reflection of X(3651) in X(3)

X(21) = isogonal conjugate of X(65)

X(21) = isotomic conjugate of X(1441)

X(21) = inverse-in-circumcircle of X(1325)

X(21) = complement of X(2475)

X(21) = anticomplement of X(442)

X(21) = X(i)-Ceva conjugate of X(j) for these (i,j): (86,81), (261,333)

X(21) = cevapoint of X(i) and X(j) for these (i,j): (1,3), (9,55), (1805,1806)

X(21) = X(i)-cross conjugate of X(j) for these (i,j): (1,29), (3,283), (9,333), (55,284), (58,285), (284,81), (522,100)

X(21) = crosspoint of X(i) and X(j) for these {i,j}: {86,333}, {1805,1806}

X(21) = crosssum of X(i) and X(j) for these (i,j): (1,1046), (42,1400), (1254,1425), (1402,1409)

X(21) = crossdifference of every pair of points on line X(647)X(661)

X(21) = X(i)-Hirst inverse of X(j) for these (i,j): (2,448), (3,416), (4,425)

X(21) = X(i)-beth conjugate of X(j) for these (i,j): (21,58), (99,21), (643,21), (1043,1043), (1098,21)

X(21) = intersection of tangents at X(1) and X(3) to the Stammler hyperbola

X(21) = X(54)-of-2nd-circumperp-triangle

X(21) = X(3574)-of-excentral-triangle

X(21) = crosspoint of X(1) and X(3) wrt the excentral triangle

X(21) = crosspoint of X(1) and X(3) wrt the tangential triangle

X(21) = trilinear pole of line X(521)X(650)

X(21) = similitude center of ABC and X(1)-Brocard triangle

X(21) = X(i)-isoconjugate of X(j) for these (i,j): (6,226), (75,1402)

X(21) = {X(1),X(63)}-harmonic conjugate of X(3868)

X(21) = perspector of 2nd circumperp triangle and cross-triangle of ABC and 2nd circumperp triangle

X(21) = perspector of ABC and cross-triangle of ABC and 1st Conway triangle

X(21) = perspector of Gemini triangles 1 and 8

X(21) = barycentric product of Feuerbach hyperbola intercepts of line X(2)X(6)

Barycentrics a

Barycentrics sin 2A - tan ω : sin 2B - tan ω : : (M. Iliev, 5/13/07)

Barycentrics tan B + tan C - tan A + tan ω : : (R. Hutson, 10/13/15)

X(22) = 3 R^2 X(2) - SW X(3)

As a point on the Euler line, X(22) has Shinagawa coefficients (E + 2F, -2E - 2F).

X(22) is the perspector of the circummedial triangle and the tangential triangle; also X(22) = X(55)-of-the-tangential triangle if ABC is acute. See the note just before X(1601) for a generalization.

Let La be the polar of X(3) wrt the A-power circle, and define Lb, Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The triangle A'B'C' is homothetic to the anticomplementary triangle, and the center of homothety is X(22). (Randy Hutson, September 5, 2015)

If you have The Geometer's Sketchpad, you can view Exeter point.

If you have GeoGebra, you can view

X(22) lies on these lines:

2,3 6,251 32,1194
35,612 36,614 51,182
56,977 69,159 76,1799
98,925 99,305 100,197
110,154 155,1614 157,183
160,325 161,343 184,511
187,1196 232,577 264,1629
347,1617 675,1305 991,1790
1184,1627 1294,1302 1486,1621
1602,1626

X(22) is the {X(3),X(25)}-harmonic conjugate of X(2). For a list of
other harmonic conjugates of X(22), click **Tables** at the top of
this page.

X(22) = reflection of X(378) in X(3)

X(22) = isogonal conjugate of X(66)

X(22) = isotomic conjugate of X(18018)

X(22) = inverse-in-circumcircle of X(858)

X(22) = anticomplement of X(427)

X(22) = X(76)-Ceva conjugate of X(6)

X(22) = cevapoint of X(3) and X(159)

X(22) = crosspoint of X(99) and X(250)

X(22) = crosssum of X(125) and X(512)

X(22) = crossdifference of every pair of points on the line X(647)X(826)

X(22) = X(i)-beth conjugate of X(j) for these (i,j): (643,345), (833,22)

X(22) = complement of X(7391)

X(22) = pole, with respect to circumcircle, of the de Longchamps line

X(22) = isotomic conjugate of the isogonal conjugate of X(206)

X(22) = tangential isogonal conjugate of X(6)

X(22) = crosspoint of X(3) and X(159) wrt both the excentral and tangential triangles

X(22) = homothetic center of the tangential triangle and the orthic triangle of the anticomplementary triangle

X(22) = exsimilicenter of circumcircle and tangential circle

X(22) = inverse-in-de-Longchamps-circle of X(5189)

X(22) = inverse-in-{circumcircle, nine-point circle}-inverter of X(2072)

X(22) = X(75)-isoconjugate of X(2353)

X(22) = trilinear pole of line X(2485)X(8673)

X(22) = homothetic center of anticomplementary and Ara triangles

X(22) = Thomson-isogonal conjugate of X(5654)

X(22) = Lucas-isogonal conjugate of X(11459)

Barycentrics 2 sin 2A - 3 tan ω : 2 sin 2B - 3 tan ω : 2 sin 2C - 3 tan ω (M. Iliev, 5/13/07)

Tripolars Sqrt[2(b^2 + c^2) - a^2] : :

X(23) = 9 R^2 X(2) - 2 SW X(3)

As a point on the Euler line, X(23) has Shinagawa coefficients (E + 4F, -4E - 4F).

Let A'B'C' be the antipedal triangle of X(3) (the tangential triangle). The circumcircles of AA'X(3), BB'X(3), CC'X(3) concur in two points: X(3) and X(23). (Randy Hutson, Octobe3r 13, 2015)

Let A'B'C' be the anti-orthocentroidal triangle. Let A" be the reflection of A' in line BC, and define B" and C" cyclically. Then X(23) is the centroid of A"B"C"; see X(9140), X(11002). (Randy Hutson, December 10, 2016)

If you have The Geometer's Sketchpad, you can view Far-out point.

If you have GeoGebra, you can view

X(23) lies on the Parry circle, anti-Brocard circle, anti-McCay circumcircle, and these lines:

2,3 6,353 51,575
52,1614 94,98 105,1290
110,323 111,187 143,1199
159,193 184,576 232,250
251,1194 324,1629 385,523
477,1302 895,1177 1196,1627
1297,1804

X(23) is the {X(22),X(25)}-harmonic conjugate of X(2). For a list of
other harmonic conjugates of X(23), click **Tables** at the top of
this page.

X(23) = reflection of X(i) in X(j) for these (i,j): (110,1495),
(323,110), (691,187), (858,468)

X(23) = isogonal conjugate of X(67)

X(23) = isotomic conjugate of X(18019)

X(23) = inverse-in-circumcircle of X(2)

X(23) = anticomplement of X(858)

X(23) = anticomplementary conjugate of X(2892)

X(23) = crosspoint of X(111) and X(251)

X(23) = crosssum of X(i) and X(j) for these (i,j): (125,690), (141,524)

X(23) = crossdifference of every pair of points on line X(39)X(647)

X(23) = complement of X(5189)

X(23) = perspector of ABC and reflection of circummedial triangle in the Euler line

X(23) = antigonal image of X(316)

X(23) = trilinear pole of line X(2492)X(6593)

X(23) = reflection of X(858) in the orthic axis

X(23) = reflection of X(110) in the Lemoine axis

X(23) = polar conjugate of isotomic conjugate of X(22151)

X(23) = X(352)-of-circumsymmedial-triangle

X(23) = X(110)-of-1st-anti-Brocard-triangle

X(23) = crosspoint of X(3) and X(2930) wrt both the excentral and tangential triangles

X(23) = inverse-in-circumcircle of X(2)

X(23) = inverse-in-polar-circle of X(427)

X(23) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5)

X(23) = inverse-in-de-Longchamps-circle of X(1370)

X(23) = X(75)-isoconjugate of X(3455)

X(23) = common radical trace of similitude circles of pairs of the Stammler circles

X(23) = one of two harmonic traces of Ehrmann circles; the other is X(6)

X(23) = X(111)-of-anti-McCay-triangle

X(23) = X(691)-of-1st-Parry-triangle

X(23) = X(842)-of-2nd-Parry-triangle

X(23) = X(1296)-of-3rd-Parry-triangle

X(23) = inverse-in-Parry-isodynamic-circle of X(352) (see X(2))

X(23) = X(111)-of-4th-anti-Brocard-triangle

X(23) = similitude center of anti-McCay and 4th anti-Brocard triangles

X(23) = anti-Artzt-to-4th-anti-Brocard similarity image of X(110)

X(23) = intersection of de Longchamps lines of 1st and 2nd Ehrmann circumscribing triangles

X(23) = intersection of orthic axes of antipedal triangles of PU(1)

X(23) = intersection of de Longchamps lines of anticevian triangles of PU(4)

X(23) = circumtangential isogonal conjugate of X(32305)

X(23) = inverse of X(33502) in the Lucas circles radical circle

X(23) = inverse of X(33503) in the Lucas(-1) circles radical circle

X(23) = trilinear pole, wrt 1st Parry triangle, of line X(110)X(1296)

Tllrilinears sec A - 2 cos A : sec B - 2 cos B : sec C - 2 cos C

Barycentrics tan A cos 2A : tan B cos 2B : tan C cos 2C

Barycentrics tan A - sin 2A : tan B - sin 2B : tan C - sin 2C

Barycentrics a(sec A - 2 cos A) : :

Barycentrics a^2 (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2)/(a^2 - b^2 - c^2) : :

X(24) = 6 X(2) + (J^2 - 5) X(3)

X(24) = (a^2 - b^2 - c^2) (a^2 + b^2 - c^2) (a^2 - b^2 + c^2) X(3) + 2 a^2 b^2 c^2 X(4)

As a point on the Euler line, X(24) has Shinagawa coefficients (2F, -E - 2F).

Let A'B'C' be the orthic triangle. Let A" = inverse-in-circumcircle of A', and define B'' and C'' cyclically. The lines AA", BB", CC" concur in X(24). (Randy Hutson, September 5, 2015)

X(24) = homothetic center of the tangential triangle and the triangle obtained by reflecting X(4) in the sidelines of ABC.

If you have The Geometer's Sketchpad, you can view X(24).

If you have GeoGebra, you can view **X(24)**.

X(24) lies on these lines:

1,1061 2,3 6,54 32,232
33,35 34,36 49,568
51,578 52,1147 56,1870
64,74 96,847 98,1289
107,1093 108,915 110,155
154,1181 182,1843 183,1235
184,389 185,1495 242,1602
254,393 264,1078 511,1092
573,1474 602,1395 944,1610
1063,1775 1112,1511 1192,1511
1324,1603 1385,1829

X(24) is the {X(3),X(4)}-harmonic conjugate of X(378). For a list of
other harmonic conjugates of X(24), click **Tables** at the top of
this page.

X(24) = reflection of X(4) in X(235)

X(24) = isogonal conjugate of X(68)

X(24) = isotomic conjugate of X(20563)

X(24) = complement of complement of X(31304)

X(24) = anticomplement of X(11585)

X(24) = perspector of ABC and reflection of X(4) in orthic triangle

X(24) = {X(3),X(25)}-harmonic conjugate of X(4)

X(24) = trilinear pole of line X(924)X(6753)

X(24) = inverse-in-circumcircle of X(403)

X(24) = inverse-in-orthocentroidal circle of X(1594)

X(24) = X(249)-Ceva conjugate of X(112)

X(24) = X(52)-cross conjugate of X(4)

X(24) = crosspoint of X(107) and X(250)

X(24) = crosssum of X(i) and X(j) for these (i,j): (6,161), (125,520), (637,638)

X(24) = X(4)-Hirst inverse of X(421)

X(24) = X(46)-of-orthic-triangle if ABC is acute

X(24) = X(56)-of-the-tangential triangle if ABC is acute

X(24) = tangential isogonal conjugate of X(1498)

X(24) = insimilicenter of circumcircle and tangential circle

X(24) = inverse-in-polar-circle of X(2072)

X(24) = homothetic center of tangential and circumorthic triangles

X(24) = homothetic center of orthic and Kosnita triangles

X(24) = X(i)-isoconjugate of X(j) for these (i,j): (75,2351), (91,3)

Trilinears a/(b

Trilinears cos A - sec A : :

Barycentrics tan A - tan ω : :

Barycentrics sec A sin(A - ω) : :

X(25) = 6 R^2 X(2) - SW X(3)

As a point on the Euler line, X(25) has Shinagawa coefficients (F, -E - F).

Constructed as indicated by the name; also X(25) is the pole of the orthic axis (the line having trilinear coefficients cos A : cos B : cos C) with respect to the circumcircle.

If you have The Geometer's Sketchpad, you can view X(25).

If you have GeoGebra, you can view **X(25)**.

Let A' be the center of the conic through the contact points of the incircle and the A-excircle with the sidelines of ABC. Define B' and C' cyclically. Let A" be the center of the conic through the contact points of the B- and C- excircles with the sidelines of ABC. Define B" and C" cyclically. Let A* be the midpoint of A' and A", and define B* and C* cyclically. The triangle A*B*C* is perspective to ABC at X(25). See also X(6), X(218), X(222), X(940), X(1743). (Randy Hutson, July 23, 2015)

Let A' be the radical center of the nine-point circle and the B- and C-power circles. efine B' and C' cyclically. The triangle A'B'C' is homothetic with the orthic triangle, and the center of homothety is X(25). Also X(25) is the point of intersection of these two lines: isotomic conjugate of polar conjugate of van Aubel line (i.e., line X(2)X(3)), and polar conjugate of isotomic conjugate of van Aubel line (i.e., line X(25)X(393)). Also, X(25) is the trilinear pole of line X(512)X(1692), this line being the isogonal conjugate of the isotomic conjugate of the orthic axis; the line X(512)X(1692) is also the polar of X(76) wrt polar circle, and the line is also the radical axis of circumcircle and 2nd Lemoine circle. (Randy Hutson, September 5, 2015)

Let A'B'C' be the orthic triangle. Let A" be the barycentric product of the (real or imaginary) circumcircle intercepts of line B'C'. Define B", C" cyclically. The lines AA", BB", CC" concur in X(25). (Randy Hutson, October 27, 2015)

The 2nd Ehrmann triangle, defined in the preamble to X(8537), can be generalized as follows. Let P be a point in the plane of ABC and not on BC∪CA∪AB. Let Ab the the point of intersection of the circle {{P,B,C}} and the line AB, and define Bc and Ca cyclically. Define Ac symmetrically, and define Ba and Cb cyclically. Let A' = BcBa∩CaCb, and define B' and C' cyclically. Triangle A'B'C', here introduced as the **P-Ehrmann triangle**, is homothetic to the orthic triangle. The X(1)-Ehrmann triangle is the intangents triangle, and the X(6)-Ehrmann triangle is the 2nd Ehrmann triangle. If P lies on the circumcircle, the P-Ehrmann triangle is the tangential triangle. If P is on the Brocard 2nd cubic K018 or the circumcircle, then the P-Ehrmann triangle is perspective to ABC. The homothetic center of the orthic triangle and the X(4)-Ehrmann triangle is X(25). (Randy Hutson, February 10, 2016)

X(25) lies on these lines:

1,1036 2,3 6,51 19,33
31,608 32,1184 34,56
35,1900 36,1878 40,1902
41,42 52,155 53,157
57,1473 58,967 64,1192
65,1452 76,1241 92,242
98,107 100,1862 105,108
110,1112 111,112 114,135
125,1853 132,136 143,156
182,3066 183,264 185,1498 221,1425
225,1842 226,1892 262,275
273,1447 286,1218 317,325
339,1289 343,1352 371,493
372,494 389,1181 393,1033
394,511 669,878 692,913
694,1613 842,1304 847,1179
941,1172 958,1891 999,1870
1001,1848 1073,1297 1096,1402
1235,1239 1300,1302 1324,1785
1376,1861 1470,1877 1503,1619
1604,1863 1631,1826 1726,1736
1730,1754

X(25) is the {X(5),X(26)}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(25), click **Tables** at the top of this page.

X(25) = reflection of X(i) in X(j) for these (i,j): (4,1596), (1370,1368)

X(25) = isogonal conjugate of X(69)

X(25) = isotomic conjugate of X(305)

X(25) = inverse-in-circumcircle of X(468)

X(25) = inverse-in-orthocentroidal-circle of X(427)

X(25) = complement of X(1370)

X(25) = anticomplement of X(1368)

X(25) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,6), (28,19), (250,112)

X(25) = X(32)-cross conjugate of X(6)

X(25) = crosspoint of X(i) and X(j) for these (i,j): (4,393), (6,64), (19,34), (112,250)

X(25) = crosssum of X(i) and X(j) for these (i,j): (2,20), (3,394), (6,159), (8,329), (63,78), (72,306), (125,525)

X(25) = crossdifference of every pair of points on line X(441)X(525)

X(25) = X(i)-Hirst inverse of X(j) for these (i,j): (4,419), (6,232)

X(25) = X(i)-beth conjugate of X(j) for these (i,j): (33,33), (108,25), (162,278)

X(25) = crosspoint of PU(4)

X(25) = barycentric product of PU(i) for these i: 4,18,23,157

X(25) = barycentric product of vertices of half-altitude triangle

X(25) = barycentric product of vertices of orthocentroidal triangle

X(25) = perspector of circumconic centered at X(3162)

X(25) = center of circumconic that is locus of trilinear poles of lines passing through X(3162)

X(25) = X(2)-Ceva conjugate of X(3162)

X(25) = pole, wrt circumcircle, of orthic axis

X(25) = pole, wrt polar circle, of de Longchamps line

X(25) = X(i)-isoconjugate of X(j) for these (i,j): (6,304), (48,76), (75,3), (92,394), (1101,339)

X(25) = tangential isogonal conjugate of X(159)

X(25) = insimilicenter of nine-point circle and tangential circle

X(25) = orthic isogonal conjugate of X(6)

X(25) = homothetic center of ABC and the 2nd pedal triangle of X(4)

X(25) = homothetic center of ABC and the 2nd antipedal triangle of X(3)

X(25) = homothetic center of the medial triangle and the 3rd pedal triangle of X(4)

X(25) = homothetic center of the anticomplementary triangle and the 3rd antipedal triangle of X(3)

X(25) = homothetic center of reflection of orthic triangle in X(4) and reflection of tangential triangle in X(3)

X(25) = homothetic center of reflections of orthic and tangential triangles in their respective Euler lines

X(25) = inverse-in-polar-circle of X(858)

X(25) = inverse-in-{circumcircle, nine-point circle}-inverter of X(403)

X(25) = inverse-in-circumconic-centered-at-X(4) of X(450)

X(25) = Danneels point of X(4)

X(25) = Danneels point of X(1113)

X(25) = Danneels point of X(1114)

X(25) = X(2)-vertex conjugate of X(2)

X(25) = vertex conjugate of X(8105) and X(8106)

X(25) = vertex conjugate of foci of orthic inconic

X(25) = vertex conjugate of PU(112)

X(25) = Zosma transform of X(63)

X(25) = X(57)-of-the-tangential triangle if ABC is acute

X(25) = perspector of ABC and the (pedal triangle of X(4) in the orthic triangle)

X(25) = X(57) of orthic triangle if ABC is acute

X(25) = intersection of tangents at X(371) and X(372) to the orthocubic K006

X(25) = insimilicenter of circumcircle and incircle of orthic triangle if ABC is acute; the exsimilicenter is X(1593)

X(25) = perspector of ABC and circummedial tangential triangle

X(25) = homothetic center of ABC and orthocevian triangle of X(2)

X(25) = homothetic center of orthocevian triangle of X(2) and Ara triangle

X(25) = {X(8880),X(8881)}-harmonic conjugate of X(184)

X(25) = homothetic center of medial triangle and cross-triangle of ABC and Ara triangle

X(25) = perspector of ABC and cross-triangle of ABC and 4th Brocard triangle

X(25) = harmonic center of circumcircle and circle O(PU(4))

X(25) = Thomson-isogonal conjugate of X(5656)

X(25) = homothetic center of Aries and 2nd Hyacinth triangles

X(25) = intersection of tangents to Hatzipolakis-Lozada hyperbola at X(4) and X(193)

X(25) = crosspoint, wrt orthic triangle, of X(4) and X(193)

X(25) = barycentric product of (real or nonreal) circumcircle intercepts of orthic axis

X(25) = vertex conjugate of X(24007) and X(24008) (the Kiepert hyperbola intercepts of the orthic axis)

X(25) = excentral-to-ABC functional image of X(57)

X(25) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,7484), (2,4,427), (2,5,7539), (3,4,1593), (3,5,7395), (4,5,7507), (4,24,3)

Trilinears (J

Barycentrics a

Barycentrics a^2 (a^8 - 2 a^6 (b^2 + c^2) + 2 a^2 (b^6 + c^6) - (b^2 - c^2)^2 (b^4 + c^4)) : :

X(26) = 6 X(2) + (J^2 - 7) X(3)

X(26) = (J^2 - 3) X(3) + 2 X(4)

As a point on the Euler line, X(26) has Shinagawa coefficients (E + 4F, -3E - 4F).

If you have The Geometer's Sketchpad, you can view X(26).

If you have GeoGebra, you can view **X(26)**.

Theorems involving X(26), published in 1889 by A. Gob, are discussed in**Roger A. Johnson,** *Advanced Euclidean Geometry,* Dover, 1960, 259-260.

X(26) lies on these lines: 2,3 6,143 52,184 68,161 98,1286 154,155 206,511 1605,1607 1606,1608

X(26) is the {X(154),X(155)}-harmonic conjugate of X(156). For a
list of other harmonic conjugates of X(26), click **Tables** at the
top of this page.

X(26) = reflection of X(155) in X(156)

X(26) = isogonal conjugate of X(70)

X(26) = isotomic conjugate of X(20564)

X(26) = tangential isogonal conjugate of X(155)

X(26) = inverse-in-circumcircle of X(2072)

X(26) = crosssum of X(125) and X(924)

X(26) = complement of X(14790)

X(26) = anticomplement of X(13371)

X(26) = intouch-to-ABC functional image of X(3)

Barycentrics (tan A)/(b + c) : (tan B)/(c + a) : (tan C)/(a + b)

X(27) = 3 SA SB SC X(2) - 2 S^2 s^2 X(3)

As a point on the Euler line, X(27) has Shinagawa coefficients (F, -E - F - $bc$).

If you have The Geometer's Sketchpad, you can view X(27).

If you have GeoGebra, you can view

X(27) lies on these lines:

2,3 6,1246 7,81 19,63
57,273 58,270 84,1896
86,1474 103,107 110,917
226,284 239,1829 243,1859
295,335 306,1043 393,967
579,1751 648,903 662,913
1014,1440 1088,1434 1268,1796
1719,1733 1730,1746 1770,1780

X(27) is the {X(2),X(4)}-harmonic conjugate of X(469). For a list of
other harmonic conjugates of X(27), click **Tables** at the top of
this page.

X(27) = isogonal conjugate of X(71)

X(27) = isotomic conjugate of X(306)

X(27) = inverse-in-circumcircle of X(2073)

X(27) = inverse-in-orthocentroidal-circle of X(469)

X(27) = complement of X(3151)

X(27) = anticomplement of X(440)

X(27) = X(286)-Ceva conjugate of X(29)

X(27) = cevapoint of X(i) and X(j) for these (i,j): (4,19), (57,278)

X(27) = X(i)-cross conjugate of X(j) for these (i,j): (4,286), (19,28), (57,81), (58,86)

X(27) = crossdifference of every pair of points on line X(647)X(810)

X(27) = X(i)-Hirst inverse of X(j) for these (i,j): (2,447), (4,423)

X(27) = X(i)-beth conjugate of X(j) for these (i,j): (648,27), (923,27)

X(27) = trilinear pole of line X(242)X(514) (the polar of X(10) wrt polar circle)

X(27) = pole wrt polar circle of trilinear polar of X(10) (line X(523)X(661))

X(27) = polar conjugate of X(10)

X(27) = X(6)-isoconjugate of X(72)

X(27) = X(75)-isoconjugate of X(2200)

X(27) = crosspoint of X(4) and X(19) wrt excentral triangle

X(27) = trilinear product X(2)*X(28)

Barycentrics (sin A tan A)/(b + c) : (sin B tan B)/(c + a) : (sin C tan C)/(a + b)

X(28) = 8 sa sb sc X(4) + a b c (3 + J^2) X(25)

As a point on the Euler line, X(28) has Shinagawa coefficients ($a$F, -$a$(E + F) - abc).

If you have The Geometer's Sketchpad, you can view X(28).

If you have GeoGebra, you can view **X(28)**.

1,19 2,3 10,1891 11,1852 33,975 34,57 35,1869 36,1838 46,1780 54,1243 56,278 60,81 65,1175 72,1257 88,162 104,107 105,112 108,225 110,915 142,1890 228,943 242,261 272,273 279,1014 281,958 291,1783 501,1831 579,1724 580,1730 607,1002 608,959 614,1472 956,1219 957,1191 961,1169 1104,1333 1125,1848 1155,1888 1170,1876 1178,1432 1224,1826 1255,1824 1295,1301 1385,1871 1412,1422 1633,1770 1710,1725

X(28) is the {X(27),X(29)}-harmonic conjugate of X(4). For a list of
other harmonic conjugates of X(28), click **Tables** at the top of
this page.

X(28) = isogonal conjugate of X(72)

X(28) = isotomic conjugate of X(20336)

X(28) = trilinear pole of line X(513)X(1430) (the polar of X(321) wrt polar circle)

X(28) = polar conjugate of X(321)

X(28) = X(6)-isoconjugate of X(306)

X(28) = X(75)-isoconjugate of X(228)

X(28) = inverse-in-circumcircle of X(2074)

X(28) = X(i)-Ceva conjugate of X(j) for these (i,j): (270,58), (286,81)

X(28) = cevapoint of X(i) and X(j) for these (i,j): (19,25), (34,56)

X(28) = X(i)-cross conjugate of X(j) for these (i,j): (19,27), (58,58)

X(28) = crossdifference of every pair of points on line X(647)X(656)

X(28) = X(4)-Hirst inverse of X(422)

X(28) = X(i)-beth conjugate of X(j) for these (i,j): (29,29), (107,28), (162,28), (270,28)

Barycentrics (tan A)/(cos B + cos C) : (tan B)/(cos C + cos A) : (tan C)/(cos A + cos B)

Barycentrics (a - b - c)/((b + c) (a^2 - b^2 - c^2)) : :

As a point on the Euler line, X(29) has Shinagawa coefficients (F*S^{2}, $bcS_{B}S_{C}$ - F*S^{2}).

If you have The Geometer's Sketchpad, you can view X(29).

If you have GeoGebra, you can view

X(29) lies on these lines:

1,92 2,3 8,219 10,1794
33,78 34,77 58,162
65,296 81,189 102,107
112,1311 226,951 242,257
270,283 284,950 314,1039
388,1037 392,1871 497,1036
515,947 648,1121 662,1800
758,1844 894,1868 960,1859
1056,1059 1057,1058 1125,1838
1220,1474 1737,1780 1807,1897
1842,1848

X(29) is the {X(3),X(4)}-harmonic conjugate of X(412). For a list of
other harmonic conjugates of X(29), click **Tables** at the top of
this page.

X(29) = isogonal conjugate of X(73)

X(29) = isotomic conjugate of X(307)

X(29) = inverse-in-circumcircle of X(2075)

X(29) = complement of X(3152)

X(29) = anticomplement of X(18641)

X(29) = X(286)-Ceva conjugate of X(27)

X(29) = cevapoint of X(i) and X(j) for these (i,j): (1,4), (33,281)

X(29) = X(i)-cross conjugate of X(j) for these (i,j): (1,21), (284,333), (497,314)

X(29) = crosssum of X(i) and X(j) for these (i,j): (1,1047), (228,1409)

X(29) = crossdifference of every pair of points on line X(647)X(822)

X(29) = X(4)-Hirst inverse of X(415)

X(29) = X(i)-beth conjugate of X(j) for these (i,j): (29,28), (811,29)

X(29) = intersection of tangents at X(1) and X(4) to hyperbola passing through X(1), X(4) and the excenters

X(29) = crosspoint of X(1) and X(4) wrt the excentral triangle

X(29) = trilinear pole of line X(243)X(522) (the polar of X(226) wrt polar circle)

X(29) = pole wrt polar circle of trilinear polar of X(226) (line X(523)X(656))

X(29) = polar conjugate of X(226)

X(29) = X(6)-isoconjugate of X(1214)

Trilinears bc[2a

Trilinears 2 sec A - sec B sec C : :

Trilinears sin B sin C - 3 cos B cos C : :

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2a

Barycentrics S^2 - 3 SB SC : :

X(30) = X(2) - X(3); if (i) and X(j) are on the Euler line, then X(30) = X(i) - X(j)

As a point on the Euler line, X(30) has Shinagawa coefficients (1, -3).

Let A'B'C' be the reflection triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" are parallel to the Euler line, and therefore concur in X(30). (Randy Hutson, December 10, 2016)

X(30) is the point of intersection of the Euler line and the line at infinity. Thus, each of the lines listed below is parallel to the Euler line.

If you have The Geometer's Sketchpad, you can view Euler Infinity Point.

X(30) lies on the Neuberg cubic, the Darboux quintic, and these (parallel) lines:

{1, 79}, {2, 3}, {6, 2549}, {7, 3488}, {8, 3578}, {9, 3587}, {10, 3579}, {11, 36}, {12, 35}, {13, 15}, {14, 16}, {17, 5238}, {18, 5237}, {32, 5254}, {33, 1060}, {34, 1062}, {40, 191}, {46, 1837}, {49, 1614}, {50, 1989}, {51, 5946}, {52, 185}, {53, 577}, {54, 3521}, {55, 495}, {56, 496}, {57, 3586}, {58, 1834}, {61, 397}, {62, 398}, {63, 3419}, {64, 68}, {65, 1770}, {69, 3426}, {74, 265}, {80, 484}, {84, 3928}, {98, 671}, {99, 316}, {100, 2687}, {101, 2688}, {102, 2689}, {103, 2690}, {104, 1290}, {105, 2691}, {106, 2692}, {107, 2693}, {108, 2694}, {109, 2695}, {110, 477}, {111, 2696}, {112, 2697}, {113, 1495}, {114, 2482}, {115, 187}, {119, 2077}, {128, 6592}, {133, 3184}, {137, 6150}, {141, 3098}, {143, 389}, {146, 323}, {148, 385}, {154, 5654}, {155, 1498}, {156, 1147}, {165, 5587}, {182, 597}, {226, 4304}, {250, 6530}, {262, 598}, {284, 1901}, {298, 616}, {299, 617}, {315, 1975}, {329, 3940}, {340, 1494}, {371, 3070}, {372, 3071}, {388, 3295}, {390, 1056}, {485, 1151}, {486, 1152}, {489, 638}, {490, 637}, {497, 999}, {498, 5217}, {499, 5204}, {511, 512}, {551, 946}, {553, 942}, {567, 5012}, {568, 3060}, {574, 3815}, {582, 1724}, {590, 6200}, {599, 1350}, {615, 6396}, {618, 623}, {619, 624}, {620, 625}, {664, 5195}, {841, 1302}, {908, 5440}, {910, 5179}, {925, 5962}, {935, 1297}, {938, 5708}, {944, 962}, {956, 3434}, {993, 2886}, {1043, 1330}, {1058, 3600}, {1117, 5671}, {1125, 3824}, {1131, 6407}, {1132, 6408}, {1141, 1157}, {1145, 5176}, {1146, 5011}, {1155, 1737}, {1160, 5860}, {1161, 5861}, {1213, 4877}, {1216, 5907}, {1285, 5304}, {1292, 2752}, {1293, 2758}, {1294, 1304}, {1295, 2766}, {1296, 2770}, {1319, 1387}, {1337, 3479}, {1338, 3480}, {1351, 1353}, {1376, 3820}, {1465, 1877}, {1490, 5763}, {1565, 4872}, {1587, 3311}, {1588, 3312}, {1625, 3289}, {1691, 6034}, {1699, 3576}, {1750, 5720}, {1754, 5398}, {1765, 5755}, {1768, 5535}, {1807, 3465}, {1838, 1852}, {1865, 2193}, {1870, 3100}, {1990, 3163}, {2021, 2023}, {2093, 5727}, {2094, 2095}, {2132, 2133}, {2292, 5492}, {2456, 5182}, {2548, 5013}, {2646, 4870}, {2654, 4303}, {2895, 4720}, {2931, 2935}, {2968, 5081}, {3003, 6128}, {3023, 6023}, {3027, 6027}, {3035, 3814}, {3053, 3767}, {3068, 6221}, {3069, 6398}, {3085, 5229}, {3086, 5225}, {3167, 5656}, {3255, 3577}, {3260, 6148}, {3292, 5609}, {3303, 4309}, {3304, 4317}, {3357, 5894}, {3424, 5485}, {3429, 4052}, {3436, 5687}, {3481, 3482}, {3485, 4305}, {3486, 4295}, {3487, 4313}, {3565, 5203}, {3589, 4045}, {3665, 4056}, {3703, 4680}, {3746, 4330}, {3829, 5450}, {3911, 5122}, {3917, 5891}, {3925, 5251}, {4030, 4692}, {4252, 5292}, {4296, 6198}, {4298, 5045}, {4301, 5882}, {4325, 4857}, {4421, 6256}, {4424, 5724}, {4511, 5057}, {4669, 5493}, {4677, 5881}, {4999, 5267}, {5008, 5355}, {5010, 5432}, {5032, 5093}, {5103, 5149}, {5107, 5477}, {5119, 5252}, {5180, 6224}, {5188, 6248}, {5207, 6393}, {5418, 6409}, {5420, 6410}, {5424, 5561}, {5448, 5893}, {5459, 5478}, {5460, 5479}, {5461, 6036}, {5463, 5473}, {5464, 5474}, {5538, 6326}, {5562, 5876}, {5603, 5731}, {5657, 5790}, {5703, 5714}, {5732, 5805}, {5758, 6223}, {5759, 5779}, {5858, 5864}, {5859, 5865}, {5889, 6241}, {5892, 5943}, {6104, 6107}, {6105, 6106}, {6193, 6225}, {6237, 6254}, {6238, 6285}

X(30) = isogonal conjugate of X(74)

X(30) = isotomic conjugate of X(1494)

X(30) = anticomplementary conjugate of X(146)

X(30) = complementary conjugate of X(113)

X(30) = orthopoint of X(523)

X(30) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,113), (265,5), (476,523)

X(30) = cevapoint of X(3) and X(399)

X(30) = crosspoint of X(i) and X(j) for these (i,j): (13,14), (94,264)

X(30) = crosssum of X(i) and X(j) for these (i,j): (15,16), (50,184)

X(30) = crossdifference of every pair of points on line X(6)X(647)

X(30) = ideal point of PU(30)

X(30) = vertex conjugate of PU(87)

X(30) = perspector of circumconic centered at X(3163)

X(30) = center of circumconic that is locus of trilinear poles of lines passing through X(3163)

X(30) = X(2)-Ceva conjugate of X(3163)

X(30) = trilinear pole of line X(1636)X(1637) (the line that is the tripolar centroid of the Euler line)

X(30) = X(517)-of-orthic triangle if ABC is acute

X(30) = X(542)-of-1st Brocard triangle

X(30) = crosspoint of X(3) and X(399) wrt both the excentral and tangential triangles

X(30) = crosspoint of X(616) and X(617) wrt both the excentral and anticomplementary triangles

X(30) = cevapoint of X(616) and X(617)

X(30) = X(6)-isoconjugate of X(2349)

X(30) = perspector of 2nd isogonal triangle of X(4) and cross-triangle of ABC and 2nd isogonal triangle of X(4)

X(30) = Thomson isogonal conjugate of X(110)

X(30) = Lucas isogonal conjugate of X(110)

X(30) = homothetic center of X(20)-altimedial and X(140)-anti-altimedial triangles

X(30) = X(1154)-of-excentral-triangle

X(30) = homothetic center of Ehrmann vertex-triangle and Trinh triangle

X(30) = homothetic center of Ehrmann side-triangle and dual of orthic triangle

X(30) = homothetic center of Ehrmann mid-triangle and medial triangle

X(30) = excentral-to-ABC functional image of X(517)

Trilinears 1 - cos 2A : 1 - cos 2B : 1 - cos 2C

Trilinears cot B + cot C : :

Trilinears S

Trilinears d(a,b,c) : : , where d(a,b,c) = distance between A and de Longchamps line

Barycentrics a

X(31) = (r^{2} +
s^{2})*X(1) - 6rR*X(2) - 4r^{2}*X(3) (Peter
Moses, April 2, 2013)

Let A'B'C' be the circumsymmedial triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. Then A", B", C" are collinear on line X(667)X(788) (the trilinear polar of X(31)). The lines AA", BB", CC" concur in X(31). (Randy Hutson, February 10, 2016)

Let A'B'C' be the Apus triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(31). (Randy Hutson, February 10, 2016)

Let A'B'C' be the Ara triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(31). (Randy Hutson, February 10, 2016)

Define the **1st Kenmotu diagonals triangle** as the triangle formed by the diagonals of the squares in the Kenmotu configuration with center X(371) that do not include X(371). Define the **2nd Kenmotu diagonals triangle** as the triangle formed by the diagonals of the squares in the Kenmotu configuration with center X(372) that do not include X(372). (Randy Hutson, February 10, 2016)

Let A_{1}B_{1}C_{1} and A_{2}B_{2}C_{2} be the 1st and 2nd Kenmotu diagonals triangles. Let A' be the trilinear product A_{1}*A_{2}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(31). (Randy Hutson, February 10, 2016)

X(31) is the Brianchon point (perspector) of the inellipse that is the trilinear square of the Lemoine axis. The center of the inellipse is X(16584). (Randy Hutson, October 15, 2018)

If you have The Geometer's Sketchpad, you can view X(31) (1), X(31) (2), X(31) (3).

If you have GeoGebra, you can view

X(31) lies on these lines:

1,21 2,171 3,601 4,3072
6,42 8,987 9,612 10,964
19,204 25,608 28,2282
32,41 33,2250 34,1254
35,386 36,995 37,2214
40,580 43,100 44,210
48,560 51,181 56,154
57,105 65,1104 72,976
75,82 76,734 86,2296
91,1087 92,162 99,715
101,609 106,2163 110,593
112,2249 158,2190 163,923
165,2999 172,1613 184,604
197,2183 198,2255 199,2277
200,1261 218,1260 222,1458
226,3011 237,904 240,1748
278,1430 284,2258 292,1915
354,1279 388,1935 404,978
497,1936 516,1754 561,722
582,3579 607,2357 649,884
663,2423 669,875 678,3158
692,2877 701,789 708,1502
740,3187 743,825 745,827
759,994 775,1097 872,2220
893,1691 899,1376 901,2382
937,1103 940,1001 982,3218
984,3219 990,1709 999,1149
1066,3157 1098,2363 1124,3076
1182,3192 1210,1771 1331,2991
1335,3077 1393,1454 1403,1428
1427,1456 1438,2279 1450,1470
1474,2215 1486,2260 1572,2170
1582,1740 1616,3304 1633,3123
1820,1953 1836,3120 1910,2186
1911,1922 1917,2085 1927,1967
1932,1973 1951,3010 1974,2281
1979,2107 2003,2078 2054,2248
2083,2156 2153,2154 2188,2638
2242,3230 2264,3198 2274,3286
2318,2911 3074,3085 3075,3086
3220,3415

X(31) is the {X(1),X(63)}-harmonic conjugate of X(38). For a list of
other harmonic conjugates of X(31), click **Tables** at the top of
this page.

X(31) = isogonal conjugate of X(75)

X(31) = isotomic conjugate of X(561)

X(31) = complement of X(6327)

X(31) = anticomplement of X(2887)

X(31) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,48), (6,41), (9,205), (58,6), (82,1)

X(31) = X(213)-cross conjugate of X(6)

X(31) = crosspoint of X(i) and X(j) for these (i,j): (1,19), (6,56)

X(31) = crosssum of X(i) and X(j) for these (i,j): (1,63), (2,8), (7,347), (10,321), (239,1281), (244,514), (307,1441), (523,1086), (693,1111)

X(31) = crossdifference of every pair of points on line X(514)X(661)

X(31) = X(1403)-Hirst inverse of X(1428)

X(31) = X(i)-aleph conjugate of X(j) for these (i,j): (82,31), (83,75)

X(31) = X(i)-beth conjugate of X(j) for these (i,j): (21,993), (55,55), (109,31), (110,57), (643,31), (692,31)

X(31) = barycentric product of PU(8)

X(31) = vertex conjugate of PU(8)

X(31) = bicentric sum of PU(i) for these i: 23, 48

X(31) = PU(23)-harmonic conjugate of X(661)

X(31) = PU(48)-harmonic conjugate of X(649)

X(31) = trilinear product of PU(36)

X(31) = trilinear product X(55)*X(56)

X(31) = trilinear pole of line X(667)X(788)

X(31) = pole wrt polar circle of trilinear polar of X(1969)

X(31) = X(48)-isoconjugate (polar conjugate) of X(1969)

X(31) = X(6)-isoconjugate of X(76)

X(31) = X(92)-isoconjugate of X(63)

X(31) = trilinear square of X(6)

X(31) = trilinear cube root of X(1917)

X(31) = vertex conjugate of foci of incentral inellipse

X(31) = perspector of ABC and extraversion triangle of X(31) (which is also the anticevian triangle of X(31))

X(31) = {X(1),X(1707)}-harmonic conjugate of X(63)

X(31) = perspector of ABC and unary cofactor triangle of extraversion triangle of X(7)

X(31) = perspector of ABC and unary cofactor triangle of extraversion triangle of X(8) (2nd Conway triangle)

X(31) = perspector of ABC and unary cofactor triangle of 4th Conway triangle

X(31) = perspector of unary cofactor triangles of 2nd and 4th Conway triangles

X(31) = perspector of unary cofactor triangles of Gemini triangles 2 and 30

X(31) = perspector of ABC and cross-triangle of Gemini triangles 33 and 34

X(31) = perspector of ABC and cross-triangle of ABC and Gemini triangle 33

X(31) = perspector of ABC and cross-triangle of ABC and Gemini triangle 34

X(31) = barycentric product of vertices of Gemini triangle 33

X(31) = barycentric product of vertices of Gemini triangle 34

X(31) = barycentric product of (nonreal) circumcircle intercepts of the antiorthic axis

X(31) = center of circumconic locus of trilinear poles of lines passing through X(32664)

X(31) = perspector of circumconic centered at X(32664)

X(31) = X(2)-Ceva conjugate of X(32664)

Trilinears sin(A - ω) : sin(B - ω) : sin(C - ω)

Trilinears sin A + sin(A - 2ω) : sin B + sin(B - 2ω) : sin C + sin(C - 2ω)

Trilinears cos A - cos(A - 2ω) : cos B - cos(B - 2ω) : cos C - cos(C -2ω)

Trilinears cos A - sin A cot ω : :

Trilinears sin A - cos A tan ω : :

Trilinears a - 2R cos A tan ω : :

Barycentrics a

If you have The Geometer's Sketchpad, you can view X(32).

The 5th Brocard triangle is here introduced as the vertex triangle of the circumcevian triangles of the 1st and 2nd Brocard points. (Randy Hutson, December 26, 2015)

The 5th Brocard triangle is homothetic to ABC at X(32), homothetic to the medial triangle at X(3096), homothetic to the anticomplementary triangle at X(2896), perspective to the 1st Brocard triangle at X(2896), and perspective to the 3rd Brocard triangle at X(32).(Randy Hutson, December 26, 2015)

Let A'B'C' be the 1st Brocard triangle. Let A", B", C" be inverse-in-circumcircle of A', B' and C' resp. AA", BB", CC" concur in X(32). (Randy Hutson, July 20, 2016)

Let A'B'C' be the 1st Brocard triangle. Let A" be the cevapoint, wrt A'B'C', of B and C, and define B", C" cyclically. A'A", B'B", C'C" concur in X(32). (Randy Hutson, July 20, 2016)

X(32) is the Brianchon point (perspector) of the inellipse that is the barycentric square of the Lemoine axis. The center of this inellipse is X(8265). (Randy Hutson, October 15, 2018)

X(32) lies on these lines:

1,172 2,83 3,6 4,98
5,230 9,987 20,2549
21,981 22,1194 24,232
25,1184 31,41 35,2276
48,1472 51,2351 55,1500
56,1015 71,2273 75,746
76,384 81,980 99,194
100,713 101,595 110,729
111,1383 163,849 165,1571
184,211 218,906 220,3052
262,3406 263,1976 512,878
538,1003 560,1918 561,724
590,640 604,1106 615,639
632,3055 637,3069 638,3068
682,1974 695,3492 710,1502
731,825 733,827 902,1334
904,1933 910,1104 941,1169
958,1572 983,3495 988,1449
993,1107 1009,1724 1055,1201
1084,1576 1092,3289 1191,3207
1204,3269 1376,1574 1395,1402
1423,3500 1468,2280 1613,1915
1843,2353 1911,1932 1919,3249
1922,1923 1950,2285 1951,2082
1992,2482 1995,3291 2004,2005
2319,3494 2508,2881 2698,2715
3087,3088 3124,3457 3170,3171
3497,3512 3499,3511

X(32) is the {X(3),X(6)}-harmonic conjugate of X(39). For a list of
other harmonic conjugates of X(32), click **Tables** at the top of
this page.

X(32) = midpoint of X(371) and X(372)

X(32) = reflection of X(315) in X(626)

X(32) = isogonal conjugate of X(76)

X(32) = isotomic conjugate of X(1502)

X(32) = inverse-in-circumcircle of X(1691)

X(32) = inverse-in-Brocard-circle of X(39)

X(32) = inverse-in-1st-Lemoine-circle of X(1692)

X(32) = complement of X(315)

X(32) = anticomplement of X(626)

X(32) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,206), (6,184), (112,512), (251,6)

X(32) = crosspoint of X(i) and X(j) for these (i,j): (2,66), (6,25)

X(32) = crosssum of X(i) and X(j) for these (i,j): (2,69), (6,22), (75,312), (115,826), (311,343), (313,321), (338,850), (339,525), (349,1231), (693,1086), (1229,1233), (1230,1269)

X(32) = crossdifference of every pair of points on line X(325)X(523)

X(32) = X(184)-Hirst inverse of X(237)

X(32) = X(i)-beth conjugate of X(j) for these (i,j): (41,41), (163,56), (919,32)

X(32) = external center of similitude of circumcircle and Moses circle

X(32) = radical trace of circumcircle and circle {X(1687),X(1688),PU(1),PU(2)}

X(32) = trilinear product of vertices of circumsymmedial triangle

X(32) = trilinear product of vertices of 3rd Brocard triangle

X(32) = insimilicenter of circles O(15,16) and O(61,62); the exsimilicenter is X(39)

X(32) = insimilicenter of circles {X(15),X(62),PU(1)} and {X(16),X(61),PU(1)}; the exsimilicenter is X(182)

X(32) = intersection of tangents at PU(1) to Brocard circle

X(32) = intersection of lines P(1)U(2) and U(1)P(2)

X(32) = vertex conjugate of PU(1)

X(32) = trilinear product of PU(9)

X(32) = barycentric product of PU(36)

X(32) = bicentric sum of PU(39)

X(32) = midpoint of PU(39)

X(32) = center of circle {{X(371),X(372),PU(1),PU(39)}} (the circle orthogonal to the Brocard circle through the 1st and 2nd Brocard points)

X(32) = crosssum of polar conjugates of PU(4)

X(32) = perspector ABC and tangential triangle of 1st Brocard triangle

X(32) = trilinear cube of X(6)

X(32) = trilinear square root of X(1917)

X(32) = inverse-in-2nd-Brocard-circle of X(3094)

X(32) = perspector of circumconic centered at X(206)

X(32) = center of circumconic that is locus of trilinear poles of lines passing through X(206)

X(32) = trilinear pole of line X(669)X(688) (the isogonal conjugate of the isotomic conjugate of the Lemoine axis)

X(32) = perspector of ABC and 3rd Brocard triangle

X(32) = {X(61),X(62)}-harmonic conjugate of X(576)

X(32) = {X(1340),X(1341)}-harmonic conjugate of X(5116)

X(32) = {X(1687),X(1688)}-harmonic conjugate of X(3)

X(32) = reflection of X(5028) in X(6)

X(32) = X(32)-of-circumsymmedial-triangle

X(32) = X(75)-isoconjugate of X(2)

X(32) = X(92)-isoconjugate of X(69)

X(32) = X(1577)-isoconjugate of X(99)

X(32) = X(4048) of 1st anti-Brocard triangle

X(32) = homothetic center of circumnormal triangle and unary cofactor triangle of Stammler triangle

X(32) = perspector of ABC and cross-triangle of ABC and 1st Brocard triangle

X(32) = homothetic center of medial triangle and cross-triangle of ABC and 5th Brocard triangle

X(32) = homothetic center of medial triangle and cross-triangle of ABC and 5th anti-Brocard triangle

X(32) = Cundy-Parry Phi transform of X(511)

X(32) = Cundy-Parry Psi transform of X(98)

X(32) = X(169)-of-orthic-triangle if ABC is acute

X(32) = barycentric square of X(6)

X(32) = barycentric product of (nonreal) circumcircle intercepts of the Lemoine axis

= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b

= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec A cos

Barycentrics sin A + tan A : sin B + tan B : sin C + tan
C

=
h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = tan A
cos^{2}(A/2)

X(33) = (r + 2R - s)(r + 2R + s)*X(1) - 6rR*X(2) + 4rR*X(3) (Peter Moses, April 2, 2013)

Let L_{A} be the reflection of line BC in the internal angle
bisector of angle A, and define L_{B} and L_{C}
cyclically. Let DEF be the triangle formed by L_{A},
L_{B}, L_{C}. Then DEF (the intangents triangle) is
homothetic to the orthic triangle, and the homothetic center is X(33).
(Randy Hutson, 9/23/2011)

If you have The Geometer's Sketchpad, you can view X(33).

If you have GeoGebra, you can view

X(33) lies on these lines:

1,4 2,1040 5,1062 6,204
7,1041 8,1039 9,212
10,406 11,427 12,235
19,25 20,1038 24,35
28,975 29,78 30,1060
36,378 40,201 42,393
47,90 56,963 57,103
63,1013 64,65 79,1063
80,1061 84,603 112,609
200,281 210,220 222,971 264,350

X(33) is the {X(1),X(4)}-harmonic conjugate of X(34). For a list of
other harmonic conjugates, click **Tables** at the top of this
page.

X(33) = isogonal conjugate of X(77)

X(33) = isotomic conjugate of X(7182)

X(33) = trilinear pole of line X(657)X(4041) (the polar of X(85) wrt polar circle)

X(33) = pole wrt polar circle of trilinear polar of X(85) (line X(522)X(693))

X(33) = polar conjugate of X(85)

X(33) = perspector of ABC and extraversion triangle of X(34)

X(33) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,19), (29,281), (318,9)

X(33) = X(i)-cross conjugate of X(j) for these (i,j): (41,9), (42,55)

X(33) = crosspoint of X(i) and X(j) for these (i,j): (1,282), (4,281)

X(33) = crosssum of X(i) and X(j) for these (i,j): (1,223), (3,222), (57,1394), (73,1214)

X(33) = crossdifference of every pair of points on line X(652)X(905)

X(33) = X(33)-beth conjugate of X(25)

X(33) = homothetic center of anti-excenters-incenter reflections triangle and anti-tangential midarc triangle

Trilinears tan A tan(A/2) : tan B tan(B/2) : tan C tan(C/2)

Trilinears 1/[(b + c - a)(b

Trilinears sec A sin

Barycentrics sin A - tan A : sin B - tan B : sin C - tan C

Barycentrics tan A sin

Tripolars (pending)

X(34) = (r + 2R - s)(r + 2R + s)*X(1) + 6rR*X(2) - 4rR*X(3) (Peter Moses, April 2, 2013)

X(34) is the center of perspective of the orthic triangle and the reflection in the incenter of the intangents triangle.

If you have The Geometer's Sketchpad, you can view X(34) (1) and X(34) (2).

If you have GeoGebra, you can view

X(34) lies on these lines:

1,4 2,1038 5,1060 6,19
7,1039 8,1041 9,201
10,475 11,235 12,427
20,1040 24,36 25,56
28,57 29,77 30,1062
35,378 40,212 46,47
55,227 79,1061 80,1063
87,242 106,108 196,937
207,1042 222,942 244,1106
331,870 347,452 860,997

X(34) is the {X(1),X(4)}-harmonic conjugate of X(33). For a list of
other harmonic conjugates of X(34), click **Tables** at the top of
this page.

X(34) = isogonal conjugate of X(78)

X(34) = isotomic conjugate of X(3718)

X(34) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,207), (4,208), (28,56), (273,57), (278,19)

X(34) = X(25)-cross conjugate of X(19)

X(34) = crosssum of X(219) and X(1260)

X(34) = crossdifference of every pair of points on line X(521)X(652)

X(34) = circumcircle-inverse of X(32757)

X(34) = X(56)-Hirst inverse of X(1430)

X(34) = trilinear pole of polar of X(312) wrt polar circle (line X(649)X(4017))

X(34) = pole wrt polar circle of trilinear polar of X(312) (line X(522)X(3717))

X(34) = polar conjugate of X(312)

X(34) = perspector of ABC and extraversion triangle of X(33)

X(34) = homothetic center of intangents triangle and reflection of orthic triangle in X(4)

X(34) = homothetic center of orthic triangle and anti-tangential midarc triangle

X(34) = X(8078)-of-orthic-triangle if ABC is acute

X(34) = X(i)-beth conjugate of X(j) for these (i,j): (1,221), (4,4), (28,34), (29,1), (107,158), (108,34), (110,47), (162,34), (811,34)

X(34) = X(i)-isoconjugate of X(j) for these {i,j}: {1,78}, {31,3718}, {48,312}

Trilinears a(b

Trilinears sin(3A/2) csc(A/2) : :

Barycentrics sin A + sin 2A : :

Barycentrics a

Let A' be the inverse-in-circumcircle of the A-excenter, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(35).

Let A'B'C' be the orthic triangle. Let B'C'A" be the triangle similar to ABC such that segment A'A" crosses the line B'C', and define B" and C" cyclically. (Equivalently, A" is the reflection of A in B'C'.) Let Ia be the incenter of B'C'A", and define Ib and Ic cyclically. The lines AIa, BIb, CIc concur in X(35). (Randy Hutson, November 18, 2015)

X(35) lies on these lines:

1,3 4,498 8,993 9,90
10,21 11,140 12,30
22,612 24,33 31,386
34,378 37,267 42,58
43,1011 47,212 71,284
72,191 73,74 79,226
172,187 228,846 255,991
376,388 404,1125 411,516
474,1001 495,550 496,549
497,499 500,1154 595,902
950,1006 968,975 1124,1152

If you have The Geometer's Sketchpad, you can view X(35).

If you have GeoGebra, you can view

X(35) is the {X(1),X(3)}-harmonic conjugate of X(36). For a list of
other harmonic conjugates of X(35), click **Tables** at the top of
this page.

X(35) = isogonal conjugate of X(79)

X(35) = isotomic conjugate of X(20565)

X(35) = homothetic center of Trinh triangle and anti-tangential midarc triangle

X(35) = inverse-in-circumcircle of X(484)

X(35) = X(500)-cross conjugate of X(1)

X(35) = crosssum of X(481) and X(482)

X(35) = X(943)-aleph conjugate of X(35)

X(35) = X(i)-beth conjugate of X(j) for these (i,j): (100,35), (643,10)

X(35) = perspector of ABC and orthic triangle of incentral triangle

X(35) = X(2975) of X(1)-Brocard triangle

X(35) = {X(55),X(56)}-harmonic conjugate of X(3295)

X(35) = crossdifference of every pair of points on line X(650)X(4802)

X(35) = homothetic center of intangents and Kosnita triangles

X(35) = perspector of ABC and extraversion triangle of X(36)

X(35) = Hofstadter 3/2 point

X(35) = homothetic center of 2nd isogonal triangle of X(1) and cevian triangle of X(3); see X(36)

X(35) = insimilicenter of circumcircle and circumcircle of reflection triangle of X(1); exsimilicenter is X(36)

X(35) = Cundy-Parry Phi transform of X(5902)

Trilinears a(b

Trilinears sec(A/2) cos(3A/2) : :

Barycentrics sin A - sin 2A : :

Barycentrics a

Tripolars Sqrt[b c (b + c - a)] : :

Tripolars sec A' : :, where A'B'C' is the excentral triangle

If you have The Geometer's Sketchpad, you can view X(36).

If you have GeoGebra, you can view **X(36)**.

Let A' be the isogonal conjugate of A with respect to BCX(1), and define B' and C' cyclically. Let A" be the circumcenter of BCX(1), and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(36). Also, X(36) is the QA-P4 center (Isogonal Center) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/25-qa-p4.html)

Let P be a point in the plane of triangle ABC, not on a sideline of ABC. Let A1 be the isogonal conjugate of A with respect to triangle BCP, and define B1, C1 cyclically. Call triangle A1B1C1 the 1st isogonal triangle of P. A1B1C1 is also the reflection triangle of the isogonal conjugate of P. A1B1C1 is perspective to ABC iff P lies on the Neuberg cubic. The perspector lies on cubic K060 (pK(X1989, X265), O(X5) orthopivotal cubic). Let A2 be the isogonal conjugate of A1 with respect to triangle B1C1P, and define B2, C2 cyclically. Call triangle A2B2C2 the 2nd isogonal triangle of P. Continuing, let An be the isogonal conjugate of A(n-1) with respect to triangle B(n-1)C(n-1)P, and define B(n), C(n) cyclically. Call triangle AnBnCn the nth isogonal triangle of P. For n >= 2, all triangles AnBnCn are perspective to A(n-1)B(n-1)C(n-1). Call the perspector, Pn, the nth isogonal perspector of P. Pn is the orthocenter of A(n-1)B(n-1)C(n-1) and either the incenter or an excenter of AnBnCn. The triangles AnBnCn are all concyclic, with P as center. Call the circle the isogonal circle of P. For P = X(1), the 2nd isogonal triangle of X(1) is homothetic to ABC at X(36); see also X(35), X(1478), X(1479), X(3583), X(3585), X(5903), X(7741), X(7951). (Randy Hutson, November 18, 2015)

Let A'B'C' be the incentral triangle. Let A" be the reflection of A in line B'C', and define B", C" cyclically. The lines A'A", B'B", C'C" concur in X(36). (Randy Hutson, June 27, 2018)

X(36) is the {X(3),X(56)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(36), click **Tables** at the top of this page.

X(36) lies on these lines:

1,3 2,535 4,499 6,609
10,404 11,30 12,140
15,202 16,203 21,79
22,614 24,34 31,995
33,378 39,172 47,602
48,579 54,73 58,60
59,1110 63,997 80,104
84,90 99,350 100,519
101,672 106,901 109,953
187,1015 191,960 214,758
226,1006 238,513 255,1106
376,497 388,498 474,958
495,549 496,550 573,604
1030,1100

X(36) = midpoint of X(1) and X(484)

X(36) = reflection of X(i) in X(j) for these (i,j): (1,1319), (484,1155) (2077,3)

X(36) = isogonal conjugate of X(80)

X(36) = isotomic conjugate of X(20566)

X(36) = polar conjugate of isotomic conjugate of X(22128)

X(36) = complement of X(5080)

X(36) = anticomplement of X(3814)

X(36) = inverse-in-circumcircle of X(1)

X(36) = inverse-in-incircle of X(942)

X(36) = inverse-in-Bevan-circle of X(46)

X(36) = X(i)-Ceva conjugate of X(j) for these (i,j): (88,6), (104,1)

X(36) = crosspoint of X(58) and X(106)

X(36) = crosssum of X(i) and X(j) for these (i,j): (1,484), (10,519), (11,900)

X(36) = crossdifference of every pair of points on line X(37)X(650)

X(36) = X(104)-aleph conjugate of X(36)

X(36) = X(i)-beth conjugate of X(j) for these (i,j): (21,36), (100,36), (643,519)

X(36) = X(2070)-of-intouch-triangle

X(36) = X(186)-of-2nd circumperp-triangle

X(36) = {X(55),X(56)}-harmonic conjugate of X(999)

X(36) = reflection of X(484) in the antiorthic axis

X(36) = inverse-in-{circumcircle, nine-point circle}-inverter of X(354)

X(36) = perspector of ABC and extraversion triangle of X(35)

X(36) = homothetic center of intangents and Trinh triangles

X(36) = perspector of ABC and the reflection of the 2nd circumperp triangle in line X(1)X(3)

X(36) = X(186)-of-reflection-triangle-of-X(1)

X(36) = exsimilicenter of circumcircle and circumcircle of reflection triangle of X(1); insimilicenter is X(35)

X(36) = homothetic center of medial triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)

X(36) = perspector of ABC and the reflection of the excentral triangle in the antiorthic axis (the reflection of the anticevian triangle of X(1) in the trilinear polar of X(1))

X(36) = Cundy-Parry Phi transform of X(5903)

X(36) = homothetic center of Kosnita triangle and anti-tangential midarc triangle

Trilinears ar - S : br - S : cr - S

Trilinears semi-major axis of A-Soddy ellipse : :

Barycentrics a(b + c) : b(c + a) : c(a + b)

Tripolars (pending)

Let A'B'C' be the cevian triangle of X(1). Let A" be the centroid of triangle AB'C', and define B" and C" cyclically. Then the lines AA", BB", CC" concur in X(37). (Eric Danneels, Hyacinthos 7892, 9/13/03)

A simple construction of X(37) as a crosspoint can be generalized as follows: let DEF be the medial triangle of ABC and let A'B'C' be the cevian triangle of a point U other than the centroid, X(2). The crosspoint of X(2) and U is then the point of concurrence of lines LD,ME,NF, where L,M,N are the respective midpoints of AA', BB', CC'. If U=u : v : w (trilinears), then crosspoint(X(2),U) = b/w+c/v : c/u+a/w : a/v+b/u, assuming that uvw is nonzero. In particular, if U=X(1), then the crosspoint is X(37). (Seiichi Kirikami, July 10, 2011)

X(37) = perspector of ABC and the medial triangle of the incentral triangle of ABC. (Randy Hutson, August 23, 2011)

X(37) = center of the Hofstadter ellipse E(1/2); see X(359). This is the incentral inellipse, which is the trilinear square of the antiorthic axis. (Randy Hutson, August 9, 2014)

Let A' be the trilinear pole of the tangent to the Apollonius circle where it meets the A-excircle, and define B' and C' cyclically. The triangle A'B'C' is homothetic to ABC at X(37). (Randy Hutson, April 9, 2016)

Let Oa be the A-extraversion of the Conway circle (the circle centered at the A-excenter and passing through A, with radius sqrt(r_{a}^2 + s^2), where r_{a} is the A-exradius). Let La be the radical axis of the circumcircle and Oa. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The lines AA', BB', CC' concur in X(37). (Randy Hutson, April 9, 2016)

If you have The Geometer's Sketchpad, you can view X(37).

If you have GeoGebra, you can view **X(37)**.

X(37) lies on these lines:

1,6 2,75 3,975 7,241
8,941 10,594 12,225
19,25 21,172 35,267
38,354 39,596 41,584
48,205 63,940 65,71
73,836 78,965 82,251
86,190 91,498 100,111
101,284 141,742 142,1086
145,391 158,281 171,846
226,440 256,694 347,948
513,876 517,573 537,551
579,942 626,746 665,900 971,991
1953,2183

X(37) is the {X(1),X(9)}-harmonic conjugate of X(6). For a list of
other harmonic conjugates of X(37), click **Tables** at the top of
this page.

X(37) = midpoint of X(i) and X(j) for these (i,j): (75,192), (190,335)

X(37) = isogonal conjugate of X(81)

X(37) = isotomic conjugate of X(274)

X(37) = complement of X(75)

X(37) = complementary conjugate of X(2887)

X(37) = anticomplement of X(3739)

X(37) = circumcircle-inverse of X(32758)

X(37) = X(i)-Ceva conjugate of X(j) for these (i,j):

(1,42), (2,10), (4,209), (9,71), (10,210), (190,513), (226,65),
(321,72), (335,518)

X(37) = cevapoint of X(213) and X(228)

X(37) = X(i)-cross conjugate of X(j) for these (i,j): (42,65), (228,72)

X(37) = crosspoint of X(i) and X(j) for these (i,j): (1,2), (9,281), (10,226)

X(37) = X(1)-line conjugate of X(238)

X(37) = crosssum of X(i) and X(j) for these (i,j): (1,6), (57,222), (58,284), (1333,1437)

X(37) = crossdifference of every pair of points on line X(36)X(238)

X(37) = X(10)-Hirst inverse of X(740)

X(37) = X(1)-aleph conjugate of X(1051)

X(37) = X(i)-beth conjugate of X(j) for these (i,j): (9,37), (644,37), (645,894), (646,37), (1018,37)

X(37) = midpoint of PU(i), for these i: 6, 52, 53

X(37) = bicentric sum of PU(i), forthese i: 6, 52, 53

X(37) = trilinear product of PU(32)

X(37) = center of circumconic that is locus of trilinear poles of lines passing through X(10)

X(37) = perspector of circumconic centered at X(10)

X(37) = trilinear pole of line X(512)X(661) (polar of X(286) wrt polar circle)

X(37) = trilinear pole wrt medial triangle of Gergonne line

X(37) = pole wrt polar circle of trilinear polar of X(286) (line X(693)X(905))

X(37) = X(48)-isoconjugate (polar conjugate) of X(286)

X(37) = {X(6),X(9)}-harmonic conjugate of X(44)

X(37) = X(160)-of-intouch triangle

X(37) = perspector of incentral triangle and tangential triangle, wrt incentral triangle, of circumconic of incentral triangle centered at X(1) (the bicevian conic of X(1) and X(57))

X(37) = inverse-in-circumconic-centered-at-X(9) of X(1757)

X(37) = complement wrt incentral triangle of X(2667)

X(37) = perspector of ABC and unary cofactor triangle of 2nd circumperp triangle

X(37) = perspector of medial triangle and Gergonne line extraversion triangle

X(37) = trilinear pole, wrt Gergonne line extraversion triangle, of Gergonne line

X(37) = perspector of ABC and cross-triangle of Gemini triangles 3 and 4

X(37) = perspector of ABC and cross-triangle of ABC and Gemini triangle 3

X(37) = perspector of ABC and cross-triangle of ABC and Gemini triangle 4

X(37) = perspector of ABC and cross-triangle of ABC and Gemini triangle 16

X(37) = center of the {ABC, Gemini 17}-circumconic

X(37) = perspector of ABC and unary cofactor triangle of Gemini triangle 23

X(37) = incentral-to-ABC barycentric image of X(37)

= csc A sin(A + ω) : csc B sin(B + ω) : csc C sin(C + ω)

= S

Barycentrics a(b^{2} + c^{2}) :
b(c^{2} + a^{2}) : c(a^{2} + b^{2})

=
sin(A + ω) : sin(B + ω) : sin(C + ω)**X(38) = 3r ^{2} + 8rR - s^{2})*X(1) - 6rR*X(2) -
4r^{2}*X(3) (Peter Moses, April 2, 2013) **

X(38) lies on these lines:

1,21 2,244 3,976 8,986
9,614 10,596 37,354
42,518 56,201 57,612
75,310 78,988 92,240
99,745 210,899 321,726
869,980 912,1064 1038,1106

X(38) is the {X(1),X(63)}-harmonic conjugate of X(31). For a list of
other harmonic conjugates of X(38), click **Tables** at the top of
this page.

X(38) = isogonal conjugate of X(82)

X(38) = isotomic conjugate of X(3112)

X(38) = anticomplement of X(1215)

X(38) = crosspoint of X(1) and X(75)

X(38) = crosssum of X(1) and X(31)

X(38) = crossdifference of every pair of points on line X(661)X(830)

X(38) = X(643)-beth conjugate of X(38)

X(38) = bicentric sum of PU(35)

X(38) = PU(35)-harmonic conjugate of X(661)

X(38) = trilinear pole of line X(2084)X(2530)

X(38) = perspector of ABC and extraversion triangle of X(38) (which is also the anticevian triangle of X(38))

X(38) = barycentric square root of X(8041)

Trilinears sin(A + ω) : sin(B + ω) : sin(C + ω)

Trilinears sin A + sin(A + 2ω) : sin B + sin(B + 2ω) : sin C + sin(C + 2ω)

Trilinears cos A - cos(A + 2ω) : cos B - cos(B + 2ω) : cos C - cos(C + 2ω)

Trilinears sin A + cos A tan ω : :

Trilinears cos A + sin A cot ω : :

Trilinears a + 2R cos A tan ω : :

Barycentrics a

X(39) = P(1) + U(1)

X(39) is the midpoint of the 1st and 2nd Brocard points, given by trilinears c/b : a/c : b/a and b/c : c/a : a/b. The third and fourth trilinear representations were given by Peter J. C. Moses (10/3/03); cf. X(511), X(32), X(182).

The locus of the nine-point center in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is the circle through X(5) with center X(39). (Randy Hutson, August 29, 2018)

**Ross Honsberger,** *Episodes in Nineteenth and Twentieth Century Euclidean Geometry,* Mathematical Association of America, 1995. Chapter 10: The Brocard Points.

X(39) lies on the bicevian conic of X(2) and X(99) and on these lines:

1,291 2,76 3,6 4,232
5,114 9,978 10,730
36,172 37,596 51,237
54,248 83,99 110,755
140,230 141,732 185,217
213,672 325,626 395,618
493,494 512,881 588,589
590,642 597,1084 615,641

X(39) is the {X(3),X(6)}-harmonic conjugate of X(32). For a list of
other harmonic conjugates of X(39), click **Tables** at the top of
this page.

X(39) = midpoint of X(76) and X(194)

X(39) = reflection of X(5052) in X(6)

X(39) = isogonal conjugate of X(83)

X(39) = isotomic conjugate of X(308)

X(39) = inverse-in-circumcircle of X(2076)

X(39) = inverse-in-Brocard-circle of X(32)

X(39) = inverse-in-1st-Lemoine-circle of X(2458)

X(39) = complement of X(76)

X(39) = complementary conjugate of X(626)

X(39) = eigencenter of anticevian triangle of X(512)

X(39) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,141), (4,211), (99,512)

X(39) = crosspoint of X(i) and X(j) for these (i,j): (2,6), (141,427)

X(39) = crosssum of X(i) and X(j) for these (i,j): (2,6), (251,1176)

X(39) = crossdifference of every pair of points on line X(23)X(385)

X(39) = radical trace of 1st and 2nd Brocard circles

X(39) = exsimilicenter of circles O(15,16) and O(61,62); the insimilicenter is X(32)

X(39) = radical trace of circles {X(15),X(62),PU(1)} and {X(16),X(61),PU(1)}

X(39) = anticenter of cyclic quadrilateral PU(1)PU(39)

X(39) = bicentric sum of PU(i) for these i: 1, 67

X(39) = midpoint of PU(1)

X(39) = PU(67)-harmonic conjugate of X(351)

X(39) = X(5007) of 5th Brocard triangle

X(39) = X(5026) of 6th Brocard triangle

X(39) = center of Moses circle

X(39) = center of Gallatly circle

X(39) = inverse-in-2nd-Brocard-circle of X(6)

X(39) = inverse-in-Kiepert-hyperbola of X(5)

X(39) = {X(61),X(62)}-harmonic conjugate of X(575)

X(39) = {X(1687),X(1688)}-harmonic conjugate of X(3398)

X(39) = {X(2009),X(2010)}-harmonic conjugate of X(5)

X(39) = Brocard axis intercept of radical axis of nine-point circles of ABC and circumsymmedial triangle

X(39) = intersection of tangents to hyperbola {{A,B,C,X(2),X(6)}} at X(2) and X(6)

X(39) = perspector of circumconic centered at X(141)

X(39) = center of circumconic that is locus of trilinear poles of lines passing through X(141)

X(39) = trilinear pole, wrt medial triangle, of orthic axis

X(39) = trilinear pole of line X(688)X(3005)

X(39) = perspector of medial triangle of ABC and medial triangle of 1st Brocard triangle

X(39) = X(2029)-of-2nd-Brocard triangle

X(39) = X(39)-of-circumsymmedial-triangle

X(39) = perspector, wrt symmedial triangle, of bicevian conic of X(6) and X(25)

X(39) = intersection of Brocard axes of ABC and 5th Euler triangle

X(39) = X(92)-isoconjugate of X(1176)

X(39) = X(1577)-isoconjugate of X(827)

X(39) = eigencenter of Steiner triangle

X(39) = perspector of ABC and unary cofactor triangle of circummedial triangle

X(39) = center of (equilateral) unary cofactor triangle of Stammler triangle

X(39) = X(7753)-of-4th-anti-Brocard-triangle

X(39) = X(11)-of-X(3)PU(1)

X(39) = X(115)-of-X(3)PU(1)

X(39) = X(125)-of-X(3)PU(1)

X(39) = homothetic center of Kosnita triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles

X(39) = Cundy-Parry Phi transform of X(182)

X(39) = Cundy-Parry Psi transform of X(262)

X(39) = endo-similarity image of antipedal triangles of PU(1); the similitude center of these triangles is X(3)

X(39) = orthoptic-circle-of-Steiner-inellipse-inverse of X(32526)

X(39) = QA-P42 (QA-Orthopole Center) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/index.php/15-mathematics/encyclopedia-of-quadri-figures/quadrangle-objects/artikelen-qa/228-qa-p42.html)

Trilinears b/(c + a - b) + c/(a + b - c) - a/(b + c -a) : :

Trilinears sin

Trilinears a^3 + a^2(b + c) - a(b + c)^2 - (b + c)(b - c)^2 : :

Trilinears b(tan B/2) + c(tan C/2) - a(tan A/2) : :

Trilinears (r/R) - 2 cos A : :

X(40) = X(1) - 2X(3) = 2R*X(4) - (r + 4R)*X(9)

X(40) = X[1] - 3 X[165] = 2 X[3] - 3 X[165] = 3 X[376] - X[944] = 5 X[631] - 4 X[1125] = 3 X[1] - 4 X[1385] = 9 X[165] - 4 X[1385] = 3 X[3] - 2 X[1385] = 3 X[1] - 2 X[1482] = 9 X[165] - 2 X[1482] = 3 X[3] - X[1482] = 4 X[5] - 5 X[1698] = 4 X[5] - 3 X[1699] = 5 X[1698] - 3 X[1699] = 2 X[3095] - 3 X[3097] = X[145] - 5 X[3522] = 2 X[551] - 3 X[3524] = 2 X[3244] - 7 X[3528] = 8 X[1385] - 9 X[3576] = 4 X[1482] - 9 X[3576] = 2 X[1] - 3 X[3576] = 4 X[3] - 3 X[3576] = 3 X[3576] - 8 X[3579]

If you have GeoGebra, you can view **X(40)**.

This point is mentioned in a problem proposal by Benjamin Bevan, published in Leybourn's *Mathematical Repository*, 1804, p. 18.

Constructions received from Randy Hutson, January 29, 2015:

(1) Let A'B'C' be the extangents triangle. Let A" be the reflection of A' in BC, and define B", C" cyclically. A'A", B'B", C'C" concur in X(40).

(2) Let A'B'C' be the extangents triangle. Let A" be the cevapoint of B' and C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(40).

(3) Let A'B'C' be the hexyl triangle and A"B"C" be the side-triangle of ABC and hexyl triangle. Let A* be the {B,C}-harmonic conjugate of A", and define B*, C* cyclically. The lines A'A*, B'B*, C'C* concur in X(40).

(4) Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib, Ic cyclically. X(40) = X(104)-of-IaIbIc.

(5) Let A'B'C' be the cevian triangle of X(189). Let A" be the orthocenter of AB'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(40).
(6) Let A'B'C' be the mixtilinear incentral triangle. Let A" be the trilinear product B'*C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(40).

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is homothetic to the Aquila triangle at X(40). (Randy Hutson, December 2, 2017)

Let A'B'C' be the excentral triangle. Let A" be the isogonal conjugate, wrt A'BC, of A. Define B" and C" cyclically. (A" is also the reflection of A' in BC, and cyclically for B" and C"). The lines A'A", B'B", C'C" concur in X(40). (Randy Hutson, December 2, 2017)

X(40) lies on the following curves: Q010, Q122, K004, K033, K100, K133, K179, K199, K269, K308, K333, K338, K343, K384, K414, K619, K667, K679, K692, K710, K736, K750, K806, K807, K815, K826, the Jerabek hyhperbola of the excentral triangle, the Mandart hyperbola, and these lines:

{1,3}, {2,946}, {4,9}, {5,1698}, {6,380}, {7,7160}, {8,20}, {11,6922}, {12,1836}, {15,10636}, {16,10637}, {21,3577}, {22,9626}, {24,9625}, {25,1902}, {26,9590}, {28,2328}, {30,191}, {31,580}, {32,9620}, {33,201}, {34,212}, {39,1571}, {41,2301}, {42,581}, {43,970}, {47,1774}, {49,9586}, {58,601}, {64,72}, {74,6011}, {75,10444}, {77,947}, {78,100}, {79,4338}, {80,90}, {92,412}, {98,6010}, {101,972}, {103,1292}, {104,1293}, {108,207}, {109,255}, {140,3624}, {144,5815}, {145,3218}, {149,10265}, {151,2816}, {153,3648}, {154,7973}, {164,188}, {168,9837}, {170,1282}, {173,8351}, {182,1700}, {184,9622}, {185,3611}, {187,10988}, {190,341}, {196,208}, {197,3556}, {198,2324}, {209,3293}, {210,1750}, {214,10698}, {219,610}, {220,910}, {221,223}, {226,3085}, {228,3191}, {238,1722}, {256,989}, {258,8092}, {269,8829}, {307,4329}, {322,8822}, {329,6260}, {347,7013}, {371,5415}, {372,5416}, {376,519}, {381,7989}, {382,5790}, {386,1064}, {387,579}, {388,3474}, {389,11435}, {390,938}, {392,474}, {402,12696}, {404,3877}, {405,1730}, {442,5715}, {483,3645}, {485,13893}, {486,13947}, {495,4312}, {497,1210}, {498,5219}, {499,6891}, {500,8274}, {511,1045}, {518,1071}, {542,9881}, {548,3633}, {549,3656}, {550,952}, {551,3524}, {572,1449}, {574,9619}, {576,8539}, {578,11428}, {595,602}, {612,2292}, {614,3915}, {631,1125}, {653,1895}, {659,2821}, {664,7183}, {672,2082}, {726,12251}, {728,1018}, {730,11257}, {738,1323}, {758,3158}, {774,4319}, {813,2724}, {846,9840}, {901,2716}, {902,3924}, {908,5552}, {912,5534}, {920,4302}, {936,960}, {943,5665}, {950,1708}, {953,2743}, {954,12560}, {956,3916}, {958,1012}, {971,2951}, {978,1050}, {979,9359}, {984,1721}, {990,7174}, {993,6906}, {997,3878}, {1000,4315}, {1001,3812}, {1006,3754}, {1043,7415}, {1054,11512}, {1056,4298}, {1058,11019}, {1066,4306}, {1104,3052}, {1118,1785}, {1130,6585}, {1145,2829}, {1147,9621}, {1151,7969}, {1152,7968}, {1154,6255}, {1160,5588}, {1161,5589}, {1181,2323}, {1191,3752}, {1253,1254}, {1256,9376}, {1320,11715}, {1329,1532}, {1330,2792}, {1334,7390}, {1386,5085}, {1419,3157}, {1421,1772}, {1423,13161}, {1448,7273}, {1473,8192}, {1475,11200}, {1478,1770}, {1479,1737}, {1480,5315}, {1483,3655}, {1496,4320}, {1503,1761}, {1519,6834}, {1537,3035}, {1587,13883}, {1588,13936}, {1593,1829}, {1621,6986}, {1630,2289}, {1633,2823}, {1656,7988}, {1657,4668}, {1707,5247}, {1712,3176}, {1723,2955}, {1724,3073}, {1725,1775}, {1726,2949}, {1727,4324}, {1728,1837}, {1736,4907}, {1739,12659}, {1743,2264}, {1745,2818}, {1746,4714}, {1748,5174}, {1765,3696}, {1769,9525}, {1777,1935}, {1783,7156}, {1790,3193}, {1817,1819}, {1834,2245}, {1859,1872}, {1870,4347}, {1888,3074}, {2066,2362}, {2123,3421}, {2130,3354}, {2131,3472}, {2177,2650}, {2218,4674}, {2222,2745}, {2254,2814}, {2266,4251}, {2269,2285}, {2294,3247}, {2331,3194}, {2717,2742}, {2771,5531}, {2777,10119}, {2778,2915}, {2782,9860}, {2784,4050}, {2794,4769}, {2796,12243}, {2801,5528}, {2807,5562}, {2835,3939}, {2886,5705}, {2900,10605}, {2939,2947}, {2940,2948}, {2945,2953}, {2946,2952}, {2956,9370}, {2957,14026}, {2975,3872}, {3008,7397}, {3062,4866}, {3065,12747}, {3068,13912}, {3069,13975}, {3070,13911}, {3071,13973}, {3086,3911}, {3090,3634}, {3091,3305}, {3095,3097}, {3098,9941}, {3099,9821}, {3100,9610}, {3146,3219}, {3160,7177}, {3169,5847}, {3182,3346}, {3185,7420}, {3207,6603}, {3208,3509}, {3220,9798}, {3241,10304}, {3243,3874}, {3244,3528}, {3306,3523}, {3309,4063}, {3348,3353}, {3355,3473}, {3358,5787}, {3398,10789}, {3434,6734}, {3436,6256}, {3452,6848}, {3467,5560}, {3476,4311}, {3485,5218}, {3486,4304}, {3487,3671}, {3488,4314}, {3515,11363}, {3516,11396}, {3526,11230}, {3529,3626}, {3534,4677}, {3545,3828}, {3555,10167}, {3560,5251}, {3575,5090}, {3583,6928}, {3585,6923}, {3622,5734}, {3636,10299}, {3640,11825}, {3641,11824}, {3647,11530}, {3653,12100}, {3661,6999}, {3663,10521}, {3681,3951}, {3683,3698}, {3689,3962}, {3690,11381}, {3692,5279}, {3697,5927}, {3710,10327}, {3711,4005}, {3714,5695}, {3715,3983}, {3719,7270}, {3729,4385}, {3781,5907}, {3784,13348}, {3814,6941}, {3822,6937}, {3825,6963}, {3827,12329}, {3839,10248}, {3841,6829}, {3844,10516}, {3868,3870}, {3873,12005}, {3880,12513}, {3884,6940}, {3886,10449}, {3890,5253}, {3914,5230}, {3918,6920}, {3947,5714}, {3955,13346}, {3980,12545}, {3984,4420}, {4026,5799}, {4047,5776}, {4084,12559}, {4187,7681}, {4293,10106}, {4299,6948}, {4326,5728}, {4333,5841}, {4384,6996}, {4413,6918}, {4414,10459}, {4421,12635}, {4450,5016}, {4511,4855}, {4654,10056}, {4662,5220}, {4663,11477}, {4669,11001}, {4678,5059}, {4816,12103}, {4847,5082}, {4880,13369}, {5013,9592}, {5044,8580}, {5056,9779}, {5057,6932}, {5067,10171}, {5084,7682}, {5088,9312}, {5171,11364}, {5180,6960}, {5234,9708}, {5252,7354}, {5259,6883}, {5260,6912}, {5261,8545}, {5267,6950}, {5274,5704}, {5281,5703}, {5295,5774}, {5312,5396}, {5314,7503}, {5316,6964}, {5330,13587}, {5426,5428}, {5432,11375}, {5433,11376}, {5435,9785}, {5439,10582}, {5440,5730}, {5442,6713}, {5445,6882}, {5550,10303}, {5554,6872}, {5559,7284}, {5561,7161}, {5688,5870}, {5689,5871}, {5692,5720}, {5722,10384}, {5726,9654}, {5729,9844}, {5735,5880}, {5744,6705}, {5745,6847}, {5763,11374}, {5768,5853}, {5791,8727}, {5804,8257}, {5805,8728}, {5806,11108}, {5905,10528}, {5909,10374}, {5918,12680}, {6043,11991}, {6048,9566}, {6068,6259}, {6198,9611}, {6200,9615}, {6221,9618}, {6237,9928}, {6241,11460}, {6246,10724}, {6265,13253}, {6407,9584}, {6700,6927}, {6759,10536}, {6764,12516}, {6842,7951}, {6889,10198}, {6890,10527}, {6897,10532}, {6899,10916}, {6943,11680}, {6947,10531}, {6949,11813}, {6967,10200}, {6990,12558}, {6998,9746}, {7082,12953}, {7387,8185}, {7413,12544}, {7589,7590}, {7596,8231}, {7672,7675}, {7673,7677}, {7970,11711}, {7978,11720}, {7983,11710}, {7984,11709}, {7993,12773}, {8075,8081}, {8076,8082}, {8078,8091}, {8089,8099}, {8090,8100}, {8107,8111}, {8108,8112}, {8140,12488}, {8144,9576}, {8188,10669}, {8189,10673}, {8197,9834}, {8204,9835}, {8214,9838}, {8215,9839}, {8224,8234}, {8244,12490}, {8245,9959}, {8423,12491}, {8616,13732}, {8981,13888}, {8983,9540}, {9521,10015}, {9751,12264}, {9786,10382}, {9857,9873}, {9896,12417}, {9906,12662}, {9907,12663}, {10087,11570}, {10090,12758}, {10197,11263}, {10373,11347}, {10386,12433}, {10404,11246}, {10436,10446}, {10437,10447}, {10578,11036}, {10606,12262}, {10695,11714}, {10696,11700}, {10697,11712}, {10703,11713}, {10705,12265}, {10738,12619}, {10791,12110}, {10899,10900}, {10912,11194}, {10915,12115}, {11251,11852}, {11445,12111}, {11722,13099}, {11754,11756}, {11763,11765}, {11772,11774}, {11781,11783}, {11828,12440}, {11829,12441}, {11900,12113}, {11919,12648}, {11920,12649}, {12059,12665}, {12247,13199}, {12387,12398}, {12407,12661}, {12408,13221}, {12517,12843}, {12518,12844}, {12519,12845}, {12530,12683}, {12556,12660}, {12653,12737}, {12670,12671}, {12756,12757}, {12786,13465}, {13935,13971}, {13942,13966}

X(40) = midpoint of X(i) and X(j) for these {i,j}: {1, 7991}, {3, 12702}, {4, 6361}, {8, 20}, {10, 5493}, {65, 7957}, {944, 12245}, {1768, 5541}, {2093, 7994}, {2100, 2101}, {2136, 6762}, {2448, 2449}, {2948, 9904}, {2951, 5223}, {3245, 5537}, {5531, 12767}, {6764, 12632}, {9860, 13174}, {9961, 12528}, {11826, 11827}, {12247, 13199}, {12408, 13221}, {12488, 12489}, {12526, 12565}, {12697, 12698}

X(40) = reflection of X(i) in X(j) for these {i,j}: {1, 3}, {3, 3579}, {4, 10}, {8, 11362}, {57, 3359}, {84, 1158}, {145, 5882}, {149, 10265}, {355, 5690}, {944, 4297}, {946, 6684}, {962, 946}, {1012, 4640}, {1071, 9943}, {1320, 11715}, {1482, 1385}, {1490, 11500}, {1537, 3035}, {1768, 12515}, {1836, 6907}, {2077, 13528}, {2948, 12778}, {3062, 5779}, {3555, 12675}, {3576, 165}, {3655, 8703}, {3656, 549}, {3679, 3654}, {3811, 8715}, {3868, 5884}, {4297, 12512}, {4301, 1125}, {5531, 12331}, {5535, 484}, {5587, 5657}, {5603, 10164}, {5691, 355}, {5693, 72}, {5732, 11495}, {5735, 5880}, {5881, 8}, {6210, 573}, {6261, 6796}, {6264, 104}, {6282, 6244}, {6326, 100}, {6361, 5493}, {6765, 3913}, {6769, 10306}, {7688, 7964}, {7701, 191}, {7970, 11711}, {7971, 6261}, {7978, 11720}, {7982, 1}, {7983, 11710}, {7984, 11709}, {7991, 12702}, {7993, 12773}, {8148, 10222}, {9579, 6850}, {9580, 6827}, {9589, 12699}, {9799, 9948}, {9812, 10175}, {9845, 9841}, {9856, 5044}, {10222, 13624}, {10695, 11714}, {10696, 11700}, {10697, 11712}, {10698, 214}, {10703, 11713}, {10705, 12265}, {10724, 6246}, {10738, 12619}, {10864, 84}, {10912, 11260}, {11014, 11012}, {11224, 10246}, {11372, 9}, {11477, 4663}, {11523, 3811}, {11531, 1482}, {12398, 12387}, {12407, 13211}, {12520, 12511}, {12629, 12513}, {12650, 12114}, {12651, 11496}, {12653, 12737}, {12665,14740}, {12672, 960}, {12688, 5777}, {12696, 402}, {12699, 5}, {12701, 6922}, {12703, 5119}, {12704, 46}, {12705, 12514}, {12717, 1766}, {12751, 1145}, {12842, 12516}, {12843, 12517}, {12844, 12518}, {12845, 12519}, {13099, 11722}, {13253, 6265}

X(40) = isogonal conjugate of X(84)

X(40) = isotomic conjugate of X(309)

X(40) = inverse-in-circumcircle of X(2077)

X(40) = complement of X(962)

X(40) = anticomplement of X(946)

X(40) = X(963)-complementary conjugate of X(10)

X(40) = X(947)-anticomplementary conjugate of X(8)

X(40) = X(i)-Ceva conjugate of X(j) for these (i,j): (4, 2910), (8, 1), (20, 1490), (63, 9), (329, 2324), (347, 223), (515, 6326), (1817, 198), (7080, 1103), (7128, 101), (8822, 329), (9369, 1050), (9778, 2951)

X(40) = X(i)-cross conjugate of X(j) for these (i,j): (64, 3354), (198, 223), (208, 3342), (221, 1), (227, 7952), (2187, 2331), (7074, 2324)

X(40) = crosspoint of X(i) and X(j) for these (i,j): (329,347)

X(40) = crosssum of X(i) and X(j) for these (i,j): {19, 7008}, {56, 1413}, {513, 7004}, {649, 2310}, {1436, 2192}, {1903, 2357}

X(40) = crossdifference of every pair of points on line X(650)X(1459)

X(40) = cevapoint of X(i) and X(j) for these (i,j): {1, 2956}, {19, 8802}, {55, 3197}, {65, 8803}, {71, 3198}, {198, 7074}

X(40) = crosspoint of X(i) and X(j) for these (i,j): {8, 7080}, {63, 7013}, {100, 7012}, {190, 7045}, {329, 347}, {1817, 8822}

X(40) = trilinear pole of line {6129, 10397}

X(40) = point of concurrence of the perpendiculars from the excenters to the respective sides

X(40) = circumcenter of the excentral triangle

X(40) = incenter of the extangents triangle if triangle ABC is acute

X(40) = perspector of the excentral and extangents triangles

X(40) = perspector of the excentral and extouch triangles

X(40) = X(4)-of-hexyl-triangle

X(40) = X(4)-of-1st-circumperp-triangle

X(40) = X(20)-of-2nd-circumperp-triangle

X(40) = Miquel associate of X(8)

X(40) = perspector of hexyl triangle and cevian triangle of X(63)

X(40) = perspector of hexyl triangle and anticevian triangle of X(9)

X(40) = perspector of hexyl triangle and antipedal triangle of X(84)

X(40) = perspector of ABC and the reflection in X(57) of the antipedal triangle of X(57)

X(40) = excentral isogonal conjugate of X(1)

X(40) = excentral isotomic conjugate of X(1742)

X(40) = hexyl isogonal conjugate of X(1)

X(40) = perspector of ABC and extraversion triangle of X(84)

X(40) = trilinear product of extraversions of X(84)

X(40) = homothetic center of extangents triangle and reflection of intangents triangle in X(3)

X(40) = trilinear product of centers of mixtilinear incircles

X(40) = intangents-to-extangents similarity image of X(1)

X(40) = X(26)-of-reflection-triangle of X(1)

X(40) = {X(56),X(3057)}-harmonic conjugate of X(1)

X(40) = perspector of extangents triangle and cross-triangle of ABC and extangents triangle

X(40) = perspector of ABC and cross-triangle of ABC and hexyl triangle

X(40) = circumcircle-inverse of X(2077)

X(40) = inverse-in-incircle-of-anticomplementary-triangle of X(10538)

X(40) = X(1)-Hirst inverse of X(9371)

X(40) = outer-Garcia-to-ABC similarity image of X(4)

X(40) = Cundy-Parry Phi transform of X(57)

X(40) = Cundy-Parry Psi transform of X(9)

X(40) = anticevian isogonal conjugate of X(1)

X(40) = X(i)-vertex conjugate of X(j) for these (i,j): {3, 3345}, {513, 2077}, {2077, 513}, {3345, 3}

X(40) = X(5)-of-6th-mixtilinear-triangle

X(40) = orthic-to-ABC functional image of X(3), if ABC is acute

X(40) = intouch-to-ABC barycentric image of X(4)

X(40) = intouch-to-excentral similarity image of X(1)

X(40) = ABC-to-excentral barycentric image of X(1)

X(40) = antipode of X(12665) in the Mandart hyperbola

X(40) = extouch-isogonal conjugate of X(14872)

X(40) = endo-homothetic center of Ehrmann side-triangle and anti-excenters-incenter reflections triangle; the homothetic center is X(382).

X(40) = X(i)-isoconjugate of X(j) for these (i,j): {1, 84}, {2, 1436}, {4, 1433}, {6, 189}, {7, 2192}, {8, 1413}, {9, 1422}, {31, 309}, {34, 271}, {40, 1256}, {55, 1440}, {56, 280}, {57, 282}, {63, 7129}, {65, 285}, {69, 7151}, {75, 2208}, {77, 7008}, {81, 1903}, {85, 7118}, {86, 2357}, {222, 7003}, {268, 278}, {273, 2188}, {279, 7367}, {284, 8808}, {346, 6612}, {348, 7154}, {513, 13138}, {522, 8059}, {603, 7020}, {1174, 13156}, {1812, 2358}, {3341, 3345}, {3346, 8886}, {7054, 13853}, {9375, 9376}

X(40) = X(i)-aleph conjugate of X(j) for these (i,j): {1, 978}, {2, 57}, {8, 40}, {9, 1742}, {100, 4551}, {188, 1}, {259, 1740}, {366, 1743}, {522, 2957}, {556, 63}, {4146, 1445}, {4182, 165}, {6728, 1052}, {6731, 2951}, {7025, 361}, {7043, 7150}

X(40) = X(i)-beth conjugate of X(j) for these (i,j): {8, 4}, {21, 1420}, {40, 221}, {100, 40}, {643, 78}, {644, 728}, {13138, 3341}

X(40) = X(i)-gimel conjugate of X(j) for these (i,j): {8, 84}, {78, 40}, {521, 10085}, {522, 40}, {3717, 40}, {4041, 40}, {4086, 40}, {4147, 40}, {4163, 40}, {4391, 40}, {4397, 40}, {4723, 40}, {4768, 40}, {4811, 40}, {4985, 40}, {6615, 40}, {6735, 40}, {7628, 40}, {7629, 40}, {7646, 40}, {7647, 40}

X(40) = X(i)-he conjugate of X(j) for these (i,j): {2, 516}, {190, 40}, {312, 6211}, {645, 40}, {646, 40}, {3699, 40}, {4518, 1766}, {4554, 40}, {4582, 40}, {4621, 40}, {4633, 40}, {4876, 165}, {4997, 40}, {6335, 40}, {6559, 10860}, {8707, 40}, {9365, 1}, {11609, 3}, {13136, 40}

X(40) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 84}, {8, 40}, {10, 1158}, {11, 1768}, {21, 7701}, {55, 1709}, {65, 7992}, {72, 1490}, {145, 10864}, {200, 10860}, {210, 165}, {390, 11372}, {497, 57}, {517, 6001}, {518, 971}, {519, 515}, {521, 513}, {522, 3667}, {950, 4}, {952, 2829}, {958, 7330}, {960, 3}, {1001, 3358}, {1145, 2950}, {1697, 12705}, {1837, 46}, {1864, 1750}, {2098, 10085}, {2321, 1766}, {2802, 2800}, {2804, 900}, {3036, 12515}, {3057, 1}, {3058, 1699}, {3059, 2951}, {3239, 649}, {3271, 9355}, {3678, 6796}, {3680, 6762}, {3686, 573}, {3688, 1742}, {3706, 1764}, {3717, 6211}, {3738, 2827}, {3877, 3576}, {3878, 6261}, {3880, 517}, {3883, 6210}, {3884, 5450}, {3885, 7982}, {3886, 12717}, {3893, 7991}, {3900, 3309}, {3907, 6002}, {4046, 2941}, {4111, 2938}, {4520, 3294}, {4534, 5540}, {4662, 3579}, {4673, 10476}, {4847, 63}, {5119, 12686}, {5245, 1277}, {5246, 1276}, {5289, 7171}, {5697, 7971}, {5795, 10}, {5837, 12514}, {5853, 516}, {5854, 952}, {5856, 5851}, {6366, 2826}, {6737, 20}, {6738, 9948}, {8058, 522}, {8275, 7966}, {9119, 5776}, {9785, 3333}, {9898, 7160}, {9957, 12114}, {10106, 12246}, {10866, 3361}, {10950, 5691}, {12448, 8158}, {12527, 6223}, {12541, 6766}, {12572, 6260}, {12575, 946}

X(40) = barycentric product X(i)*X(j) for these {i,j}: {1, 329}, {6, 322}, {7, 2324}, {8, 223}, {9, 347}, {10, 1817}, {37, 8822}, {57, 7080}, {63, 7952}, {69, 2331}, {75, 198}, {76, 2187}, {78, 196}, {85, 7074}, {92, 7078}, {189, 1103}, {190, 6129}, {208, 345}, {219, 342}, {221, 312}, {227, 333}, {281, 7013}, {304, 3195}, {306, 3194}, {318, 7011}, {321, 2360}, {341, 6611}, {651, 8058}, {1088, 7368}, {2199, 3596}, {3209, 3718}, {5514, 7045}, {7017, 7114}, {7128, 7358}

X(40) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 189}, {2, 309}, {6, 84}, {9, 280}, {25, 7129}, {31, 1436}, {32, 2208}, {33, 7003}, {41, 2192}, {42, 1903}, {48, 1433}, {55, 282}, {56, 1422}, {57, 1440}, {65, 8808}, {101, 13138}, {196, 273}, {198, 1}, {208, 278}, {212, 268}, {213, 2357}, {219, 271}, {221, 57}, {223, 7}, {227, 226}, {281, 7020}, {284, 285}, {322, 76}, {329, 75}, {342, 331}, {347, 85}, {354, 13156}, {604, 1413}, {607, 7008}, {1103, 329}, {1106, 6612}, {1253, 7367}, {1254, 13853}, {1415, 8059}, {1436, 1256}, {1817, 86}, {1819, 1812}, {1973, 7151}, {2175, 7118}, {2187, 6}, {2199, 56}, {2212, 7154}, {2324, 8}, {2331, 4}, {2360, 81}, {3194, 27}, {3195, 19}, {3197, 3341}, {3209, 34}, {6129, 514}, {6611, 269}, {7011, 77}, {7013, 348}, {7074, 9}, {7078, 63}, {7080, 312}, {7114, 222}, {7368, 200}, {7952, 92}, {8058, 4391}, {8822, 274}, {10397, 521}

X(40) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3, 3576), (1, 35, 3601), (1, 36, 1420), (1, 46, 57), (1, 57, 3333), (1, 65, 11529), (1, 165, 3), (1, 484, 46), (1, 1764, 10476), (1, 2093, 65), (1, 3336, 3338), (1, 3339, 942), (1, 3361, 999), (1, 3612, 13384), (1, 3670, 3677), (1, 3746, 10389), (1, 5010, 3612), (1, 5119, 1697), (1, 5264, 5269), (1, 5697, 7962), (1, 5709, 12704), (1, 5902, 11518), (1, 5903, 3340), (1, 7987, 1385), (1, 7994, 6769), (1, 9819, 9957), (1, 10268, 10902), (1, 10980, 5045), (1, 11010, 5119), (1, 11224, 10222), (1, 11531, 1482), (2, 946, 8227), (2, 962, 946), (3, 55, 10902), (3, 942, 8726), (3, 1385, 7987), (3, 1482, 1385), (3, 2095, 9940), (3, 3428, 11012), (3, 3579, 165), (3, 5584, 7688), (3, 5708, 11227), (3, 5709, 57), (3, 6244, 10310), (3, 7991, 7982), (3, 8148, 10246), (3, 8158, 999), (3, 8251, 10319), (3, 9940, 10857), (3, 10246, 13624), (3, 10306, 55), (3, 10310, 2077), (3, 10679, 10267), (3, 10680, 10269), (3, 11248, 35), (3, 11249, 36), (4, 10, 5587), (4, 5657, 10), (4, 6197, 19), (4, 12705, 11372), (5, 12699, 1699), (8, 9778, 20), (8, 10860, 10864), (9, 1706, 10), (10, 573, 9548), (10, 12514, 9), (10, 12572, 2551), (11, 12701, 9614), (12, 1836, 9612), (19, 71, 9), (19, 11471, 4), (20, 63, 84), (20, 9537, 3101), (20, 11362, 5881), (35, 3245, 5903), (35, 5537, 11248), (35, 5903, 1), (36, 5697, 1), (39, 1571, 9574), (39, 1572, 9575), (42, 4300, 581), (43, 1695, 970), (46, 484, 5128), (46, 1697, 3333), (46, 3338, 3336), (46, 5119, 1), (46, 5709, 5535), (46, 11010, 1697), (55, 65, 1), (55, 2093, 11529), (55, 5183, 2093), (55, 5584, 3), (55, 7957, 6769), (55, 7964, 165), (56, 3057, 1), (57, 1697, 1), (57, 5128, 46), (57, 7991, 6766), (63, 9778, 10860), (64, 3198, 1490), (65, 3579, 7688), (65, 6769, 7982), (65, 7964, 5584), (65, 10268, 3576), (72, 5687, 200), (72, 7580, 1490), (78, 6261, 6326), (100, 411, 6796), (100, 3869, 78), (140, 5886, 3624), (145, 3522, 5731), (145, 5731, 5882), (164, 505, 188), (165, 484, 3359), (165, 5538, 5010), (165, 6282, 2077), (165, 6769, 10902), (165, 7991, 1), (165, 7994, 55), (165, 11531, 7987), (165, 12702, 7982), (169, 3730, 9), (191, 2960, 1710), (191, 5691, 7330), (200, 12526, 72), (200, 12565, 1490), (210, 12688, 5777), (221, 227, 223), (221, 7074, 7078), (227, 7074, 1103), (354, 3303, 1), (355, 3654, 5690), (355, 5690, 3679), (376, 944, 4297), (376, 12245, 944), (381, 9956, 7989), (388, 3474, 4292), (392, 474, 8583), (411, 3869, 6261), (484, 5119, 57), (484, 7991, 5709), (484, 11010, 1), (497, 1788, 1210), (498, 12047, 5219), (573, 1766, 9), (595, 13329, 602), (631, 4301, 9624), (631, 5603, 1125), (942, 3295, 1), (946, 6684, 2), (950, 1708, 10396), (956, 10914, 4853), (958, 5836, 9623), (960, 1376, 936), (962, 6684, 8227), (986, 5255, 1), (999, 9957, 1), (1125, 4301, 5603), (1125, 5603, 9624), (1125, 10164, 631), (1151, 7969, 9583), (1155, 3057, 56), (1210, 10624, 497), (1276, 1277, 9), (1276, 6192, 6191), (1277, 6191, 6192), (1319, 2098, 1), (1381, 1382, 2077), (1385, 1482, 1), (1385, 7987, 3576), (1388, 5048, 1), (1402, 10480, 1), (1420, 7962, 1), (1467, 10388, 1), (1478, 1770, 9579), (1478, 10039, 9578), (1479, 1737, 9581), (1482, 11531, 7982), (1490, 12526, 5693), (1571, 1572, 39), (1656, 9955, 7988), (1697, 5128, 57), (1698, 1699, 5), (1698, 9589, 1699), (1699, 9589, 12699), (1700, 1701, 182), (1702, 1703, 6), (1704, 1705, 182), (1706, 12705, 5587), (1709, 7330, 7701), (1750, 7995, 12688), (1754, 5264, 3072), (1770, 10039, 1478), (1837, 6284, 3586), (2017, 2018, 39), (2077, 11012, 3), (2093, 10306, 7982), (2098, 5204, 1319), (2099, 2646, 1), (2099, 5217, 2646), (2136, 3928, 6762), (2136, 9841, 944), (2292, 4220, 8235), (2551, 5698, 12572), (2572, 2573, 3), (2975, 6909, 5450), (3057, 10270, 3576), (3085, 4295, 226), (3091, 9780, 10175), (3158, 11523, 3811), (3303, 5221, 354), (3304, 5919, 1), (3336, 3338, 57), (3340, 3601, 1), (3359, 3587, 165), (3359, 5119, 3576), (3359, 5709, 46), (3359, 7991, 12704), (3361, 9819, 1), (3428, 6244, 2077), (3428, 6282, 3576), (3428, 10310, 3), (3428, 13528, 165), (3436, 6925, 6256), (3485, 5218, 13411), (3496, 3501, 9), (3523, 3616, 10165), (3555, 10167, 12675), (3576, 5535, 57), (3576, 7982, 1), (3576, 12704, 3333), (3579, 7957, 10902), (3579, 7991, 3576), (3579, 10306, 10268), (3579, 12702, 1), (3587, 5709, 3), (3587, 12702, 1697), (3624, 11522, 5886), (3634, 3817, 3090), (3666, 5710, 1), (3671, 13405, 3487), (3679, 5691, 355), (3681, 9961, 12528), (3681, 11684, 3951), (3730, 5011, 169), (3746, 5902, 1), (3811, 8715, 3158), (3868, 3871, 3870), (3868, 7411, 10884), (3869, 6796, 6326), (3872, 4652, 2975), (3890, 9352, 5253), (3895, 5731, 7966), (3911, 12053, 3086), (3916, 10914, 956), (3931, 5711, 1), (4297, 6762, 9845), (4297, 12512, 376), (4301, 10164, 1125), (4302, 10573, 10572), (4314, 6738, 3488), (4424, 5264, 1), (4512, 12651, 11496), (4640, 5836, 958), (4853, 10914, 11525), (4855, 11682, 4511), (5044, 9709, 8580), (5045, 5708, 10980), (5045, 6767, 1), (5119, 5128, 3333), (5119, 5709, 7982), (5221, 8273, 9940), (5252, 7354, 9613), (5535, 7982, 12704), (5536, 7987, 3338), (5541, 6763, 3632), (5552, 11415, 908), (5584, 6769, 3576), (5584, 7957, 1), (5584, 7991, 11529), (5584, 10306, 10902), (5657, 6361, 4), (5687, 7580, 11500), (5708, 6767, 5045), (5709, 11010, 12703), (5714, 8164, 3947), (5758, 6908, 226), (5812, 6907, 9612), (5887, 11499, 5720), (6191, 6192, 9), (6210, 6211, 9), (6210, 12717, 11372), (6212, 6213, 9), (6252, 6404, 3779), (6736, 12527, 3421), (6769, 10268, 55), (6838, 11415, 12608), (6922, 12700, 9614), (6923, 10526, 3585), (6928, 10525, 3583), (7589, 12445, 7590), (7672, 7676, 7675), (7688, 10902, 3), (7742, 11508, 2078), (7957, 7964, 3), (7987, 7991, 11531), (7987, 11531, 1), (7991, 7994, 7957), (8075, 8093, 8081), (8076, 8094, 8082), (8107, 9805, 8111), (8108, 9806, 8112), (8148, 10222, 11224), (8148, 10246, 10222), (8148, 11224, 7982), (8148, 13624, 1), (8158, 9819, 7982), (8224, 9808, 8234), (9572, 9573, 8141), (9574, 9575, 39), (9576, 9577, 8144), (9578, 9579, 1478), (9580, 9581, 1479), (9582, 9583, 1151), (9584, 9585, 6407), (9586, 9587, 49), (9588, 9589, 5), (9590, 9591, 26), (9780, 9812, 3091), (9955, 11231, 1656), (10222, 10246, 1), (10222, 13624, 10246), (10267, 10679, 3746), (10269, 10680, 5563), (10306, 12702, 7957), (10389, 11518, 1), (10434, 12435, 1), (10470, 11521, 1), (10572, 10573, 5727), (10912, 11194, 11260), (11019, 12575, 1058), (11822, 11823, 55), (12000, 13373, 1), (12703, 12704, 7982)

Trilinears a

Trilinears a

Trilinears a tan A' : : , where A'B'C' is the excentral triangle

Barycentrics a

If you have The Geometer's Sketchpad, you can view X(41).

X(41) lies on these lines: 1,101 3,218 6,48 9,21 25,42 31,32 37,584 55,220 58,609 65,910 219,1036 226,379 560,872 601,906 603,911 663,884

X(41) is the {X(32),X(213)}-harmonic conjugate of X(31). For a list
of other harmonic conjugates of X(41), click **Tables** at the top
of this page.

X(41) = isogonal conjugate of X(85)

X(41) = isotomic conjugate of X(20567)

X(41) = anticomplement of X(17046)

X(41) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,31), (9,212), (284,55)

X(41) = crosspoint of X(i) and X(j) for these (i,j): (6,55), (9,33)

X(41) = crosssum of X(i) and X(j) for these (i,j): (1,169), (2,7), (57,77), (92,342), (226,1441), (514,1111)

X(41) = crossdifference of every pair of points on line X(522)X(693)

X(41) = X(i)-beth conjugate of X(j) for these (i,j): (41,32), (101,41), (220,220)

X(41) = X(75)-isoconjugate of X(57)

X(41) = X(92)-isoconjugate of X(77)

X(41) = trilinear product of vertices of 4th mixtilinear triangle

X(41) = trilinear product of vertices of 5th mixtilinear triangle

X(41) = trilinear product of PU(93)

X(41) = barycentric product of PU(104)

X(41) = PU(93)-harmonic conjugate of X(663)

X(41) = bicentric sum of PU(93)

X(41) = perspector of unary cofactor triangles of Gemini triangles 1 and 13

Trilinears (1 + cos A)(cos B + cos C) : (1 + cos B)(cos C + cos A) : (1 + cos C)(cos A + cos B)

Trilinears a(ar - S) : b(br - S): c(cr - S)

Trilinears csc B + csc C : :

Barycentrics a

Tripolars (pending)

If you have The Geometer's Sketchpad, you can view X(42).

Let A'B'C' be the extangents triangle. Let La be the trilinear polar of A', and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines AA", BB", CC" concur in X(42). (Randy Hutson, December 26, 2015)

Let A'B'C' be the extangents triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(42).(Randy Hutson, December 26, 2015)

Let Ab, Ac, Bc, Ba, Ca, Cb be as defined at X(3588). Let A* be the intersection of the tangents to the Myakishev conic at Ba and Ca, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(42).(Randy Hutson, December 26, 2015)

X(42) lies on these lines:

1,2 3,967 6,31 9,941
25,41 33,393 35,58
37,210 38,518 40,581
48,197 57,1001 65,73
81,100 101,111 165,991
172,199 181,228 244,354
308,313 321,740 517,1064
560,584 649,788 694,893
748,1001 750,940 894,1045
942,1066

X(42) is the {X(1),X(43)}-harmonic conjugate of X(2). For a list of
other harmonic conjugates of X(42), click **Tables** at the top of
this page.

X(42) = reflection of X(321) in X(1215)

X(42) = isogonal conjugate of X(86)

X(42) = isotomic conjugate of X(310)

X(42) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,37), (6,213), (10,71), (55,228)

X(42) = crosspoint of X(i) and X(j) for these (i,j): (1,6), (33,55), (37,65)

X(42) = crosssum of X(i) and X(j) for these (i,j): (1,2), (7,77), (21,81)

X(42) = crossdifference of every pair of points on line X(239)X(514)

X(42) = circumcircle-inverse of X(32759)

X(42) = X(1)-line conjugate of X(239)

X(42) = X(i)-beth conjugate of X(j) for these (i,j): (21,551), (55,42), (100,226), (210,210), (643,171)

X(42) = bicentric sum of PU(8)

X(42) = PU(8)-harmonic conjugate of X(649)

X(42) = barycentric product of PU(32)

X(42) = trilinear product of PU(85)

X(42) = trilinear pole of line X(512)X(798)

X(42) = Danneels point of X(1)

X(42) = {X(1),X(2)}-harmonic conjugate of X(3720)

X(42) = X(75)-isoconjugate of X(58)

X(42) = X(92)-isoconjugate of X(1790)

X(42) = trilinear square root of X(872)

X(42) = perspector of ABC and unary cofactor triangle of 1st Conway triangle

X(42) = perspector of ABC and unary cofactor triangle of 5th Conway triangle

X(42) = perspector of unary cofactor triangles of 1st and 5th Conway triangles

X(42) = perspector of ABC and unary cofactor triangle of Gemini triangle 2

X(42) = barycentric product of vertices of Gemini triangle 15

Trilinears csc B + csc C - csc A : csc C + csc A - csc B : csc A + csc B - csc C

Barycentrics a(ab + ac - bc) : b(bc + ba - ca) : c(ca + cb - ab)

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B' and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle at X(43). (Randy Hutson, November 30, 2018)

X(43) lies on the Kiepert hyperbola of the excentral triangle and these lines:

1,2 6,87 9,256 31,100
35,1011 40,970 46,851
55,238 57,181 58,979
72,986 75,872 81,750
165,573 170,218 210,984
312,740 518,982

X(43) is the {X(2),X(42)}-harmonic conjugate of X(1). For a list of
other harmonic conjugates of X(43), click **Tables** at the top of
this page. X(43) is the external center of similitude of the Bevan
circle and Apollonius circle; the internal center is X(1695).

X(43) = reflection of X(1) in X(995)

X(43) = isogonal conjugate of X(87)

X(43) = isotomic conjugate of X(6384)

X(43) = anticomplement of X(3840)

X(43) = X(6)-Ceva conjugate of X(1)

X(43) = X(192)-cross conjugate of X(1)

X(43) = crosssum of X(2) and X(330)

X(43) = X(55)-Hirst inverse of X(238)

X(43) = inverse-in-excircles-radical-circle of X(5121)

X(43) = perspector of ABC and extraversion triangle of X(87)

X(43) = trilinear product of extraversions of X(87)

X(43) = excentral-isogonal conjugate of X(1766)

X(43) = polar conjugate of isotomic conjugate of X(22370)

X(43) = perspector of Gemini triangle 5 and cross-triangle of Gemini triangles 3 and 5

X(43) = X(i)-aleph conjugate of X(j) for these (i,j):

(1,9), (6,43), (55,170), (100,1018), (174,169), (259,165), (365,1), (366,63), (507,362), (509,57)

X(43) = X(660)-beth conjugate of X(43)

X(43) = {X(2),X(8)}-harmonic conjugate of X(3741)

Barycentrics a(b + c - 2a) : b(c + a - 2b) : c(a + b - 2c)

X(44) lies on these lines: 1,6 2,89 10,752 31,210 51,209 65,374 88,679 181,375 190,239 193,344 214,1017 241,651 292,660 354,748 513,649 527,1086 583,992 678,902

X(44) is the {X(1),X(9)}-harmonic conjugate of X(45). For a list of
other harmonic conjugates of X(44), click **Tables** at the top of
this page.

X(44) = midpoint of X(i) and X(j) for these (i,j): (190,239),
(3218,3257)

X(44) = reflection of X(1279) in X(238)

X(44) = isogonal conjugate of X(88)

X(44) = isotomic conjugate of X(20568)

X(44) = complement of X(320)

X(44) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,214), (88,1), (104,55)

X(44) = crosspoint of X(i) and X(j) for these (i,j): (1,88), (2,80)

X(44) = crosssum of X(i) and X(j) for these (i,j): (1,44), (6,36), (57,1465)

X(44) = crossdifference of every pair of points on line X(1)X(513)

X(44) = X(6)-line conjugate of X(1)

X(44) = X(88)-cross conjugate of X(44)

X(44) = X(i)-beth conjugate of X(j) for these (i,j): (9,44), (644,44), (645,239), (44,44)

X(44) = bicentric sum of PU(i) for these i: 33, 50

X(44) = midpoint of PU(i) for these i: 33, 50

X(44) = crossdifference of PU(55)

X(44) = perspector of circumconic centered at X(214)

X(44) = center of circumconic that is locus of trilinear poles of lines passing through X(214)

X(44) = {X(6),X(9)}-harmonic conjugate of X(37)

X(44) = inverse-in-circumconic-centered-at-X(9) of X(1)

X(44) = trilinear pole of line line X(678)X(1635)

X(44) = polar conjugate of isogonal conjugate of X(23202)

Barycentrics a(2b + 2c - a) : b(2c + 2a - b) : c(2a + 2b - c)

X(45) lies on these lines: 1,6 2,88 53,281 55,678 141,344 198,1030 210,968 346,594

X(45) is the {X(1),X(9)}-harmonic conjugate of X(44). For a list of
other harmonic conjugates of X(45), click **Tables** at the top of
this page.

X(45) = isogonal conjugate of X(89)

X(45) = isotomic conjugate of X(20569)

X(45) = crosssum of X(6) and X(999)

X(45) = anticomplement of isotomic conjugate of X(32013)

X(45) = X(i)-beth conjugate of X(j) for these (i,j): (9,1), (644,45)

X(45) = complement of polar conjugate of isogonal conjugate of X(22129)

X(45) = anticomplement of anticomplement of X(31285)

Trilinears a^3 + a^2(b + c) - a(b^2 + c^2) - (b - c)^2(b + c) : :

Trilinears r

Barycentrics a(cos B + cos C - cos A) : :

Tripolars (pending)

Let Ja' be the reflection of the A-excenter in BC, and define Jb', Jc' cyclically. Let Oa be the circumcenter of AJb'Jc', and define Ob, Oc cyclically. OaObOc and ABC are perspective at X(46). (Randy Hutson, July 20, 2016)

Let A' be the inverse-in-Bevan-circle of the A-vertex of the hexyl triangle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(46). (Randy Hutson, July 20, 2016)

Let JaJbJc be the excentral triangle. Let A" be the inverse-in-Bevan-circle of A, and define B", C" cyclically. The lines JaA", JbB", JcC" concur in X(46). (Randy Hutson, July 20, 2016)

X(46) lies on these lines:

1,3 4,90 9,79 10,63
19,579 34,47 43,851
58,998 78,758 80,84
100,224 158,412 169,672
200,1004 218,910 222,227
225,254 226,498 269,1103
404,997 474,960 499,946
595,614 750,975 978,1054

X(46) is the {X(3),X(65)}-harmonic conjugate of X(1). For a list of
other harmonic conjugates of X(46), click **Tables** at the top of
this page.

X(46) = reflection of X(i) in X(j) for these (i,j): (1,56), (1479,1210)

X(46) = isogonal conjugate of X(90)

X(46) = isotomic conjugate of X(20570)

X(46) = circumcircle-inverse of X(32760)

X(46) = Bevan-circle-inverse of X(36)

X(46) = X(4)-Ceva conjugate of X(1)

X(46) = crosssum of X(3) and X(1069)

X(46) = X(i)-aleph conjugate of X(j) for these (i,j): (4,46), (174,223), (188,1079), (366,610), (653, 1020)

X(46) = X(100)-beth conjugate of X(46)

X(46) = perspector of excentral and orthic triangles

X(46) = orthic isogonal conjugate of X(1)

X(46) = excentral isogonal conjugate of X(1490)

X(46) = X(24)-of-excentral-triangle

X(46) = {X(1),X(3)}-harmonic conjugate of X(3612)

X(46) = {X(1),X(40)}-harmonic conjugate of X(5119)

X(46) = perspector of ABC and extraversion triangle of X(90)

X(46) = trilinear product of extraversions of X(90)

X(46) = X(24) of reflection triangle of X(1)

X(46) = homothetic center of ABC and orthic triangle of reflection triangle of X(1)

X(46) = Cundy-Parry Phi transform of X(46)

X(46) = Cundy-Parry Psi transform of X(90)

X(46) = {X(1),X(57)}-harmonic conjugate of X(3338)

f(a,b,c) = a

Trilinears a

Barycentrics a cos 2A : b cos 2B : c cos 2C

Trilinears tan A cot 2A : :

Trilinears cos^2 A - sin^2 A : :

Trilinears 1 - 2 sin^2 A : :

Trilinears 1 - 2 cos^2 A : :

X(47) = (r^{2} - R^{2} + s^{2})*X(1) - 6rR*X(2) - 4r^{2}*X(3)
(Peter Moses, April 2, 2013)

Let A'B'C' be the Kosnita triangle. Let A" be the trilinear product B'*C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(47). (Randy Hutson, March 21, 2019)

Let A'B'C' and A"B"C" be the Lucas and Lucas(-1) tangents triangles. Let A* be the trilinear product A'*A", and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(47). (Randy Hutson, March 21, 2019)

X(47) lies on these lines:

1,21 19,921 33,90 34,46
35,212 36,602 91,92
158,162 171,498 238,499

X(47) is the {X(91),X(92)}-harmonic conjugate of X(564). For a list
of other harmonic conjugates of X(47), click **Tables** at the top
of this page.

X(47) = isogonal conjugate of X(91)

X(47) = isotomic conjugate of X(20571)

X(47) = trilinear product X(371)*X(372)

X(47) = X(92)-isoconjugate of X(1820)

X(47) = perspector of ABC and extraversion triangle of X(47) (which is also the anticevian triangle of X(47))

X(47) = eigencenter of cevian triangle of X(92)

X(47) = eigencenter of anticevian triangle of X(48)

X(47) = X(92)-Ceva conjugate of X(48)

X(47) = crosssum of X(i) and X(j) for these (i,j): (656,1109)

X(47) = X(275)-aleph conjugate of X(92)

X(47) = X(i)-beth conjugate of X(j) for these (i,j): (110,34), (643,47)

X(47) = trilinear product of X(371) and X(372)

Trilinears a

Trilinears S

Trilinears 1 - cot B cot C : :

Trilinears sin 2A : :

X(48) = (r^{2} + 4rR^{2} +
4R^{2} + s^{2})*X(1) - 6R(2R + r)*X(2) -
2(r^{2} + 2rR - s^{2})*X(3) (Peter Moses,
April 2, 2013)

Let A'B'C' be the hexyl triangle. Let A" be the barycentric product of the circumcircle intercepts of line B'C'. Define B", C" cyclically. The lines AA", BB", CC" concur in X(48). (Randy Hutson, July 31 2018)

X(48) lies on these lines:

1,19 3,71 6,41 9,101
31,560 36,579 37,205
42,197 55,154 63,326
75,336 163,1094 184,212
220,963 255,563 281,944
282,947 354,584 577,603
692,911 949,1037 958,965

X(48) is the {X(41),X(604)}-harmonic conjugate of X(6). For a list
of other harmonic conjugates of X(48), click **Tables** at the top
of this page.

X(48) = isogonal conjugate of X(92)

X(48) = isotomic conjugate of X(1969)

X(48) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,31), (3,212), (63,255), (92,47), (284, 6)

X(48) = X(228)-cross conjugate of X(3)

X(48) = complement of X(21270)

X(48) = crosspoint of X(i) and X(j) for these (i,j): (1,63), (3,222), (91,92), (219,268)

X(48) = crosssum of X(i) and X(j) for these (i,j): (1,19), (4,281), (47,48), (196,278), (523, 1146), (661,1109)

X(48) = crossdifference of every pair of points on line X(240)X(522)

X(48) = X(1)-line conjugate of X(240)

X(48) = X(i)-beth conjugate of X(j) for these (i,j): (101,48), (219,219), (284,604), (906,48)

X(48) = barycentric product of PU(16)

X(48) = vertex conjugate of PU(18)

X(48) = bicentric sum of PU(22)

X(48) = PU(22)-harmonic conjugate of X(656)

X(48) = trilinear pole of line X(810)X(822)

X(48) = X(2)-isoconjugate of X(4)

X(48) = X(75)-isoconjugate of X(19)

X(48) = X(91)-isoconjugate of X(1748)

X(48) = perspector of ABC and extraversion triangle of X(48) (which is also the anticevian triangle of X(48))

X(48) = crosspoint of X(2066) and X(5414)

X(48) = {X(1),X(19)}-harmonic conjugate of X(1953)

Trilinears cos(B - C) - cos(C - A) cos(A - B) : :

Barycentrics sin A cos 3A : sin B cos 3B : sin C cos 3C

Barycentrics a^4 (a^2 - b^2 - c^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - b^2 c^2) : :

**
V. Thebault,** "Sine-triple-angle-circle," *Mathesis* 65 (1956) 282-284.

X(49) lies on these lines: 1,215 3,155 4,156 5,54 24,568 52,195 93,94 381,578

Suppose that P and Q are distinct points in the plane of a triangle ABC . Let P_{A} = reflection of P in line AQ, let Q_{A} = reflection of Q in line AP, and let M_{A} = midpoint of segment P_{A}Q_{A}. Define M_{B} and M_{C} cyclically. César Lozada found that if Q = isogonal conjugate of P, then the locus of P for which M_{A}M_{B}M_{C} is perspective to ABC is the union of a cubic and 6 circles: specifically, the McCay cubic (K003), the circles {{B,C,B',C'}}, {{C,A,C',A'}}, {{A,B,A',B'}}, and the circles {{B,C,A'}}, {{C,A,B'}}, {{A,B,C'}}, where A',B',C' are the excenters of ABC. Moreover, if P = X(3) and Q = X(4), then M_{A}M_{B}M_{C} is not only perspective, but homothetic, to ABC, and the center of homothety is X(49).
See Hyacinthos 23265, June 1, 2015.

X(49) is the {X(54),X(110)}-harmonic conjugate of X(5). For a list of other harmonic conjugates of X(49), click **Tables** at the top of this page.

X(49) = isogonal conjugate of X(93)

X(49) = isotomic conjugate of X(20572)

X(49) = X(4)-isoconjugate of X(2962)

X(49) = X(92)-isoconjugate of X(2963)

X(49) = eigencenter of cevian triangle of X(94)

X(49) = eigencenter of anticevian triangle of X(50)

X(49) = X(94)-Ceva conjugate of X(50)

Trilinears cos A sin 2A + sin A cos 2A : :

Trilinears sin A + cos A cot D/2 : : , where cot D/2 = (4*area)/(6R

Trilinears a(1 - 4 cos

Trilinears a(1 + 2 cos 2A) : b(1 + 2 cos 2B) : c(1 + 2 cos 2C)

Barycentrics sin A sin 3A : sin B sin 3B : sin C sin 3C

Barycentrics a^4 ((a^2 - b^2 - c^2)^2 - b^2 c^2) : :

Tripolars (pending)

X(50) = -(r

Let DEF be any equilateral triangle inscribed in the circumcircle of ABC. Let D' be the barycentric product E*F, and define E', F' cyclically. Then D',E',F' all line on a line passing through X(50). In the special case that DEF is the circumtangential triangle, the points D',E',F' lie on the Brocard axis, and in case DEF is the circumnormal triangle, the points D',E'F' lie on the line X(50)X(647). See also X(6149). (Randy Hutson, January 29, 2015)

Let A'B'C' and A"B"C" be the (equilateral) circumcevian triangles of X(15) and X(16). Let A* be the barycentric product A'*A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(50). See also X(6149). (Randy Hutson, January 29, 2015)

Let AA_{1}A_{2}, BB_{1}B_{2}, CC_{1}C_{2} be the circumcircle-inscribed equilateral triangles used in the construction of the Trinh triangle. Let A' be the barycentric product A_{1}*A_{2}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(50); see also X(6149). (Randy Hutson, October 13, 2015)

Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. Let A' be the crossdifference of A1 and A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(50). (Randy Hutson, June 27, 2018)

X(50) lies on these lines: 3,6 67,248 112,477 115,231 230,858 338,401 647,654

X(50) is the {X(3),X(6)}-harmonic conjugate of X(566). For a list of other harmonic conjugates of X(40), click **Tables** at the top of this page.

X(50) = isogonal conjugate of X(94)

X(50) = isotomic conjugate of X(20573)

X(50) = circumcircle-inverse of X(32761)

X(50) = Brocard-circle-inverse of X(566)

X(50) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,215), (74,184), (94,49)

X(50) = crosspoint of X(i) and X(j) for these (i,j): (93,94), (186,323)

X(50) = crosssum of X(49) and X(50)

X(50) = crossdifference of every pair of points on line X(5)X(523)

X(50) = barycentric product of X(15) and X(16)

X(50) = X(i)-isoconjugate of X(j) for these (i,j): (92,265), (1577,476)

X(50) = Cundy-Parry Phi transform of X(568)

X(50) = perspector of circumconic passing through X(110) and the isogonal conjugates of PU(5)

X(50) = X(2)-Ceva conjugate of X(11597)

X(50) = perspector of ABC and unary cofactor triangle of Ehrmann vertex-triangle

X(50) = barycentric product X(35)*X(36)

X(50) = crossdifference of PU(173)

Trilinears a[a

Trilinears sin A (sin 2B + sin 2C) : :

Trilinears sec A (csc 2B + csc 2C) : :

Barycentrics a

X(51) = (r^{2} + 2rR +
s^{2})*X(1) + 6R(R - r)*X(2) - (r^{2} + 4rR -
s^{2})*X(3) (Peter Moses, April 2, 2013)

Let A'B'C' be the anticomplementary triangle and let Ba and Ca be the orthogonal projections of B' and C' on BC, respectively. Define Cb and Ac cyclically, and define Ab and Bc cyclically. Then X(51) is the centroid of BaCaCbAbAcBc. (Randy Hutson, April 9, 2016)

Let L be the van Aubel line. Let U = X(6)X(25), the isogonal conjugate of polar conjugate of L; let V = X(4)X(51), the polar conjugate of the isogonal conjugate of L. Then X(51) = U∩V. (Randy Hutson, April 9, 2016)

Let A'B'C' be the orthic triangle. Let Oa be the A-McCay circle of triangle AB'C', and define Ob, Oc cyclically. X(51) is the radical center of circles Oa, Ob, Oc. (Randy Hutson, July 31 2018)

X(51) lies on these lines:

2,262 4,185 5,52 6,25
21,970 22,182 23,575
24,578 26,569 31,181
39,237 44,209 54,288
107,275 125,132 129,137
130,138 199,572 210,374
216,418 381,568 397,462
398,463 573,1011

X(51) is the {X(5),X(143)}-harmonic conjugate of X(52). For a list
of other harmonic conjugates of X(51), click **Tables** at the top
of this page.

X(51) = reflection of X(210) in X(375)

X(51) = isogonal conjugate of X(95)

X(51) = complement of X(2979)

X(51) = anticomplement of X(3819)

X(51) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,53), (5,216), (6,217)

X(51) = X(217)-cross conjugate of X(216)

X(51) = crosspoint of X(i) and X(j) for these (i,j): (4,6), (5,53)

X(51) = crosssum of X(i) and X(j) for these (i,j): (2,3), (6,160), (54,97)

X(51) = crossdifference of every pair of points on line X(323)X(401)

X(51) = inverse-in-orthosymmedial-circle of X(125)

X(51) = X(2) of tangential triangle of Johnson circumconic

X(51) = trilinear pole of polar of X(276) wrt polar circle

X(51) = pole wrt polar circle of trilinear polar of X(276) (line X(340)X(520))

X(51) = X(48)-isoconjugate (polar conjugate) of X(276)

X(51) = X(92)-isoconjugate of X(97)

X(51) = Zosma transform of X(92)

X(51) = perspector of 1st & 2nd orthosymmedial triangles

X(51) = bicentric sum of PU(157)

X(51) = PU(157)-harmonic conjugate of X(647)

X(51) = perspector of orthic-of-orthocentroidal triangle and orthocentroidal-of-orthic triangle

X(51) = centroid of reflection triangle of X(125)

X(51) = excentral-to-ABC functional image of X(2)

X(51) = centroid of anticomplementary circle intercepts of sidelines of ABC

X(51) = {X(34221),X(34222)}-harmonic conjugate of X(5480)

Trilinears sec A (sec 2B + sec 2C) : :

Trilinears cos(A - 2B) + cos(A - 2C) : :

Barycentrics tan A (sec 2B + sec 2C) : tan B (sec 2C + sec 2A) : tan C (sec 2A + sec 2B)

Barycentrics a^2(a^4 + b^4 + c^4 - 2a^2b^2 - 2a^2c^2)[a^2(b^2 + c^2) - (b^2 - c^2)^2] : :

Tripolars (pending)

X(52) = (r

Let Ha be the foot of the A-altitude. Let Ba be the foot of the perpendicular from Ha to CA, and define Cb and Ac cyclically. Let Ca be the foot of the perpendicular from Ha to AB, and define Ab and Bc cyclically. Let A' be the orthocenter of HaBaCa, and define B' and C' cyclically. The lines HaA', HbB', HcC' concur in X(52). (Randy Hutson, December 10, 2016)

X(52) lies on these lines:

3,6 4,68 5,51 25,155
26,184 30,185 49,195
113,135 114,211 128,134 129,139

X(52) is the {X(5),X(143)}-harmonic conjugate of X(51). For a list
of other harmonic conjugates of X(52), click **Tables** at the top
of this page.

X(52) = reflection of X(i) in X(j) for these (i,j): (3,389), (5,143), (113,1112), (1209,973)

X(52) = isogonal conjugate of X(96)

X(52) = anticomplement of X(1216)

X(52) = circumcircle-inverse of X(32762)

X(52) = Brocard-circle-inverse of X(569)

X(52) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,5), (317,467), (324,216)

X(52) = crosspoint of X(4) and X(24)

X(52) = crosssum of X(3) and X(68)

X(52) = {X(3),X(6)}-harmonic conjugate of X(569)

X(52) = orthic isogonal conjugate of X(5)

X(52) = X(20)-of-2nd Euler triangle

X(52) = perspector of ABC and cross-triangle of ABC and 2nd Euler triangle

X(52) = perspector of ABC and cross-triangle of ABC and Kosnita triangle

X(52) = antipode of X(113) in Hatzipolakis-Lozada hyperbola

X(52) = Cundy-Parry Phi transform of X(571)

X(52) = Cundy-Parry Psi transform of X(5392)

X(52) = X(1577)-isoconjugate of X(32692)

X(52) = excentral-to-ABC functional image of X(4)

Barycentrics a tan A cos(B - C) : b tan B cos(C - A) : c tan C cos(A - B)

Barycentrics (a^2 (b^2 + c^2) - (b^2 - c^2)^2)/(a^2 - b^2 - c^2) : :

Let A'B'C' be the Euler triangle. Let L_{A} be the trilinear polar of A', and define L_{B} and L_{C} cyclically. Let A" = L_{B}∩L_{C}, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(53). (Randy Hutson, June 7, 2019)

X(53) lies on these lines:

4,6 5,216 25,157 30,577
45,281 115,133 128,139
137,138 141,264 232,427
273,1086 275,288 311,324
317,524 318,594 395,472 396,473

X(53) is the {X(4),X(393)}-harmonic conjugate of X(6). For a list of
other harmonic conjugates of X(53), click **Tables** at the top of
this page.

X(53) = isogonal conjugate of X(97)

X(53) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,51), (324,5)

X(53) = X(51)-cross conjugate of X(5)

X(53) = crosssum of X(3) and X(577)

X(53) = Kosnita(X(4),X(6)) point

X(53) = trilinear pole of line X(12077)X(15451) (the polar of X(95) wrt polar circle)

X(53) = pole wrt polar circle of trilinear polar of X(95) (line X(323)X(401))

X(53) = polar conjugate of X(95)

X(53) = excentral-to-ABC functional image of X(6)

Trilinears a/(b cos B + c cos C) : :

Barycentrics sin A sec(B - C) : sin B sec(C - A) : sin C sec(A - B)

Barycentrics a^2/(S^2 + SB SC) : :

Barycentrics a^2/(b^4 + c^4 - a^2b^2 - a^2c^2 - 2b^2c^2) : :

**John Rigby,** "Brief notes on some forgotten geometrical
theorems," *Mathematics and Informatics Quarterly* 7 (1997)
156-158.

Let O be the circumcenter of triangle ABC, and O_{A} the
circumcenter of triangle BOC. Define O_{B} and O_{C}
cyclically. Then the lines AO_{A}, BO_{B},
CO_{C} concur in X(54). For details and generalization, see

**Darij Grinberg,**
A New
Circumcenter Question

The above construction of X(54) generalizes. Suppose that P and Q are points (as functions of a,b,c). Let A' = Q-of-BCP, B' = Q-of-CAP, C' = Q-of-ABP. If the lines AA', BB', CC' concur, the perspector is called the Kosnita(P,Q) point, denoted by K(P,Q). (Randy Hutson, 9/23/2011)

Let N_{A}N_{B}N_{C} be the reflection triangle of X(5). Let O_{A} be the circumcenter of AN_{B}N_{C}, and define O_{B} and O_{C} cyclically. Triangle O_{A}O_{B}O_{C} is perspective to ABC at X(54), homothetic to the orthic-of-orthocentroidal triangle at X(54), and orthologic to the reflection triangle at X(54). (Randy Hutson, June 7, 2019)

X(3) = K(X(20),X(2)) | X(4) = K(X(20,X(20) | X(5) = K(X(4),X(2)) |

X(13) = K(X(13),X(1)) | X(17) = K(X(13),X(3)) | X(18) = K(X(14),X(3)) |

X(140) = K(X(3), X(2)) | X(251) = K(X(6), X(6)) | |

X(481) = K(X(175),X(1)) | X(482) = K(X(176),X(1)) |

X(54) lies on the Napoleon cubic and these lines:

1,3460 2,68 3,97 4,184
5,49 6,24 12,215 36,73
39,248 51,288 64,378
69,95 71,572 72,1006
74,185 112,217 140,252
156,381 186,389 276,290
575,895 826,879 3336,3468

X(54) is the {X(5),X(49)}-harmonic conjugate of X(110). For a list
of other harmonic conjugates of X(54), click **Tables** at the top
of this page.

X(54) = midpoint of X(3) and X(195)

X(54) = reflection of X(195) in X(1493)

X(54) = isogonal conjugate of X(5)

X(54) = isotomic conjugate of X(311)

X(54) = inverse-in-circumcircle of X(1157)

X(54) = complement of X(2888)

X(54) = anticomplement of X(1209)

X(54) = X(i)-Ceva conjugate of X(j) for these (i,j): (5,2120), (95,97), (288,6)

X(54) = cevapoint of X(6) and X(184)

X(54) = X(i)-cross conjugate of X(j) for these (i,j): (3,96), (6,275), (186,74), (389,4), (523,110)

X(54) = crosspoint of X(i) and X(j) for these {i,j}: {4,3459}, {95,275}

X(54) = crosssum of X(i) and X(j) for these (i,j): (3,195), (51,216), (627,628)

X(54) = antigonal conjugate of X(33565)

X(54) = X(24)-of-intouch-triangle

X(54) = trilinear pole of line X(50)X(647) (the polar of X(324) wrt polar circle)

X(54) = pole wrt polar circle of trilinear polar of X(324)

X(54) = X(48)-isoconjugate (polar conjugate) of X(324)

X(54) = X(92)-isoconjugate of X(216)

X(54) = intersection of tangents to hyperbola {{A,B,C,X(4),X(5)}} at X(4) and X(3459)

X(54) = {X(2595),X(2596)}-harmonic conjugate of X(1087)

X(54) = trilinear product of vertices of circumnormal triangle

X(54) = intersection of tangents at X(3) and X(4) to Neuberg cubic K001

X(54) = exsimilicenter of nine-point circle and sine-triple-angle circle

X(54) = homothetic center of orthocevian triangle of X(3) and circumorthic triangle

X(54) = perspector of ABC and unary cofactor triangle of reflection triangle

X(54) = X(3)-of-reflection-triangle-of-X(5)

X(54) = perspector of ABC and cross-triangle of ABC and circumorthic triangle

X(54) = perspector of ABC and Hatzipolakis-Moses triangle

X(54) = X(191)-of-orthic-triangle if ABC is acute

X(54) = trilinear product of vertices of X(4)-altimedial triangle

Trilinears 1 + cos A : 1 + cos B : 1 + cos C

Trilinears cos

Trilinears tan(B/2) + tan(C/2) : tan(C/2) + tan(A/2) : tan(A/2) + tan(B/2)

Trilinears a(a - s) : b(b - s) : c(c - s)

Trilinears a(cot A/2) : :

Trilinears a

Trilinears a(2ar - S) : :

Barycentrics a

Barycentrics area(A'BC) : : , where A'B'C' = 1st circumperp triangle

X(55) = R*X(1) + r*X(3)

X(55) = (Ra+Rb+Rc)*X(1) + r*Ja + r*Jb + r*Jc, where Ja, Jb, Jc are excenters, and Ra, Rb, Rc are the exradii

X(55) = center of homothety of three triangles: tangential, intangents, and extangents. Also, X(55) is the pole-with-respect-to-the-circumcircle of the trilinear polar of X(1). These properties and others are given in

**O. Bottema and J. T. Groenman,** "De gemene raaklijnen van de vier raakcirkels van een driehoek, twee aan twee," *Nieuw Tijdschrift voor Wiskunde* 67 (1979-80) 177-182.

Let A', B', C' be the second points of intersection of the angle bisectors of triangle ABC with its incircle. Let Oa be the circle externally tangent to the incircle at A', and internally tangent to the circumcircle; define Ob and Oc cyclically. Then X(55) is the radical center of circles Oa, Ob, Oc. Let A" be the touchpoint of Oa and the circumcircle, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(55). Let Ba, Ca be the intersections of lines CA, AB, respectively, and the antiparallel to BC through a point P. Define Cb, Ab, Ac, Bc cyclically. Triangles ABaCa, AbBCb, AcBcC are congruent only when P = X(55) or one of its 3 extraversions. Let A*B*C* be the incentral triangle. Let La be the reflection of line BC in line AA*, and define Lb and Lc cyclically. Let A''' = Lb∩Lc, and define B''' and C'''. The lines A*A''', B*B''', C*C''' concur in X(55). (Randy Hutson, November 18, 2015)

Let A'B'C' be the extouch triangle. Let A" be the barycentric product of the circumcircle intercepts of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(55). (Randy Hutson, July 31 2018)

Let (Oa) be the circumcircle of BCX(1). Let Pa be the perspector of (Oa). Let La be the polar of Pa wrt (Oa). Define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(55). (Randy Hutson, July 31 2018)

X(55) lies on these lines:

1,3 2,11 4,12 5,498
6,31 7,2346 8,21 9,200 10,405
15,203 16,202 19,25
20,388 30,495 34,227
41,220 43,238 45,678
48,154 63,518 64,73
77,1037 78,960 81,1002
92,243 103,109 104,1000
108,196 140,496 181,573
182,613 183,350 184,215
192,385 199,1030 201,774
204,1033 219,284 223,1456 226,516
255,601 256,983 329,1005
376,1056 386,595 392,997
411,962 511,611 515,1012
519,956 574,1015 603,963
631,1058 650,884 654,926
748,899 840,901 846,984
869,893 1026,1083 1070,1076
1072,1074 2195,5452

X(55) is the {X(1),X(3)}-harmonic conjugate of X(56). For a list of other harmonic conjugates of X(55), click **Tables** at the top of this page.

X(55) = reflection of X(i) in X(j) for these (i,j): (1478,495), (2099,1)

X(55) = isogonal conjugate of X(7)

X(55) = isotomic conjugate of X(6063)

X(55) = centroid of curvatures of circumcircle and excircles

X(55) = inverse-in-circumcircle of X(1155)

X(55) = complement of X(3434)

X(55) = anticomplement of X(2886)

X(55) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,6), (3,198), (7,218), (8,219), (9,220), (21,9), (59,101), (104,44), (260,259), (284,41)

X(55) = cevapoint of X(42) and X(228) for these (i,j)

X(55) = X(i)-cross conjugate of X(j) for these (i,j): (41,6), (42,33), (228,212)

X(55) = crosspoint of X(i) and X(j) for these (i,j): (1,9), (3,268), (7,277), (8,281), (21,284), (59,101)

X(55) = crosssum of X(i) and X(j) for these (i,j): (1,57), (2,145), (4,196), (11,514), (55,218), (56,222), (63,224), (65,226), (81,229), (177,234), (241,1362), (513,1086), (905,1364), (1361,1465)

X(55) = crossdifference of every pair of points on line X(241)X(514)

X(55) = X(i)-Hirst inverse of X(j) for these (i,j): (6,672), (43,241)

X(55) = X(1)-line conjugate of X(241)

X(55) = X(i)-beth conjugate of X(j) for these (i,j): (21,999), (55,31), (100,55), (200,200), (643,2), (1021,1024)

X(55) = insimilicenter of the intangents and extangents circles

X(55) = insimilicenter of the intangents and tangential circles

X(55) = exsimilicenter of then extangents and tangential circles

X(55) = X(22)-of-intouch-triangle

X(55) = trilinear pole of line X(657)X(663) (polar of X(331) wrt polar circle)

X(55) = pole wrt polar circle of trilinear polar of X(331)

X(55) = X(48)-isoconjugate (polar conjugate) of X(331)

X(55) = homothetic center of ABC and Mandart-incircle triangle

X(55) = inverse-in-Feuerbach-hyperbola of X(1001)

X(55) = inverse-in-circumconic-centered-at-X(1) of X(1936)

X(55) = {X(1),X(40)}-harmonic conjugate of X(65)

X(55) = trilinear square of X(259)

X(55) = Danneels point of X(100)

X(55) = vertex conjugate of PU(48)

X(55) = vertex conjugate of foci of Mandart inellipse

X(55) = excentral isotomic conjugate of X(2942)

X(55) = homothetic center of the reflections of the intangents and extangents triangles in their respective Euler lines

X(55) = perspector of ABC and extraversion triangle of X(56)

X(55) = trilinear product of PU(104)

X(55) = barycentric product of PU(112)

X(55) = bicentric sum of PU(112)

X(55) = PU(112)-harmonic conjugate of X(650)

X(55) = perspector of ABC and unary cofactor triangle of 7th mixtilinear triangle

X(55) = perspector of 4th mixtilinear triangle and unary cofactor triangle of 7th mixtilinear triangle

X(55) = perspector of unary cofactor triangles of 3rd, 4th and 5th extouch triangles

X(55) = {X(3513),X(3514)}-harmonic conjugate of X(56)

X(55) = perspector of ABC and cross-triangle of ABC and extangents triangle

X(55) = perspector of ABC and cross-triangle of ABC and Hutson extouch triangle

X(55) = homothetic center of ABC and cross-triangle of ABC and 2nd Johnson-Yff triangle

X(55) = Thomson-isogonal conjugate of X(5657)

X(55) = homothetic center of midarc triangle and 2nd-circumperp-of-1st-circumperp triangle (which is also 1st-circumperp-of-2nd-circumperp triangle)

X(55) = homothetic center of 2nd midarc triangle and 1st-circumperp-of-1st-circumperp triangle (which is also 2nd-circumperp-of-2nd-circumperp triangle)

X(55) = Cundy-Parry Phi transform of X(942)

X(55) = Cundy-Parry Psi transform of X(943)

X(55) = X(4)-of-1st-Johnson-Yff-triangle

X(55) = homothetic center of anti-Hutson intouch triangle and anti-tangential midarc triangle

X(55) = barycentric product of circumcircle intercepts of excircles radical circle

Trilinears 1 - cos A : 1 - cos B : 1 - cos C

Trilinears sin

Trilinears a(tan A/2) : :

Trilinears Ra - r : Rb - r : Rc - r, where Ra, Rb, Rc are the exradii

Trilinears a*Ra : b*Rb : c*Rc, where Ra, Rb, Rc are the exradii

Trilinears a cos A - (c + a) cos B - (a + b) cos C : :

Barycentrics a

Barycentrics area(A'BC) : : , where A'B'C' = 2nd circumperp triangle

X(56) is the perspector of the tangential triangle and the reflection of the intangents triangle in X(1).

Let A'B'C' be the Fuhrmann triangle. Let La be the line through A' parallel to BC, and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines AA", BB", CC" concur in X(56). Also, let AaBaCa be the poristic triangle (i.e., a triangle with common circumcircle and incircle as ABC) such that BaCa is parallel to BC. Define AbBbCb and AcBcCc cyclically. The lines AAa, BBb, CCc concur in X(56). (Randy Hutson, November 18, 2015)

Let A'B'C' be the intouch triangle. Let A" be the barycentric product of the circumcircle intercepts of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(56). (Randy Hutson, June 27, 2018)

If you have Geometer's Sketchpad, **X(56)**.

If you have GeoGebra, you can view **X(56)**.

X(56) lies on these lines:

1,3 2,12 4,11 5,499
6,41 7,21 8,404 10,474
19,207 20,497 22,977
25,34 28,278 30,496
31,154 32,1015 33,963
38,201 58,222 61,202
62,203 63,960 72,997
77,1036 78,480 81,959
85,870 87,238 100,145
101,218 105,279 106,109
140,495 181,386 182,611
197,227 212,939 219,579
220,672 223,937 226,405
255,602 266,289 269,738
330,385 376,1058 411,938
511,613 551,553 607,911
631,1056 667,764 946,1012
978,979 1025,1083 1070,1074
1072,1076 1345,2464

X(56) is the {X(1),X(3)}-harmonic conjugate of X(55). For a list of other harmonic conjugates of X(56), click **Tables** at the top of this page.

X(56) = midpoint of X(1) and X(46)

X(56) = reflection of X(i) in X(j) for these (i,j): (1479,496), (2098,1)

X(56) = isogonal conjugate of X(8)

X(56) = isotomic conjugate of X(3596)

X(56) = inverse-in-circumcircle of X(1319)

X(56) = complement of X(3436)

X(56) = anticomplement of X(1329)

X(56) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,221), (7,222), (28,34), (57,6), (59,109), (108,513)

X(56) = X(31)-cross conjugate of X(6)

X(56) = crosspoint of X(i) and X(j) for these (i,j): (1,84), (7,278), (28,58), (57,269), (59,109)

X(56) = crosssum of X(i) and X(j) for these (i,j): (1,40), (2,144), (6,197), (9,200), (10,72), (11,522), (55,219), (175,176), (519,1145)

(56) = crossdifference of every pair of points on line X(522)X(650)

X(56) = X(i)-Hirst inverse of X(j) for these (i,j): (6,1458), (34,1430), (57,1429), (604,1428), (1416,1438)

X(56) = X(266)-aleph conjugate of X(1050)

X(56) = X(i)-beth conjugate of X(j) for these (i,j):

(1,1), (21,3), (56,1106), (58,56), and (P,57) for all P on the circumcircle

X(56) = homothetic center of the intouch triangle and the circumcevian triangle of X(1)

X(56) = trilinear pole of line X(649)X(854) (the isogonal conjugate of the isotomic conjugate of the Gergonne line)

X(56) = homothetic center of ABC and the reflection of the Mandart-incircle triangle in X(1)

X(56) = {X(1),X(40)}-harmonic conjugate of X(3057)

X(56) = {X(1),X(57)}-harmonic conjugate of X(65)

X(56) = trilinear square of X(266)

X(56) = trilinear square root of X(1106)

X(56) = X(92)-isoconjugate of X(219)

X(56) = vertex conjugate of PU(93)

X(56) = inverse-in-{circumcircle, incircle}-inverter of X(3660)

X(56) = perspector of ABC and extraversion triangle of X(55)

X(56) = perspector of ABC and unary cofactor triangle of Gemini triangle 15

X(56) = homothetic center of intangents triangle and reflection of tangential triangle in X(3)

X(56) = homothetic center of tangential triangle and reflection of intangents triangle in X(1)

X(56) = Brianchon point (perspector) of inellipse that is isogonal conjugate of isotomic conjugate of incircle

X(56) = pole wrt polar circle of trilinear polar of X(7017) (line X(2804)X(4397))

X(56) = X(48)-isoconjugate (polar conjugate) of X(7017)

X(56) = barycentric product of PU(46)

X(56) = bicentric sum of PU(60)

X(56) = PU(60)-harmonic conjugate of X(650)

X(56) = trilinear product of PU(92)

X(56) = perspector of ABC and cross-triangle of ABC and Apus triangle

X(56) = perspector of ABC and cross-triangle of ABC and Hutson intouch triangle

X(56) = homothetic center of ABC and cross-triangle of ABC and 1st Johnson-Yff triangle

X(56) = homothetic center of midarc triangle and 1st-circumperp-of-1st-circumperp triangle (which is also 2nd-circumperp-of-2nd-circumperp triangle)

X(56) = homothetic center of 2nd midarc triangle and 2nd-circumperp-of-1st-circumperp triangle (which is also 1st-circumperp-of-2nd-circumperp triangle)

X(56) = Cundy-Parry Phi transform of X(517)

X(56) = Cundy-Parry Psi transform of X(104)

X(56) = {X(3513),X(3514)}-harmonic conjugate of X(55)

X(56) = X(4)-of-2nd-Johnson-Yff-triangle

X(56) = homothetic center of tangential triangle and anti-tangential midarc triangle

X(56) = Ursa-major-to-Ursa-minor similarity image of X(4)

X(56) = barycentric product of (nonreal) circumcircle intercepts of the Gergonne line

Trilinears tan(A/2) : tan(B/2) : tan(C/2)

Trilinears 1 + cos B + cos C - cos A

Trilinears 1 + sin(A/2)csc(B/2)csc(C/2) : :

Trilinears cos

Trilinears S

Trilinears csc A - cot A : :

Trilinears (1 - cos A) csc A : :

Trilinears b(cot B/2) + c(cot C/2) - a(cot A/2) : :

Trilinears cot A' : :, where A'B'C' is the excentral triangle

Trilinears |AA'|/|AX(1)| : |BB'|/|BX(1)| : |CC'|/|CX(1)|, where A'B'C' is the excentral triangle

Trilinears Ra : Rb : Rc, where Ra, Rb, Rc are the exradii

Barycentrics: Ra - r : Rb - r : Rc - r, where Ra, Rb, Rc are the exradii

Barycentrics a/(b + c - a) : b/(c + a - b) : c/(a + b - c)

Barycentrics 1 - cos A : 1 - cos B : 1 - cos C

Barycentrics area(A'BC) : : , where A'B'C' = excentral triangle

Let Ja, Jb, Jc be the excenters and I the incenter of ABC. Let Ka be the symmedian point of JbJcI, and define Kb, Kc cyclically. Then KaKbKc is perspective to ABC at X(57). (Randy Hutson, September 14, 2016)

Let A' be the perspector of the circumconic centered at the A-excenter, and define B' and C'cyclically. The lines AA', BB', CC' concur in X(57). (Randy Hutson, September 14, 2016)

Let A'B'C' be the mixtilinear incentral triangle. Let A" be the trilinear pole of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(57). (Randy Hutson, September 14, 2016)

Let A' be the perspector of the A-mixtilinear incircle, and define B' and C'cyclically. The lines AA', BB', CC' concur in X(57). (Randy Hutson, September 14, 2016)

Let A', B' and C'be the inverse-in-{circumcircle, incircle}-inverter of A, B, C. Let A"B"C" be the tangential triangle of A'B'C'. A"B"C" is perspective to the intouch triangle at X(57). (Randy Hutson, September 14, 2016)

Let A'B'C' be the orthic triangle. Let La be the reflection of line B'C' in the internal angle bisector of A, and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. Triangle A"B"C" is homothetic to ABC, with center of homothety X(57). (Randy Hutson, September 14, 2016)

Let Oa be the circle passing through B and C, and tangent to the incircle. Define Ob and Oc cyclically. Let A' be the point of tangency of Oa and the incircle, and define B' and C' cyclically. Triangle A'B'C' is perspective to the intouch triangle at X(57). Also, X(57) is the radical center of circles Oa, Ob, Oc. (Randy Hutson, July 31 2018)

Let A'B'C' be the intouch triangle. Let A" be the trilinear product of the circumcircle intercepts of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(57). (Randy Hutson, July 31 2018)

Let A_{1}B_{1}C_{1} be Gemini triangle 1. Let A' be the perspector of conic {{A,B,C,B_{1},C_{1}}}, and define B' and C' cyclically. Triangle A'B'C' is the tangential of excentral triangle. The lines AA', BB', CC' concur in X(57). (Randy Hutson, January 15, 2019)

Let V_{a}, V_{b}, V_{c} be the antipodes of V=X(40) in the circles (VBC), (VCA), (VAB), respectively. The lines AV_{a}, BV_{b}, CV_{c} concur in X(57). (Angel Montesdeoca, October 14, 2019)

X(57) is the perspector of the intouch triangle and excentral triangle.

X(57) lies on the Thomson cubic and these lines:

1,3 2,7 4,84 6,222
10,388 19,196 20,938
27,273 28,34 31,105
33,103 38,612 42,1002
43,181 72,474 73,386
77,81 78,404 79,90
85,274 88,651 92,653
164,177 169,277 173,174
200,518 201,975 234,362
239,330 255,580 279,479
282,3343 345,728 497,516
499,920 649,1024 658,673
748,896 758,997 955,991
957,995 959,1042 961,1106
978,1046 1020,1086 1073,3351
3342,3350

X(57) is the {X(2),X(7)}-harmonic conjugate of X(226). For a list of other harmonic conjugates of X(57), click **Tables** at the top of this page.

X(57) = midpoint of X(497) and X(3474)

X(57) = reflection of X(i) in X(j) for these (i,j): (1,999), (200,1376)

X(57) = isogonal conjugate of X(9)

X(57) = isotomic conjugate of X(312)

X(57) = inverse-in-circumcircle of X(2078)

X(57) = inverse-in-Bevan-circle of X(1155)

X(57) = complement of X(329)

X(57) = anticomplement of X(3452)

X(57) = trilinear product of PU(46)

X(57) = trilinear pole of PU(96) (line X(513)X(663), the polar of X(318) wrt polar circle, and the Monge line of the mixtilinear incircles)

X(57) = barycentric product of PU(94)

X(57) = pole wrt polar circle of trilinear polar of X(318)

X(57) = X(48)-isoconjugate (polar conjugate) of X(318)

X(57) = X(6)-isoconjugate of X(8)

X(57) = X(75)-isoconjugate of X(41)

X(57) = X(92)-isoconjugate of X(212)

X(57) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,223), (7,1), (27,278), (81,222), (85,77), (273,34), (279,269)

X(57) = cevapoint of X(i) and X(j) for these (i,j): (6,56), (19,208)

X(57) = X(i)-cross conjugate of X(j) for these (i,j): (6,1), (19,84), (56,269), (65,7)

X(57) = crosspoint of X(i) and X(j) for these (i,j): (2,189), (7,279), (27,81), (85,273)

X(57) = crosssum of X(i) and X(j) for these (i,j): (1,165), (6,198), (31,205), (37,71), (41,212), (55,220), (210,1334)

X(57) = crossdifference of every pair of points on line X(650)X(663)

X(57) = X(i)-Hirst inverse of X(j) for these (i,j): (1,241), (7,1447), (56,1429), (105,1462), (910,1419)

X(57) = perspector of ABC and unary cofactor triangle of 6th mixtilinear triangle

X(57) = perspector of ABC and antipedal triangle of X(40)

X(57) = homothetic center of: ABC; orthic triangle of intouch triangle; tangential triangle of excentral triangle

X(57) = X(25)-of-excentral-triangle

X(57) = X(25)-of-intouch-triangle

X(57) = pole wrt Bevan circle of antiorthic axis

X(57) = perspector of Bevan circle

X(57) = perspector of circumconic centered at X(223)

X(57) = center of circumconic that is locus of trilinear poles of lines passing through X(223)

X(57) = {X(1),X(3)}-harmonic conjugate of X(3601)

X(57) = {X(1),X(40)}-harmonic conjugate of X(1697)

X(57) = {X(2),X(63)}-harmonic conjugate of X(9)

X(57) = {X(55),X(56)}-harmonic conjugate of X(1617)

X(57) = {X(56),X(65)}-harmonic conjugate of X(1)

X(57) = {X(3513),X(3514)}-harmonic conjugate of X(1)

X(57) = perspector of pedal and antipedal (or anticevian) triangles of X(1)

X(57) = perspector of ABC and medial triangle of pedal triangle of X(84)

X(57) = inverse-in-incircle of X(3660)

X(57) = inverse-in-circumconic-centered-at-X(9) of X(3911)

X(57) = orthocorrespondent of X(1)

X(57) = Danneels point of X(7)

X(57) = vertex conjugate of X(55) and X(57)

X(57) = perspector of ABC and extraversion triangle of X(9)

X(57) = crosssum of X(2066) and X(5414)

X(57) = trilinear product of extraversions of X(9)

X(57) = SS(A->A') of X(63), where A'B'C' is the excentral triangle

X(57) = Cundy-Parry Phi transform of X(40)

X(57) = Cundy-Parry Psi transform of X(84)

X(57) = perspector of ABC and cross-triangle of Gemini triangles 9 and 10

X(57) = perspector of ABC and cross-triangle of ABC and Gemini triangle 9

X(57) = perspector of ABC and cross-triangle of ABC and Gemini triangle 10

X(57) = barycentric product of vertices of Gemini triangle 9

X(57) = barycentric product of vertices of Gemini triangle 10

X(57) = perspector of ABC and tangential triangle, wrt Gemini triangle 2, of {ABC, Gemini 2}-circumconic

X(57) = perspector of Gemini triangle 36 and cross-triangle of ABC and Gemini triangle 36

X(57) = perspector of ABC and unary cofactor triangle of Gemini triangle 36

X(57) = X(i)-aleph conjugate of X(j) for these (i,j): (2,40), (7,57), (57,978), (174,1), (366,165), (507,503), (508,9), (509,43)

X(57) = X(i)-beth conjugate of X(j) for these (i,j):

(2,2), (81,57), (88,57), (100,57), (110,31), (162,57), (190,57), (333,63), (648,92), (651,57), (653,57), (655,57), (658,57), (660,57), (662,57), (673,57), (771,57), (799,57), (823,57), (897, 57)

Trilinears (1 - cos A)/(cos B + cos C) : :

Trilinears sa

Trilinears r cos A - s sin A : : , where s = semiperimeter and r = inradius

Trilinears sin(A - U) : : , U as at X(572) and X(573)

Trilinears (R/r) - 1/(cos B + cos C) : :

Trilinears (r/R) - cos 2A + 1 : :

Trilinears eccentricity of A-Soddy ellipse : :

Barycentrics a

X(58) is the point of concurrence of the Brocard axes of triangles BIC, CIA, AIB, ABC, (where I denotes the incenter, X(1)), as proved in **Antreas P. Hatzipolakis, Floor van Lamoen, Barry Wolk, and Paul Yiu,** Concurrency of Four Euler Lines, *Forum Geometricorum* 1 (2001) 59-68.

Let (Sa) be the reflection of the Spieker circle in BC, and define (Sb), (Sc) cyclically. X(58) is the radical center of (Sa), (Sb), (Sc). (Randy Hutson, July 20, 2016)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let A"B"C" be the circumcevian triangle, wrt A'B'C', of X(1). The lines AA", BB", CC" concur in X(58). (Randy Hutson, July 20, 2016)

Let A'B'C' be the Feuerbach triangle. Let La be the tangent to the nine-point circle at A', and define Lb, Lc cyclically. Let A" be the isogonal conjugate of the trilinear pole of La, and define B", C" cyclically. Let A* = BB"∩CC", B* = CC"∩AA", C* = AA"∩BB". The lines AA*, BB*, CC* concur in X(58). (Randy Hutson, July 20, 2016)

Let A'B'C' be the 2nd circumperp triangle. Let A" be the trilinear product B'*C', and define B", C" cyclically. A", B", C" are collinear on line X(36)X(238) (the trilinear polar of X(81)). The lines AA", BB", CC" concur in X(58). (Randy Hutson, July 20, 2016)

X(58) lies on these lines:

1,21 2,540 3,6 7,272
8,996 9,975 10,171
20,387 25,967 27,270
28,34 29,162 35,42
36,60 40,601 41,609
43,979 46,998 56,222
65,109 82,596 84,990
86,238 87,978 99,727
101,172 103,112 106,110
229,244 269,1014 274,870
314,987 405,940 519,1043
942,1104 977,982 1019,1027

X(58) is the {X(3),X(6)}-harmonic conjugate of X(386). For a list of other harmonic conjugates of X(58), click **Tables** at the top of this page.

X(58) = isogonal conjugate of X(10)

X(58) = isotomic conjugate of X(313)

X(58) = inverse-in-circumcircle of X(1326)

X(58) = inverse-in-Brocard-circle of X(386)

X(58) = complement of X(1330)

X(58) = anticomplement of X(3454)

X(58) = X(i)-Ceva conjugate of X(j) for these (i,j): (81,284), (267,501), (270,28)

X(58) = cevapoint of X(6) and X(31)

X(58) = X(i)-cross conjugate of X(j) for these (i,j): (6,81), (36,106), (56,28), (513,109)

X(58) = crosspoint of X(i) and X(j) for these (i,j): (1,267), (21,285), (27,86), (60,270)

X(58) = crosssum of X(i) and X(j) for these (i,j): (1,191), (6,199), (12,201), (37,210), (42,71), (65,227), (594, 756)

X(58) = crossdifference of every pair of points on line X(523)X(661)

X(58) = X(6)-Hirst inverse of X(1326)

X(58) = X(i)-beth conjugate of X(j) for these (i,j): (21,21), (60,58), (110,58), (162,58), (643,58), (1098,283)

X(58) = barycentric product of PU(31)

X(58) = trilinear pole of line X(649)X(834)

X(58) = {X(1),X(31)}-harmonic conjugate of X(595)

X(58) = {X(21),X(283)}-harmonic conjugate of X(2328)

X(58) = X(42)-isoconjugate of X(75)

X(58) = X(71)-isoconjugate of X(92)

X(58) = X(101)-isoconjugate of X(1577)

X(58) = homothetic center of 2nd circumperp triangle and 'Hatzipolakis-Brocard triangle' (A'B'C' as defined at X(5429))

X(58) = trilinear product of vertices of 2nd circumperp triangle

X(58) = perspector of 2nd circumperp triangle and unary cofactor triangle of 1st circumperp triangle

X(58) = perspector of ABC and cross-triangle of ABC and 2nd circumperp triangle

X(58) = Cundy-Parry Phi transform of X(573)

X(58) = Cundy-Parry Psi transform of X(13478)

X(58) = perspector of ABC and unary cofactor triangle of Gemini triangle 11

X(58) = {X(1),X(21)}-harmonic conjugate of X(4653)

Trilinears csc^2 (B/2 - C/2) : :

Barycentrics a/[1 - cos(B - C)] : b/[1 - cos(C - A)] : c/[1 - cos(A - B)]

X(59) lies on these lines: 36,1110 60,1101 100,521 101,657 109,901 513,651 518,765 523,655

X(59) = isogonal conjugate of X(11)

X(59) = cevapoint of X(i) and X(j) for these (i,j): (55,101), (56,109), (1381,1382)

X(59) = X(i)-cross conjugate of X(j) for these (i,j): (1,110), (3,100), (55,101), (56,109), (182,1492)

X(59) = X(765)-beth conjugate of X(765)

X(59) = trilinear pole of line X(101)X(109)

X(59) = perspector of ABC and the reflection of the intouch triangle in line X(1)X(3)

X(59) = perspector of ABC and extraversion triangle of X(60)

X(59) = X(75)-isoconjugate of X(3271)

X(59) = trilinear square of X(6733)

Trilinears sec^2 (B/2 - C/2) : :

Barycentrics a/[1 + cos(B - C)] : b/[1 + cos(C - A)] : c/[1+ cos(A - B)]

Let A'B'C' be the cevian triangle of X(21). Let A", B", C" be the inverse-in-circumcircle of A', B', C'. The lines AA", BB", CC" concur in X(60). (Randy Hutson, October 15, 2018)

X(60) lies on these lines: 1,110 21,960 28,81 36,58 59,1101 86,272 283,284 404,662 757,1014

X(60) = isogonal conjugate of X(12)

X(60) = X(58)-cross conjugate of X(270)

X(60) = X(i)-beth conjugate of X(j) for these (i,j): (60,849), (1098,1098)

X(60) = crossdifference of every pair of points on line X(2610)X(4024)

X(60) = perspector of ABC and extraversion triangle of X(59)

X(60) = X(75)-isoconjugate of X(181)

X(60) = trilinear square of X(6727)

Trilinears cos(A - π/3) : cos(B - π/3) : cos(C - π/3)

Trilinears cos A + sqrt(3) sin A : :

Barycentrics sin A sin(A + π/6) : sin B sin(B + π/6) : sin C sin(C + π/6)

X(61) lies on the Napoleon cubic and these lines:

1,203 2,18 3,6 4,13
5,14 30,397 56,202
140,395 299,636 302,629 618,627

X(61) is the {X(3),X(6)}-harmonic conjugate of X(62). For a list of
other harmonic conjugates of X(61), click **Tables** at the top of
this page.

X(61) = reflection of X(633) in X(635)

X(61) = isogonal conjugate of X(17)

X(61) = inverse-in-Brocard-circle of X(62)

X(61) = complement of X(633)

X(61) = anticomplement of X(635)

X(61) = eigencenter of cevian triangle of X(14)

X(61) = eigencenter of anticevian triangle of X(16)

X(61) = X(14)-Ceva conjugate of X(16)

X(61) = crosspoint of X(302) and X(473)

X(61) = point of concurrence of Brocard axes of BCX(15), CAX(15), ABX(15)

X(61) = perspector of ABC and centers of circles used in construction of X(1337)

X(61) = X(61)-of-circumsymmedial-triangle

X(61) = orthocorrespondent of X(16)

X(61) = {X(15),X(62)}-harmonic conjugate of X(3)

X(61) = {X(371),X(372)}-harmonic conjugate of X(15)

X(61) = perspector of inner Napoleon triangle and orthocentroidal triangle

X(61) = Cundy-Parry Phi transform of X(15)

X(61) = Cundy-Parry Psi transform of X(13)

X(61) = reflection of X(62) in X(5007)

X(61) = Kosnita(X(15),X(3)) point

X(61) = Kosnita(X(15),X(15)) point

X(61) = antigonal conjugate of X(34219)

Trilinears cos(A + π/3) : cos(B + π/3) : cos(C + π/3)

Trilinears cos A - sqrt(3) sin A : :

Barycentrics sin A sin(A - π/6) : sin B sin(B - π/6) : sin C sin(C - π/6)

X(62) lies the Napoleon cubic and these lines:

1,202 2,17 3,6 4,14
5,13 30,398 56,203
140,396 298,635 303,630 619,628

X(62) is the {X(3),X(6)}-harmonic conjugate of X(61). For a list of
other harmonic conjugates of X(62), click **Tables** at the top of
this page.

X(62) = reflection of X(634) in X(636)

X(62) = isogonal conjugate of X(18)

X(62) = inverse-in-Brocard-circle of X(61)

X(62) = complement of X(634)

X(62) = anticomplement of X(636)

X(62) = eigencenter of cevian triangle of X(13)

X(62) = eigencenter of anticevian triangle of X(15)

X(62) = X(13)-Ceva conjugate of X(15)

X(62) = crosspoint of X(303) and X(472)

X(62) = point of concurrence of Brocard axes of BCX(16), CAX(16), ABX(16)

X(62) = perspector of ABC and centers of circles used in construction of X(1338)

X(62) = X(62)-of-circumsymmedial-triangle

X(62) = orthocorrespondent of X(15)

X(62) = reflection of X(61) in X(5007)

X(62) = {X(16),X(61)}-harmonic conjugate of X(3)

X(62) = {X(371),X(372)}-harmonic conjugate of X(16)

X(62) = perspector of outer Napoleon triangle and orthocentroidal triangle

X(62) = Cundy-Parry Phi transform of X(16)

X(62) = Cundy-Parry Psi transform of X(14)

X(62) = Kosnita(X(16),X(3)) point

X(62) = Kosnita(X(16),X(16)) point

X(62) = antigonal conjugate of X(34220)

Trilinears b

Trilinears S

Trilinears csc A - tan(A/2) : :

Trilinears csc A - cot(A/2) : :

Trilinears tan(A/2) - cot(A/2) : :

Trilinears d(a,b,c) : : , where d(a,b,c) = directed distance from A to the orthic axis

Trilinears 2 csc 2A - tan A : :

Barycentrics cos A : cos B : cos C

X(63) = (cos A)*[A] + (cos B)*[B] + (cos C)*[C], where A, B, C denote angles and [A], [B], [C] denote vertices

Let Oa be the A-extraversion of the Conway circle (the circle centered at the A-excenter and passing through A, with radius sqrt(r_a^2 + s^2), where r_a is the A-exradius). Let Pa be the perspector of Oa, and La the polar of Pa wrt Oa. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Triangle A'B'C' is perspective to the excentral triangle at X(63). (Randy Hutson, February 10, 2016)

Let A'B'C' be the 2nd Brocard triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(63). (Randy Hutson, February 10, 2016)

Let A'B'C' be the hexyl triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(63). (Randy Hutson, February 10, 2016)

Let A'B'C' be the side-triangle of ABC and hexyl triangle. Let A" be the {B,C}-harmonic conjugate of A', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(63). (Randy Hutson, February 10, 2016)

Let A'B'C' be the excentral triangle. Let A" be the isotomic conjugate, wrt triangle A'BC, of X(1). Define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(63). (Randy Hutson, July 31 2018)

X(63) lies on these lines:

1,21 2,7 3,72 6,2221 8,20 10,46 19,27 33,1013
36,997 37,940 48,326
55,518 56,960 65,958
69,71 77,219 91,921
100,103 162,204 169,379
171,612 190,312 194,239
201,603 210,1004 212,1040
213,980 220,241 223,651
238,614 240,1096 244,748
304,1102 318,412 354,1001
392,999 404,936 405,942
452,938 484,535 517,956
544,1018 561,799 654,918
750,756

X(63) is the {X(9),X(57)}-harmonic conjugate of X(2). For a list of
other harmonic conjugates of X(63), click **Tables** at the top of
this page.

X(63) = reflection of X(i) in X(j) for these (i,j): (1,993), (1478,10)

X(63) = isogonal conjugate of X(19)

X(63) = isotomic conjugate of X(92)

X(63) = complement of X(5905)

X(63) = anticomplement of X(226)

X(63) = anticomplementary conjugate of X(2893)

X(63) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,224), (69,78), (75,1), (304,326), (333,2), (348,77)

X(63) = cevapoint of X(i) and X(j) for these (i,j): (3,219), (9,40), (48,255), (71,72)

X(63) = X(i)-cross conjugate of X(j) for these (i,j): (3,77), (9,271), (48,1), (71,3), (72,69), (219,78), (255,326)

X(63) = crosspoint of X(i) and X(j) for these (i,j): (69,348), (75,304)

X(63) = crosssum of X(25) and X(607)

X(63) = crossdifference of every pair of points on line X(661)X(663)

X(63) = trilinear product X(2)*X(3)

X(63) = trilinear product of PU(22)

X(63) = bicentric sum of PU(i) for these i: 128, 129

X(63) = PU(128)-harmonic conjugate of X(661)

X(63) = midpoint of PU(129)

X(63) = {X(1),X(1707)}-harmonic conjugate of X(31)

X(63) = {X(2),X(9)}-harmonic conjugate of X(3305)

X(63) = {X(2),X(57)}-harmonic conjugate of X(3306)

X(63) = {X(92),X(1748)}-harmonic conjugate of X(19)

X(63) = trilinear pole of line X(521)X(656)

X(63) = pole wrt polar circle of trilinear polar of X(158)

X(63) = X(48)-isoconjugate (polar conjugate) of X(158)

X(63) = X(i)-isoconjugate of X(j) for these {i,j}: {4,6}, {31,92}, {75,1973}

X(63) = excentral isogonal conjugate of X(1742)

X(63) = homothetic center of excentral triangle and anticomplement of the intouch triangle

X(63) = X(161)-of-intouch-triangle

X(63) = X(184)-of-excentral-triangle

X(63) = inverse-in-circumconic-centered-at-X(9) of X(908)

X(63) = trilinear square of X(5374)

X(63) = perspector of excentral triangle and Gemini triangle 2

X(63) = homothetic center of excentral triangle and Gemini triangle 30

X(63) = perspector of ABC and cross-triangle of Gemini triangles 35 and 36

X(63) = perspector of ABC and cross-triangle of ABC and Gemini triangle 35

X(63) = perspector of ABC and cross-triangle of ABC and Gemini triangle 36

X(63) = barycentric product of vertices of Gemini triangle 35

X(63) = barycentric product of vertices of Gemini triangle 36

X(63) = X(i)-aleph conjugate of X(j) for these (i,j):

(2,1), (75,63), (92,920), (99,662), (174,978), (190,100), (333,411), (366,43), (514,1052), (556,40), (648,162), (664,651), (668,190), (670,799), (671,897), (903,88)

X(63) = X(i)-beth conjugate of X(j) for these (i,j):

(63,222), (190,63), (333,57), (345,345), (643,63), (645,312), (662,223)

X(63) = perspector of ABC and extraversion triangle of X(63), which is also the anticevian triangle of X(63)

Barycentrics a/(cos A - cos B cos C) : b/(cos B - cos C cos A) : c/(cos C - cos A cos B)

Barycentrics a^2/[3a^4 - 2a^2(b^2 + c^2) - (b^2 - c^2)^2] : :

A construction of X(64) appears in Lemoine's 1886 paper cited at X(19).

Let A'B'C' be the half-altitude triangle. Let La be the trilinear polar of A', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(64). (Randy Hutson, November 18, 2015)

Let Oa be the circle with segment BC as diameter. Let A' be the perspector of Oa. Let La be the polar of A' wrt Oa. Define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines AA", BB", CC" concur in X(64). (Randy Hutson, November 18, 2015)

Let A'B'C' be the cevian triangle of X(69). Let A" be the orthocenter of AB'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(64). (Randy Hutson, November 18, 2015)

Let A'B'C' be the reflection of the orthic triangle in X(4). Let A''B''C'' be the trangential triangle, with respect ot the orthic triangle, of the circumconic of the orthic triangle with center X(4); i.e., the bicevian conic of X(4) and X(459). Then X(64) is the perspector of A'B'C' and A''B''C''. (Randy Hutson, November 18, 2015)

The tangents at A, B, C to the Darboux cubic K004 concur in X(64). (Randy Hutson, November 18, 2015)

X(64) lies on the Darboux cubic and these lines:

1,3182 3,154 4,3183
6,185 20,69 24,74 30,68
33,65 40,72 54,378
55,73 71,198 84,3353
265,382 3345,3472 3346,3355

X(64) = reflection of X(1498) in X(3)

X(64) = isogonal conjugate of X(20)

X(64) = isotomic conjugate of X(14615)

X(64) = complement of X(6225)

X(64) = anticomplement of X(2883)

X(64) = X(25)-cross conjugate of X(6)

X(64) = X(1)-beth conjugate of X(207)

X(64) = perspector of hexyl triangle and anticevian triangle of X(2184)

X(64) = trilinear pole of line X(647)X(657)

X(64) = concurrence of normals to MacBeath circumconic at A, B, C

X(64) = isogonal conjugate, wrt tangential triangle of MacBeath circumconic (or anticevian triangle of X(3)), of X(1498)

X(64) = orthocenter of x(3)X(6)X(2435)

X(64) = orthology center of ABC and half-altitude triangle

X(64) = intersection of tangents at X(3) and X(4) to Thomson cubic K002

X(64) = intersection of tangents at X(20) and X(64) to Darboux cubic K004

X(64) = perspector of ABC and the reflection in X(3) of the antipedal triangle of X(3) (tangential triangle)

X(64) = perspector of ABC and circumcircle antipode of circumanticevian triangle of X(3)

X(64) = perspector of ABC and unary cofactor triangle of half-altitude triangle

X(64) = X(2136)-of-orthic-triangle if ABC is acute

X(64) = X(8905)-of-excentral-triangle

X(64) = X(3)-vertex conjugate of X(3)

Trilinears (b + c)/(b + c - a) : (c + a)/(c + a - b) : (a + b)/(a + b - c)

Trilinears sin(A/2) cos(B/2 - C/2) : sin(B/2) cos(C/2 - A/2) : sin(C/2) cos(A/2 - B/2)

Trilinears Ra + r : Rb + r : Rc + r, where Ra, Rb, Rc are the exradii

Barycentrics a(b + c)/(b + c - a) : b(c + a)/(c + a - b) : c(a + b)/(a + b - c)

Let A' be the intersections of the tangents to the Yiu conic at the points where they meet the A-excircle. Define B' and C' similarly. The lines AA', BB', CC' concur in X(65). (Randy Hutson, July 20, 2016)

Let Ab, Ac be the points where the A-excircle touches lines CA and AB resp., and define Bc, Ba, Ca, Cb cyclically. Let Ta be the intersection of the tangents to the Yiu conic (defined at X(478)) at Bc and Ca, and define Tb, Tc cyclically. Let Ta' be the intersection of the tangents to the Yiu conic at Ba and Cb, and define Tb', Tc' cyclically. Let Sa = TbTc∩Tb'Tc', Sb = TcTa∩Tc'Ta', Sc = TaTb∩Ta'Tb'. The lines ASa, BSb, CSc concur in X(65). (See also X(1903).) (Randy Hutson, July 20, 2016)

Let A' be the homothetic center of the orthic triangles of the intouch and A-extouch triangles, and define B' and C' cyclically. The triangle A'B'C' is perspective to the extouch triangle at X(65). (Randy Hutson, July 20, 2016)

Let A'B'C' be the orthic triangle. Let B'C'A" be the triangle similar to ABC such that segment A'A" crosses the line B'C'. Define B" and C" cyclically. Equivalently, A" is the reflection of A in B'C', and cyclically for B" and C". Let Ia be the incenter of B'C'A", and define Ib and Ic cyclically. The circumcenter of triangle IaIbIc is X(65). Let A* be the intersection of lines A"Ia and B'C', and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(65). (Randy Hutson, July 20, 2016)

Let Ab, Ac be the points where the A-excircle touches lines CA and AB resp., and define Bc, Ba, Ca, Cb cyclically. Let IaIbIc be the intouch triangle. Let Oa be the circle through Ab, Ac, Ib, Ic, and define Ob, Oc cyclically. X(65) is the radical center of Oa, Ob, Oc. (Randy Hutson, July 20, 2016)

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is perspective to the intouch triangle and 4th and 5th extouch triangles at X(65). (Randy Hutson, December 2 2017)

X(65) lies on these lines:

1,3 2,959 4,158 6,19
7,8 10,12 11,117 29,296
31,1104 33,64 37,71
41,910 42,73 44,374
58,109 63,958 68,91
74,108 77,969 79,80
81,961 110,229 169,218
172,248 224,1004 225,407
243,412 257,894 278,387
279,1002 386,994 409,1098
474,997 497,938 516,950
519,553 604,1100 651,895
1039,1041 1061,1063

X(65) is the {X(1),X(40)}-harmonic conjugate of X(55). For a list of
other harmonic conjugates of X(65), click **Tables** at the top of
this page.

X(65) = reflection of X(i) in X(j) for these (i,j): (1,942), (72,10)

X(65) = isogonal conjugate of X(21)

X(65) = isotomic conjugate of X(314)

X(65) = inverse-in-incircle of X(1319)

X(65) = complement of X(3869)

X(65) = anticomplement of X(960)

X(65) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,73), (4,225), (7,226), (10,227), (109,513), (226,37)

X(65) = X(42)-cross conjugate of X(37)

X(65) = crosspoint of X(i) and X(j) for these (i,j): (1,4), (7,57)

X(65) = crosssum of X(i) and X(j) for these (i,j): (1,3), (9,55), (56,1394), (1805,1806)

X(65) = crossdifference of every pair of points on line X(521)X(650)

X(65) = X(1284)-Hirst inverse of X(1400)

X(65) = X(i)-beth conjugate of X(j) for these (i,j): (1,65), (8,72), (10,10), (65,1042), (80,65), (100,65), (101,213), (291,65), (668,65), (1018, 65)

X(65) = bicentric sum of PU(15)

X(65) = PU(15)-harmonic conjugate of X(650)

X(65) = trilinear product of PU(81)

X(65) = trilinear pole of line X(647)X(661)

X(65) = perspector of ABC and the extangents triangle

X(65) = X(1986)-of-Fuhrmann-triangle

X(65) = X(40) of Mandart-incircle triangle

X(65) = homothetic center of intangents triangle and reflection of extangents triangle in X(40)

X(65) = homothetic center of extangents triangle and reflection of intangents triangle in X(1)

X(65) = reflection of X(3057) in X(1)

X(65) = {X(1),X(3)}-harmonic conjugate of X(2646)

X(65) = {X(1),X(57)}-harmonic conjugate of X(56)

X(65) = {P,Q}-harmonic conjugate of X(1463), where P and Q are the intersections of the incircle and line X(7)X(8)

X(65) = pairwise perspector of: intouch triangle, 4th extouch triangle, 5th extouch triangle

X(65) = perspector of [reflection of incentral triangle in X(1)] and tangential triangle, wrt incentral triangle, of circumconic of incentral triangle centered at X(1) (bicevian conic of X(1) and X(57))

X(65) = inverse-in-{incircle, circumcircle}-inverter of X(2078)

X(65) = inverse-in-circumcircle of X(5172)

X(65) = pedal-isogonal conjugate of X(1)

X(65) = X(5) of reflection triangle of X(1)

X(65) = radical trace of circumcircle and circumcircle of reflection triangle of X(1)

X(65) = X(188)-of-orthic-triangle if ABC is acute

X(65) = perspector of ABC and cross-triangle of ABC and 4th extouch triangle

X(65) = perspector of ABC and cross-triangle of ABC and 5th extouch triangle

X(65) = polar conjugate of X(31623)

X(65) = pole wrt polar circle of trilinear polar of X(31623) (line X(521)X(1948))

X(65) = perspector of ABC and anti-tangential midarc triangle

X(65) = homothetic center of extangents triangle and anti-tangential midarc triangle

X(65) = excentral-to-intouch similarity image of X(1)

Barycentrics 1/(b

X(66) lies on these lines:

2,206 3,141 6,427
68,511 73,976 193,895
248,571 290,317 879,924

X(66) = midpoint of X(2892) and X(3448)

X(66) = reflection of X(i) in X(j) for these (i,j): (159,141), (1177,125)

X(66) = isogonal conjugate of X(22)

X(66) = isotomic conjugate of X(315)

X(66) = cyclocevian conjugate of X(2998)

X(66) = anticomplement of X(206)

X(66) = cevapoint of X(125) and X(512)

X(66) = X(32)-cross conjugate of X(2)

X(66) = crosssum of X(3) and X(159)

X(66) = trilinear pole of line X(647)X(826) (radical axis of Brocard and polar circles)

X(66) = antigonal image of X(1177)

X(66) = orthocenter of X(3)X(4)X(2435)

X(66) = X(3174)-of-orthic-triangle if ABC is acute

Barycentrics 1/(b

Let A' be the reflection in BC of the A-vertex of the antipedal triangle of X(6), and define B' and C' cyclically. The circumcircles of A'BC, B'CA, C'AB concur at X(67). Also, let A' be the reflection of X(6) in BC, and define B' and C' cyclically. The circumcircles of A'BC, B'CA, C'AB concur in X(67). Note: the above 2 sets of circumcircles are identical. (Randy Hutson, November 18, 2015)

X(67) lies on these lines:

3,542 4,338 6,125
50,248 74,935 110,141
265,511 290,340 524,858 526,879

X(67) = midpoint of X(69) and X(3448)

X(67) = reflection of X(i) in X(j) for these (i,j): (6,125), (110,141)

X(67) = isogonal conjugate of X(23)

X(67) = isotomic conjugate of X(316)

X(67) = inverse-in-circumcircle of X(3455)

X(67) = cevapoint of X(141) and X(524)

X(67) = X(187)-cross conjugate of X(2)

X(67) = antigonal image of X(6)

X(67) = trilinear pole of line X(39)X(647)

X(67) = orthocenter of X(3)X(74)X(879)

X(67) = perspector of ABC and X(2)-Ehrmann triangle; see X(25)

X(67) = X(19)-isoconjugate of X(22151)

Barycentrics tan 2A : tan 2B : tan 2C

Barycentrics (b^2 + c^2 - a^2)/(a^4 + b^4 + c^4 - 2a^2b^2 - 2a^2c^2) : :

Let A'B'C' be the 2nd Euler triangle. The lines AA', BB', CC' concur in X(68), as proved in **V. V. Prasolov,** *Zadachi po planimetrii,* Moscow, 4th edition, 2001.

Coordinates for X(68) can be obtained easily from the Ceva ratios given his Prasolov's proof of concurrence.

Let Oa be the circle centered at the A-vertex of the orthic triangle and passing through A; define Ob and Oc cyclically. Then X(68) is the radical center of Oa, Ob, Oc. (Randy Hutson, November 2, 2017)

The X(3)-Fuhrmann triangle is inversely similar to ABC, with similitude center X(3), and perspective to ABC at X(68). (Randy Hutson, November 3, 2017)

X(68) lies on these lines:

2,54 3,343 4,52 5,6
11,1069 20,74 26,161
30,64 65,91 66,511
73,1060 136,254 290,315 568,973

X(68) = reflection of X(155) in X(5)

X(68) = isogonal conjugate of X(24)

X(68) = isotomic conjugate of X(317)

X(68) = anticomplement of X(1147)

X(68) = X(96)-Ceva conjugate of X(3)

X(68) = cevapoint of X(i) and X(j) for these (i,j): (6,161), (125,520)

X(68) = X(115)-cross conjugate of X(525)

X(68) = pedal antipodal perspector of X(4)

X(68) = pedal antipodal perspector of X(186)

X(68) = crossdifference of every pair of points on line X(924)X(6753)

X(68) = trilinear product of vertices of X(3)-anti-altimedial triangle

X(68) = orthic-to-ABC barycentric image of X(52)

X(68) = cyclocevian conjugate of X(34287)

Trilinears bc(b

Trilinears sec

Barycentrics cot A : cot B : cot C

Barycentrics b

Barycentrics cot B + cot C - cot ω : :

Barycentrics cot B + cot C - cot A - cot ω : :

Barycentrics SA : SB : SC

X(69) = (cot A)*[A] + (cot B)*[B] + (cot C)*[C], where A, B, C denote angles and [A], [B], [C] denote vertices

Let A'B'C' be the anticomplementary triangle. Let A" be the inverse-in-anticomplementary-circle of A, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(69). (Randy Hutson, February 10, 2016)

Let A'B'C' be the anticomplementary triangle. Let A" be the orthogonal projection of A' on line BC, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(69). (Randy Hutson, February 10, 2016)

Let A'B'C' be the half-altitude triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. Let A* be the trilinear pole of line B"C", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(69). (Randy Hutson, February 10, 2016)

Let A2B2C2 be the 2nd Conway triangle. Let A' be the cevapoint of B2 and C2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(69). (Randy Hutson, December 10, 2016)

X(69) is the barycentric multiplier for the MacBeath circumconic. (The barycentric product of X(69) and the circumcircle is the MacBeath circumconic.) (Randy Hutson, August 19, 2019)

X(69) lies on the Lucas cubic and these lines:

2,6 3,332 4,76 7,8 9,344 10,969 20,64
22,159 54,95 63,71
72,304 73,77 74,99
110,206 125,895 144,190
150,668 189,309 192,742
194,695 200,269 219,1332 248,287
263,308 265,328 274,443
290,670 297,393 347,664
350,497 404,1014 478,651
485,639 486,640 520,879 1225,2888 1369,3410

X(69) is the {X(7),X(8)}-harmonic conjugate of X(75). For a list of
other harmonic conjugates of X(69), click **Tables** at the top of
this page.

X(69) = reflection of X(i) in X(j) for these (i,j): (2,599), (4,1352), (6,141), (20,1350), (193,6), (895,125), (1351,5), (1353,140)

X(69) = isogonal conjugate of X(25)

X(69) = isotomic conjugate of X(4)

X(69) = cyclocevian conjugate of X(253)

X(69) = complement of X(193)

X(69) = anticomplement of X(6)

X(69) = anticomplementary conjugate of X(2)

X(69) = X(i)-Ceva conjugate of X(j) for these (i,j): (76,2), (304,345), (314,75), (332,326)

X(69) = cevapoint of X(i) and X(j) for these (i,j): (2,20), (3,394), (6,159), (8,329), (63,78), (72,306), (125,525)

X(69) = X(i)-cross conjugate of X(j) for these (i,j): (3,2), (63,348), (72,63), (78,345), (125,525), (306,304), (307,75), (343,76)

X(69) = crosspoint of X(i) and X(j) for these (i,j): (2,2996), (76,305), (314,332)

X(69) = X(2)-Hirst inverse of X(325)

X(69) = X(i)-beth conjugate of X(j) for these (i,j): (69,77), (99,347), (314,7), (332,69), (645,69), (668,69)

X(69) = barycentric product of PU(37)

X(69) = bicentric sum of PU(132)

X(69) = midpoint of PU(132)

X(69) = perspector of the orthic-of-medial triangle and the reference triangle

X(69) = perspector of ABC and the pedal triangle of X(20)

X(69) = perspector of ABC and (reflection in X(2) of the pedal triangle of X(2))

X(69) = intersection of extended sides P(11)U(45) and U(11)P(45) of the trapezoid PU(11)PU(45)

X(69) = perspector of ABC and 4th extouch triangle

X(69) = antipode of X(287) in hyperbola {{A,B,C,X(2),X(69)}}

X(69) = trilinear pole of line X(441)X(525)

X(69) = pole wrt polar circle of trilinear polar of X(393) (line X(460)X(512))

X(69) = X(48)-isoconjugate (polar conjugate) of X(393)

X(69) = X(6)-isoconjugate of X(19)

X(69) = X(92)-isoconjugate of X(32)

X(69) = antigonal image of X(895)

X(69) = crosssum of X(i) and X(j) for these (i,j): (3,3053), (32,1974)

X(69) = perspector of ABC and the 2nd pedal triangle of X(3)

X(69) = crosspoint of X(6) and X(159) wrt both the excentral and tangential triangles

X(69) = crosspoint of X(2) and X(20) wrt both the excentral and anticomplementary triangles

X(69) = homothetic center of anticomplementary triangle and 2nd antipedal triangle of X(4) (i.e., of 1st and 2nd antipedal triangles of X(4))

X(69) = perspector of the complement of the polar circle

X(69) = pole, wrt de Longchamps circle, of trilinear polar of X(95)

X(69) = perspector of the extraversion triangles of X(7) and X(8)

X(69) = {X(2),X(6)}-harmonic conjugate of X(3618)

X(69) = perspector of ABC and anticomplement of submedial triangle

X(69) = perspector of ABC and mid-triangle of orthic and dual of orthic triangles

X(69) = perspector of ABC and cross-triangle of ABC and 2nd Brocard triangle

X(69) = perspector of 2nd Conway triangle and cross-triangle of ABC and 2nd Conway triangle

X(69) = Lucas-isogonal conjugate of X(376)

X(69) = anticevian-isogonal conjugate of X(2)

X(69) = inverse-in-MacBeath-circumconic of X(22151)

X(69) = {X(7),X(8)}-harmonic conjugate of X(75)

X(69) = intersection of van Aubel lines of outer and inner Vecten triangles

Barycentrics 1/(a^8 - 2 a^6 (b^2 + c^2) + 2 a^2 (b^6 + c^6) - (b^2 - c^2)^2 (b^4 + c^4)) : :

X(70) lies on the Jerabek circumhyperbola and these lines:

{3,8907}, {6,1594}, {54,1899}, {64,6240}, {66,6403}, {71,2158}, {74,1288}, {265,6243}, {1176,1352}, {1177,3542}, {3448,5504}, {3527,7507}, {4846,6241}, {6145,6152}

X(70) = isogonal conjugate of X(26)

X(70) = anticomplement of X(34116)

X(70) = X(571)-crossconjugate of X(2)

X(70) = X(i)-isoconjugate of X(j) for these {i,j}: {{1,26}, {63,8746}

X(70) = reflection of the isogonal conjugate of X(2072) in X(125)

X(70) = X(125)-cevapoint of X(924)

X(70) = X(161)-crosssum of X(8553)

X(70) = barycentric product X(525) X(1288)

Barycentrics (b + c) sin 2A : (c + a) sin 2B : (a + b) sin 2C

X(71) lies on these lines:

1,579 3,48 4,9 6,31
35,284 37,65 54,572
63,69 64,198 74,101
165,610 190,290 583,1100

X(71) is the intersection of the isotomic conjugate of the polar conjugate of the Nagel line (i.e., line X(63)X(69)), and the polar conjugate of the isotomic conjugate of the Nagel line (i.e., line X(4)X(9)). (Randy Hutson, July 11, 2019)

X(71) is the {X(9),X(40)}-harmonic conjugate of X(19). For a list of
other harmonic conjugates of X(71), click **Tables** at the top of
this page.

X(71) = isogonal conjugate of X(27)

X(71) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,228), (9, 37), (10,42), (63,72)

X(71) = X(228)-cross conjugate of X(73)

X(71) = crosspoint of X(i) and X(j) for these (i,j): (3,63), (9,219), (10,306)

X(71) = crosssum of X(i) and X(j) for these (i,j): (1,579), (4,19), (28,1127), (57,278), (58,1474)

X(71) = crossdifference of every pair of points on line X(242)X(514)

X(71) = X(4)-line conjugate of X(242)

X(71) = X(i)-beth conjugate of X(j) for these (i,j): (219,71), (1018,71)

X(71) = trilinear pole of line X(647)X(810)

X(71) = X(92)-isoconjugate of X(58)

X(71) = barycentric product of Jerabek hyperbola intercepts of Nagel line

Trilinears (b + c)(b

Barycentrics (b + c) cos A : (c + a) cos B : (a + b) cos C

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(72) = X(11) of IaIbIc. (Randy Hutson, September 14, 2016)

X(72) lies on the Mandart hyperbola and these lines:

1,6 2,942 3,63 4,8
5,908 7,443 10,12
20,144 21,943 31,976
35,191 40,64 43,986
54,1006 56,997 57,474
69,304 73,201 74,100
145,452 171,1046 185,916
190,1043 222,1038 248,293
290,668 295,337 306,440
394,1060 519,950 672,1009
894,1010 940,975 978,982

X(72) is the {X(1),X(9)}-harmonic conjugate of X(405). For a list of
other harmonic conjugates of X(72), click **Tables** at the top of
this page.

X(72) is the perspector of the 1st and 2nd extouch triangles. X(72) is also the orthocenter of the 2nd extouch triangle. (Randy Hutson, August 23, 2011)

X(72) = reflection of X(i) in X(j) for these (i,j): (1,960), (65,10), (1145,14740), (3555,1)

X(72) = isogonal conjugate of X(28)

X(72) = isotomic conjugate of X(286)

X(72) = inverse-in-Fuhrmann circle of X(3419)

X(72) = anticomplement of X(942)

X(72) = X(i)-Ceva conjugate of X(j) for these (i,j): (8,10), (63,71), (69,306), (321,37)

X(72) = X(i)-cross conjugate of X(j) for these (i,j): (201,10), (228,37)

X(72) = crosspoint of X(i) and X(j) for these (i,j): (8,78), (63,69), (306,307)

X(72) = crosssum of X(i) and X(j) for these (i,j): (19,25), (34,56)

X(72) = crossdifference of every pair of points on line X(513)X(1430)

X(72) = X(i)-beth conjugate of X(j) for these (i,j): (8,65), (72,73), (78,72), (100,227), (644,72)

X(72) = trilinear pole of line X(647)X(656)

X(72) = complement of X(3868)

X(72) = X(149) of X(1)-Brocard triangle

X(72) = X(6)-isoconjugate of X(27)

X(72) = X(75)-isoconjugate of X(2203)

X(72) = X(92)-isoconjugate of X(1333)

X(72) = inverse-in-Fuhrmann-circle of X(3419)

X(72) = X(6146)-of-excentral-triangle

X(72) = perspector of ABC and cross-triangle of ABC and 2nd extouch triangle

X(72) = trilinear product of Jerabek hyperbola intercepts of Nagel line

X(72) = excentral-to-ABC barycentric image of X(4)

X(72) = antipode of X(1145) in the Mandart hyperbola

X(72) = extouch-isogonal conjugate of X(5687)

Trilinears a(b + c)(a^2 - b^2 - c^2)/(a - b - c) : :

Barycentrics (cos B + cos C) sin 2A : (cos C + cos A) sin 2B : (cos A + cos B) sin 2C

X(73) lies on these lines:

1,4 3,212 6,41 21,651
35,74 36,54 37,836
42,65 55,64 57,386
66,976 68,1060 69,77
72,201 102,947 228,408
284,951 290,336 1036,1037
1057,1059

X(73) is the {X(1064),X(1066)}-harmonic conjugate of X(1). For a
list of other harmonic conjugates of X(73), click **Tables** at the
top of this page.

X(73) = isogonal conjugate of X(29)

X(73) = X(1)-Ceva conjugate of X(65)

X(73) = X(228)-cross conjugate of X(71)

X(73) = crosspoint of X(i) and X(j) for these (i,j): (1,3), (77,222), (226,307)

X(73) = crosssum of X(i) and X(j) for these (i,j): (1,4), (33,281)

X(73) = crossdifference of every pair of points on line X(243)X(522)

X(73) = X(i)-Hirst inverse of X(j) for these (i,j): (1,243), (65,851)

X(73) = X(i)-beth conjugate of X(j) for these (i,j): (1,1042), (3,73), (21,946), (72,72), (100,10), (101,73), (295,73)

X(73) = bicentric sum of PU(16)

X(73) = PU(16)-harmonic conjugate of X(652)

X(73) = trilinear product of PU(83)

X(73) = trilinear pole of line X(647)X(822)

X(73) = X(92)-isoconjugate of X(284)

X(73) = {X(1),X(1745)}-harmonic conjugate of X(4)

Trilinears 1/(3 cos A - 2 sin B sin C) : 1/(3 cos B - 2 sin C sin A) : 1/(3 cos C - 2 sin A sin B)

Trilinears a/[2a

Barycentrics a/(cos A - 2 cos B cos C) : b/(cos B - 2 cos C cos A) : c/(cos C - 2 cos A cos B)

X(74) = (r^{2} + 2rR +
s^{2})*X(1) - R(6r + 9R)*X(2) + (r^{2} + 12rR +
18R^{2} - 3s^{2})*X(3) (Peter Moses, April
2, 2013)

As the isogonal conjugate of the point in which the Euler line meets the line at infinity, X(74) lies on the circumcircle.

Let T be the triangle formed by reflecting the orthic axis in the
sidelines of ABC; then T is perspective to ABC, and the perspector ix
X(74). Let A' be the point of intersection of the orthic axis and line
BC, and define B' and C' cyclically. Let O_{A} be the
circumcenter of AB'C', and define Let O_{B} and O_{C}
cyclically; then the lines AO_{A}, BO_{B},
CO_{C} concur in X(74). (Randy Hutson, August 26, 2014)

Let A'B'C' be the anticomplementary triangle. Let L be the line through A' parallel to the Euler line, and define M and N cyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L', M', N' concur in X(74). (Randy Hutson, August 26, 2014)

Let A'B'C' be the X(3)-Fuhrmann triangle. Let A'' be the reflection of A in line B'C', and define B'' and C'' cyclically. Then A''B''C'' is inversely similar to ABC, with similitude center X(265), and A''B''C'' is perspective to ABC at X(74), which is also the orthocenter of A''B''C''. (Randy Hutson, August 26, 2014)

In Hyacinthos 8129 (10/4/03), Floor van Lamoen noted that if X(74) is denoted by J, then each of the points A,B,C,J is J of the other three, in analogy with the well known property of orthocentric systems (that is, each of the points A,B,C,H is the orthocenter of the other three).

Let A'B'C' be the orthocentroidal triangle and A"B"C" the anti-orthocentroidal triangle. Let A* be the reflection of A" in B'C', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(74). (Randy Hutson, December 10, 2016)

Let A'B'C' be the anti-orthocentroidal triangle. Let A" be the reflection of A in line B'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(74). (Randy Hutson, January 15, 2019)

The tangents at A, B, C to the Neuberg cubic K001 concur in X(74)

X(74) lies on the circumcircle, Jerabek hyperbola, Neuberg cubic, Darboux septic curve, and these lines:

1,3464 2,113 3,110
4,107 6,112 20,68 24,64
30,265 35,73 54,185
65,108 67,935 69,99
71,101 72,100 98,690
187,248 477,523 484,3465
511,691 512,842 550,930
1157,3484

X(74) = midpoint of X(20) and X(3448)

X(74) = reflection of X(i) in X(j) for these (i,j): (4,125), (110,3), (146,113), (399,1511)

X(74) = isogonal conjugate of X(30)

X(74) = isotomic conjugate of X(3260)

X(74) = complement of X(146)

X(74) = anticomplement of X(113)

X(74) = cevapoint of X(i) and X(j) for these (i,j): (15,16), (50,184)

X(74) = crosssum of X(i) and X(j) for these (i,j): (3,399), (616),617)

X(74) = X(i)-cross conjugate of X(j) for these (i,j): (186,54), (526,110)

X(74) = circumcircle-antipode of X(110)

X(74) = trilinear pole of line X(6)X(647)

X(74) = Ψ(X(6),X(647))

X(74) = reflection of X(477) in the Euler line

X(74) = reflection of X(842) in the Brocard axis

X(74) = reflection of X(2687) in the line X(1)X(3)

X(74) = reflection of X(1296) in the line X(3)X(351)

X(74) = {X(3),X(399)}-harmonic conjugate of X(1511)

X(74) = X(128)-of-excentral-triangle

X(74) = X(137)-of-hexyl-triangle

X(74) = X(1296)-of-circumsymmedial

X(74) = inverse-in-polar-circle of X(133)

X(74) = trilinear pole wrt circumorthic triangle of van Aubel line

X(74) = inverse-in-O(15,16) of X(2715), where O(15,16) is the circle having segment X(15)X(16) as diameter

X(74) = X(1577)-isoconjugate of X(2420)

X(74) = orthocentroidal-to-ABC similarity image of X(4)

X(74) = 4th-Brocard-to-circumsymmedial similarity image of X(4)

X(74) = perspector of ABC and the reflection of the Kosnita triangle in X(3)

X(74) = orthocenter of X(3)X(67)X(879)

X(74) = intersection of tangents at X(3) and X(4) to Napoleon-Feuerbach cubic, K005

X(74) = X(1317)-of-tangential-triangle is ABC is acute

X(74) = 2nd-Parry-to-ABC similarity image of X(110)

X(74) = X(80)-of-Trinh-triangle if ABC is acute

X(74) = Trinh-isogonal conjugate of X(2071)

X(74) = trilinear product of PU(86)

X(74) = perspector of ABC and the (degenerate) side-triangle of the (equilateral) circumcevian triangles of X(15) and X(16)

X(74) = homothetic center of X(15)- and X(16)-Ehrmann triangles; see X(25)

X(74) = perspector of ABC and X(15)-Ehrmann triangle

X(74) = perspector of ABC and X(16)-Ehrmann triangle

X(74) = 3rd-Parry-to-circumsymmedial similarity image of X(23)

X(74) = perspector of ABC and unary cofactor triangle of orthocentroidal triangle

X(74) = endo-homothetic center of X(4)-altimedial and X(4)-anti-altimedial triangles

X(74) = Thomson isogonal conjugate of X(523)

X(74) = Lucas isogonal conjugate of X(523)

X(74) = X(100)-of-circumorthic-triangle if ABC is acute

X(74) = perspector of ABC and 2nd anti-Parry triangle

X(74) = X(110)-of-2nd-anti-Parry-triangle

X(74) = X(9138)-of-1st-anti-Parry-triangle

X(74) = excentral-to-ABC functional image of X(5541)

X(74) = orthic-to-ABC functional image of X(128)

X(74) = trilinear pole wrt, Thomson triangle, of line X(3)X(5646)

X(74) = trilinear pole, wrt Lucas triangle, of line X(4)X(15066)

X(74) = BSS(a->a^2) of X(1156)

Trilinears 1/(1 - cos 2A) : 1/(1 - cos 2B) : 1/(1 - cos 2C)

Trilinears S

Trilinears sec

Trilinears d(a,b,c) : : , where d(a,b,c) = distance from A to Lemoine axis

Trilinears h(a,b,c) : : , where h(a,b,c) = (distance from A to antiorthic axis)

Barycentrics 1/a : 1/b : 1/c

Barycentrics csc A : csc B : csc C

Let A2B2C2 be the 2nd Conway triangle. Let A' be the trilinear pole of line B2C2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(75). (Randy Hutson, December 10, 2016)

Let A4B4C4 be the 4th Conway triangle. Let A' be the trilinear pole of line B4C4, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(75). (Randy Hutson, December 10, 2016)

Let AaBaCa, AbBbCb, AcBcCc be the A-, B-, and C-anti-altimedial triangles, resp. Let A' be the trilinear product Ba*Ca, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(75). (Randy Hutson, November 2, 2017)

Let A_{23}B_{23}C_{23} be Gemini triangle 23. Let A' be the perspector of conic {{A,B,C,B_{23},C_{23}}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(75). (Randy Hutson, January 15, 2019)

Let A_{40}B_{40}C_{40} be Gemini triangle 40. Let A' be the perspector of conic {{A,B,C,B_{40},C_{40}}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(75). (Randy Hutson, January 15, 2019)

If you have Geometer's Sketchpad, **X(75)**.

If you have GeoGebra, you can view **X(75)**.

X(75) lies on these lines:

1,86 2,37 6,239 7,8
9,190 10,76 19,27
21,272 31,82 32,746
38,310 42,1218 43,872
47,2216 48,336 72,1246
77,664 81,2214 87,3226
99,261 100,675 101,767
141,334 142,2321 144,391
149,2805 150,2893 158,240
183,1376 194,1107 219,1944
222,1943 225,264 234,556
244,1978 255,2190 257,698
269,1222 279,1219 280,309
298,1081 299,554 325,2886
491,1659 522,3261 523,876
537,668 538,1573 560,1580
689,745 700,971 728,1223
753,789 757,1468 758,994
775,1496 799,897 811,1099
901,2863 927,2751 934,2370940,1999
958,1975 982,1920 1088,3668
1089,1268
1150,3218&bsp; 1237,1240 1332,2989
1370,3434 1444,2217 1581,1934
1812,2219 1897,2000 1928,2085
1953,1959 2167,2168 2894,2897

X(75) is the {X(7),X(8)}-harmonic conjugate of X(69). For a list of
other harmonic conjugates of X(75), click **Tables** at the top of
this page.

X(75) = reflection of X(i) in X(j) for these (i,j): (192,37), (335,1086), (984,10)

X(75) = isogonal conjugate of X(31)

X(75) = isotomic conjugate of X(1)

X(75) = complement of X(192)

X(75) = anticomplement of X(37)

X(75) = anticomplementary conjugate of X(2895)

X(75) = X(i)-Ceva conjugate of X(j) for these (i,j): (76,312), (274,2), (310,76), (314,69)

X(75) = cevapoint of X(i) and X(j) for these (i,j): (1,63), (2,8), (7,347), (10,321), (244,514)

X(75) = X(i)-cross conjugate of X(j) for these (i,j): (1,92), (2,85), (7,309), (8,312), (10,2), (38,1), (63,304), (244,514), (307,69), (321,76), (347,322), (522,190)

X(75) = crosspoint of X(i) and X(j) for these (i,j): (2,330), (274,310)

X(75) = crossdifference of every pair of points on line X(667)X(788)

X(75) = X(i)-Hirst inverse of X(j) for these (i,j): (2,350), (334,335)

X(75) = X(83)-aleph conjugate of X(31)

X(75) = X(i)-beth conjugate of X(j) for these (i,j): (8,984), (75,7), (99,77), (314,75), (522,876), (645,9), (646,75), (668,75), (811,342)

X(75) = X(37)-of-anticomplementary triangle.

X(75) = trilinear product of PU(i) for these i: 3, 35

X(75) = barycentric product of PU(10)

X(75) = trilinear product of PU(75)

X(75) = X(6752)-of-excentral-triangle

X(75) = trilinear pole of line X(514)X(661)

X(75) = pole wrt polar circle of trilinear polar of X(19) (line X(661)X(663))

X(75) = X(48)-isoconjugate (polar conjugate) of X(19)

X(75) = X(6)-isoconjugate of X(6)

X(75) = crosspoint of X(1) and X(63) with respect to the excentral triangle

X(75) = crosspoint of X(1) and X(63) with respect to the anticomplementary triangle

X(75) = trilinear square of X(2)

X(75) = trilinear square root of X(561)

X(75) = trilinear product of the four CPCC points; https://bernard-gibert.pagesperso-orange.fr/Tables/table11.html

X(75) = perspector of ABC and extraversion triangle of X(75) (which is also the anticevian triangle of X(75))

X(75) = perspector of ABC and cross-triangle of Gemini triangles 3 and 6

X(75) = perspector of Gemini triangle 13 and cross-triangle of ABC and Gemini triangle 13

X(75) = perspector of ABC and cross-triangle of ABC and Gemini triangle 21

X(75) = perspector of ABC and cross-triangle of ABC and Gemini triangle 22

X(75) = perspector of ABC and cross-triangle of Gemini triangles 21 and 22

X(75) = barycentric product of vertices of Gemini triangle 21

X(75) = barycentric product of vertices of Gemini triangle 22

X(75) = {X(7),X(8)}-harmonic conjugate of X(69)

Trilinears csc(A - ω) : csc(B - ω) : csc(C - ω)

Barycentrics 1/a

Let A' be the perspector of the A-McCay circle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(76). (Randy Hutson, April 9, 2016)

X(76) is the vertex conjugate of the foci of the inellipse that is the barycentric square of the de Longchamps line. The center of the inellipse is X(626) and its Brianchon point (perspector) is X(1502). (Randy Hutson, October 15, 2018)

Let A'B'C' be the obverse triangle of X(1). Let A"B"C" be the N-obverse triangle of X(1). Let A* be the barycentric product A'*A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(76). (Randy Hutson, October 15, 2018)

X(76) is the barycentric multiplier for the Steiner circumellipse. (The barycentric product of X(76) and the circumcircle is the Steiner circumellipse.) (Randy Hutson, August 19, 2019)

X(76) lies on these lines:

1,350 2,39 3,98 4,69
5,262 6,83 7,1240 8,668
10,75 13,299 14,298
17,303 18,302 20,3424
22,1799 25,1241 31,734
32,384 37,1218 85,226
95,96 100,767 107,2366
110,2367 115,626 141,698
148,2896 182,3406 187,3552
192,1221 251,1239 257,1926
275,276 297,343 321,561
330,1015 331,1231 333,1751
334,1089 335,871
338,599 485,491 486,492
524,598 620,1569 689,755
691,2868 693,764 761,789
799,1150 826,882 940,1509
1003,3053 1007,3090 1131,1271
1132,1270 1229,1446 1423,3403
1501,3115 1670,1677 1671,1676
1698,3097 2001,2909 2319,3500
2394,3267 3224,3225 3492,3506
3496,3512 3497,3509

X(76) is the {X(2),X(194)}-harmonic conjugate of X(39). For a list
of other harmonic conjugates of X(76), click **Tables** at the top
of this page.

X(76) = reflection of X(194) in X(39)

X(76) = isogonal conjugate of X(32)

X(76) = isotomic conjugate of X(6)

X(76) = complement of X(194)

X(76) = anticomplement of X(39)

X(76) = anticomplementary conjugate of X(2896)

X(76) = X(i)-Ceva conjugate of X(j) for these (i,j): (308,2), (310,75)

X(76) = cevapoint of X(i) and X(j) for these (i,j): (2,69), (6,22), (75,312), (311,343), (313,321), (339,525)

X(76) = X(i)-cross conjugate of X(j) for these (i,j): (2,264), (69,305), (141,2), (321,75), (343,69), (525,99)

X(76) = crosssum of X(669) and X(1084)

X(76) = crossdifference of every pair of points on line X(669)X(688)

X(76) = X(i)-beth conjugate of X(j) for these (i,j): (76,85), (799,348)

X(76) = pole wrt polar circle of trilinear polar of X(25) (line X(512)X(1692))

X(76) = X(48)-isoconjugate (polar conjugate) of X(25)

X(76) = X(6)-isoconjugate of X(31)

X(76) = trilinear product of PU(i) for these i: 10, 86

X(76) = barycentric product of PU(11)

X(76) = antigonal image of X(1916)

X(76) = cevapoint of polar conjugates of PU(4)

X(76) = trilinear product of vertices of 1st Brocard triangle

X(76) = trilinear product of vertices of 1st anti-Brocard triangle

X(76) = X(2)-Ceva conjugate of X(6374)

X(76) = X(384)-of-5th-Brocard-triangle

X(76) = X(6)-of-6th-Brocard-triangle

X(76) = perspector of ABC and 1st Brocard triangle

X(76) = trilinear pole of de Longchamps line

X(76) = bicentric sum of PU(159)

X(76) = PU(159)-harmonic conjugate of X(9494)

X(76) = perspector of conic {{A,B,C,X(670),X(689),X(1978)}} (isotomic conjugate of Lemoine axis.)

X(76) = X(1916) of 1st Brocard triangle

X(76) = crosspoint of X(6) and X(22) wrt both the anticomplementary and tangential triangles

X(76) = inverse-in-circumcircle of X(5152)

X(76) = inverse-in-2nd-Brocard circle of X(99)

X(76) = X(3094)-of-1st anti-Brocard-triangle

X(76) = trilinear product of vertices of mid-triangle of 1st Brocard and 1st anti-Brocard triangles

X(76) = perspector of ABC and cross-triangle of ABC and 3rd Brocard triangle

X(76) = trilinear product of vertices of the three anti-altimedial triangles

X(76) = Cundy-Parry Phi transform of X(98)

X(76) = Cundy-Parry Psi transform of X(511)

X(76) = barycentric product X(99)*X(850)

X(76) = intersection, other than X(4), of P(1)- and U(1)-Fuhrmann circles (aka -Hagge circles)

X(76) = {X(7737),X(14023)}-harmonic conjugate of X(20065)

X(76) = intersection of lines PU(1) of 1st and 2nd Ehrmann circumscribing triangles

X(76) = trilinear cube of X(2)

X(76) = barycentric square of X(75)

X(76) = trilinear product of vertices of Gemini triangle 19

Trilinears cos A sec

Trilinears (b

Trilinears S

Barycentrics a/(1 + sec A) : b/(1 + sec B) : c/(1 + sec C)

Barycentrics cot A (1 - cos A) : :

X(77) lies on these lines:

1,7 2,189 6,241 9,651
29,34 40,947 55,1037
56,1036 57,81 63,219
65,969 69,73 75,664
102,934 283,603 309,318
738,951 988,1106 999,1057

X(77) = isogonal conjugate of X(33)

X(77) = isotomic conjugate of X(318)

X(77) = X(i)-Ceva conjugate of X(j) for these (i,j): (85,57), (86,7), (348,63)

X(77) = cevapoint of X(i) and X(j) for these (i,j): (1,223), (3,222)

X(77) = X(i)-cross conjugate of X(j) for these (i,j): (3,63), (73,222)

X(77) = trilinear pole of line X(652)X(905)

X(77) = {X(175),X(176)}-harmonic conjugate of X(962)

X(77) = X(92)-isoconjugate of X(41)

X(77) = perspector of ABC and extraversion triangle of X(78)

X(77) = X(i)-beth conjugate of X(j) for these (i,j): (21,990), (69,69), (86,269), (99,75), (332,326), (336,77), (662,77), (664,77), (811,77)

Trilinears cos A csc

Trilinears (b + c - a)(b

Trilinears S

Trilinears (b + c - a) cot A : :

Trilinears cot A cot(A/2) : :

Barycentrics a/(1 - sec A) : b/(1 - sec B) : c/(1 - sec C)

If you have The Geometer's Sketchpad, you can view X(78).

If you have GeoGebra, you can view

X(78) lies on these lines:

1,2 3,63 4,908 9,21
20,329 29,33 37,965
38,988 40,100 46,758
55,960 56,480 57,404
69,73 101,205 207,653
210,958 212,283 220,949
226,377 271,394 273,322
280,282 345,1040 392,1057
474,942 517,945 644,728
999,1059

X(78) = isogonal conjugate of X(34)

X(78) = isotomic conjugate of X(273)

X(78) = X(i)-Ceva conjugate of X(j) for these (i,j): (69,63), (312,9), (332,345)

X(78) = X(i)-cross conjugate of X(j) for these (i,j): (3,271), (72,8), (212,9), (219,63)

X(78) = crosspoint of X(69) and X(345)

X(78) = crosssum of X(i) and X(j) for these (i,j): (25,608), (56,1406), (604,1395), (1042,1426)

X(78) = X(i)-beth conjugate of X(j) for these (i,j): (78,3), (643,40), (1043,1)

X(78) = trilinear pole of line X(521)X(652)

X(78) = {X(1),X(8)}-harmonic conjugate of X(3872)

X(78) = {X(2),X(145)}-harmonic conjugate of X(938)

X(78) = X(92)-isoconjugate of X(604)

X(78) = homothetic center of anticomplementary triangle and tangential triangle of the hexyl triangle

X(78) = perspector of ABC and extraversion triangle of X(77)

Trilinears bc/(b

Trilinears (sin A/2)(sin 3B/2)(sin 3C/2) : :

Trilinears sin(A/2) csc(3A/2) : :

Barycentrics 1/(b

Barycentrics 1/(b c + 2 SA) : :

X(79) = (2r + 3R)*X(1) + 6r*X(2) - 6r*X(3) (Peter Moses, April 2, 2013)

Let A' be the reflection of X(1) in sideline BC, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(79). (Eric Danneels, Hyacinthos 7892, 9/13/03)

A'B'C' is also the reflection triangle of X(1). The lines AA', BB', CC' concur in X(79). (Randy Hutson, July 20, 2016)

Let P and Q be the intersections of line BC and circle {X(1),2r}. Let X = X(1). Let A' be the circumcenter of triangle PQX, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(79). (Compare to X(592), where the circle is the 1st Lemoine circle) (Randy Hutson, July 20, 2016)

Let A_{25}B_{25}C_{25} be Gemini triangle 25. Let A' be the perspector of conic {{A,B,C,B_{25},C_{25}}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(79). (Randy Hutson, January 15, 2019)

X(79) lies on these lines:

1,30 2,3647 8,758 9,46 12,484
21,36 33,1063 34,1061
35,226 57,90 65,80
104,946 314,320 388,1000

X(79) = reflection of X(191) in X(442)

X(79) = isogonal conjugate of X(35)

X(79) = isotomic conjugate of X(319)

X(79) = cevapoint of X(481) and X(482)

X(79) = crosssum of X(55) and X(1030)

X(79) = anticomplement of X(3647)

X(79) = X(2914) of Fuhrmann triangle

X(79) = antigonal image of X(3065)

X(79) = trilinear pole of line X(650)X(4802)

X(79) = perspector of ABC and extraversion triangle of X(80)

X(79) = Hofstadter -1/2 point

X(79) = trilinear pole of line X(650)X(4802)

X(79) = trilinear product of vertices of reflection triangle of X(1)

X(79) = X(6152)-of-excentral-triangle

Trilinears bc/(b

Trilinears cos(A/2) sec(3A/2) : :

Barycentrics 1/(b

Barycentrics 1/(bc - 2 SA) : :

X(80) = (2r + R)*X(1)- 6r*X(2) + 2r*X(3) (Peter Moses, April 2, 2013)

Let A' be the reflection in BC of the A-vertex of the excentral triangle, and define B' and C' cyclically. The circumcircles of A'BC, B'CA, and C'AB concur in X(80). Also, the lines AA', BB', CC' concur in X(80). (Randy Hutson, December 10, 2016)

Let A'B'C' be the Fuhrmann triangle. Let La be the line through A parallel to B'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines A'A", B'B", C'C" concur in X(80). (Randy Hutson, December 10, 2016)

Let A'B'C' be the orthic triangle. Let La be the antiorthic axis of AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc. B" = Lc∩La, C" = La∩Lb. The triangle A"B"C" is inversely similar to ABC, with similitude center X(9). The incenter of triangle A"B"C" is X(80). Also, the lines AA", BB", CC" concur in X(80).(Randy Hutson, December 10, 2016)

Let A'B'C' be the excentral triangle. Let A" be the isogonal conjugate, wrt A'BC, of A. Define B", C" cyclically. (A" is also the reflection of A' in BC, and cyclically for B" and C"). The lines AA", BB", CC" concur in X(80). (Randy Hutson, January 29, 2018)

Let A'B'C' be the excentral triangle. Let Oa be the A'-Johnson circle of triangle A'BC, and define Ob and Oc cyclically. X(80) is the radical center of Oa, Ob, Oc. (Randy Hutson, June 27, 2018)

X(80) lies on these lines:

1,5 2,214 7,150 8,149
9,528 10,21 30,484
33,1061 34,1063 36,104
40,90 46,84 65,79
313,314 497,1000 499,944
516,655 519,908 943,950

X(80) = midpoint of X(8) and X(149)

X(80) = reflection of X(i) in X(j) for these (i,j): (1,11), (100,10), (1317,1387)

X(80) = isogonal conjugate of X(36)

X(80) = isotomic conjugate of X(320)

X(80) = inverse-in-incircle of X(1387)

X(80) = inverse-in-Fuhrmann-circle of X(1)

X(80) = anticomplement of X(214)

X(80) = cevapoint of X(10) and X(519)

X(80) = X(i)-cross conjugate of X(j) for these (i,j): (44,2), (517,1)

X(80) = X(8)-beth conjugate of X(100)

X(80) = antigonal image of X(1)

X(80) = syngonal conjugate of X(10)

X(80) = X(186)-of-Fuhrmann triangle

X(80) = orthology center of ABC and Fuhrmann triangle

X(80) = reflection of any vertex of ABC in the corresponding side of the Fuhrmann triangle

X(80) = perspector of ABC and reflection of Fuhrmann triangle in X(11)

X(80) = trilinear pole of line X(37)X(650)

X(80) = inverse-in-circumconic-centered-at-X(1)-of-X(1807)

X(80) = perspector of ABC and extraversion triangle of X(79)

X(80) = X(1986)-of-excentral triangle

X(80) = perspector of ABC and mid-triangle of 1st and 2nd extouch triangles

X(80) = inner-Garcia-to-outer-Garcia similarity image of X(1)

X(80) = X(100)-of-outer-Garcia-triangle

Barycentrics a/(b + c) : b/(c + a) : c/(a + b)

Barycentrics eccentricity of A-Soddy ellipse : :

X(81) = (r^{2} + 2rR + s^{2})*X(1) - 3rR*X(2) - 2r^{2}*X(3)
(Peter Moses, April 2, 2013)

Let A'B'C' be the cevian triangle of X(1). Let A" be the symmedian point of triangle AB'C', and define B" and C" cyclically. Then the lines AA", BB", CC" concur in X(81). (Eric Danneels, Hyacinthos 7892, 9/13/03)

Let A'B'C' be the incentral triangle. Let L_{A} be the
reflection of B'C' in the internal angle bisector of vertex angle A,
and define L_{B} and L_{C} cyclically. Let A'' =
L_{B}∩L_{C}, B'' = L_{C}∩L_{A},
C'' = L_{A}∩L_{B}. The lines AA'', BB'', CC''
concur in X(81). (Randy Hutson, 9/23/2011)

Let H* be the Stammler hyperbola. Let A'B'C' be the tangential triangle and A"B"C" be the excentral triangle. Let A* be the intersection of the tangents to H* at A' and A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(81). (Randy Hutson, February 10, 2016)

Let A'B'C' be the 2nd circumperp triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(81). (Randy Hutson, February 10, 2016)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let A" be BB'∩CC', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(81). (Randy Hutson, February 10, 2016)

Let Ba, Ca be the intersections of lines CA, AB, resp., and the antiparallel to BC through X(1). Define Cb, Ab, Ac, Bc cyclically. Triangles ABaCa, AbBCb, AcBcC are similar to each other and inversely similar to ABC. Let Sa be the similitude center of triangles AbBCb and AcBcC. Define Sb and Sc cyclically. The lines ASa, BSb, CSc concur in X(81). (Randy Hutson, February 10, 2016)

Let A_{10}B_{10}C_{10} be Gemini triangle 10. Let A' be the perspector of conic {{A,B,C,B_{10},C_{10}}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(81). (Randy Hutson, January 15, 2019)

Let A_{11}B_{11}C_{11} be Gemini triangle 11. Let A' be the perspector of conic {{A,B,C,B_{11},C_{11}}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(81). (Randy Hutson, January 15, 2019)

If you have The Geometer's Sketchpad, you can view X(81).

If you have GeoGebra, you can view

X(81) lies on these lines:

1,21 2,6 7,27 8,1010
19,969 28,60 29,189
32,980 42,100 43,750
55,1002 56,959 57,77
65,961 88,662 99,739
105,110 145,1043 226,651
239,274 314,321 377,387
386,404 411,581 593,757
715,932 859,957 941,967
982,985 1019,1022 1051,1054
1098,1104

X(81) = isogonal conjugate of X(37)

X(81) = isotomic conjugate of X(321)

X(81) = anticomplement of X(1211)

X(81) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,229), (86,21), (286,28)

X(81) = cevapoint of X(i) and X(j) for these (i,j): (1,6), (57,222), (58,284)

X(81) = X(i)-cross conjugate of X(j) for these (i,j): (1,86), (3,272), (6,58), (57,27), (284,21)

X(81) = crosspoint of X(274) and X(286)

X(81) = crosssum of X(i) and X(j) for these (i,j): (1,846), (6,1030), (42,1334), (213,228)

X(81) = crossdifference of every pair of points on line X(512)X(661)

X(81) = X(i)-beth conjugate of X(j) for these (i,j): (333,333), (643,81), (645,81), (648,81), (662,81), (931,81)

X(81) = trilinear product of PU(31)

X(81) = intersection of tangents at X(1) and X(6) to the Stammler hyperbola

X(81) = crosspoint of X(1) and X(6) wrt both the excentral and tangential triangles

X(81) = trilinear pole of line X(36)X(238) (the polar of X(1) wrt the circumcircle)

X(81) = {X(1),X(31)}-harmonic conjugate of X(1621)

X(81) = X(6)-isoconjugate of X(10)

X(81) = X(92)-isoconjugate of X(228)

X(81) = perspector of ABC and cross-triangle of Gemini triangles 1 and 2

X(81) = barycentric product of vertices of Gemini triangle 1

X(81) = barycentric product of vertices of Gemini triangle 2

X(81) = barycentric product of vertices of Gemini triangle 3

X(81) = barycentric product of vertices of Gemini triangle 4

X(81) = perspector of Gemini triangles 2 and 7

X(81) = perspector of ABC and cross-triangle of ABC and Gemini triangle 1

X(81) = perspector of ABC and cross-triangle of ABC and Gemini triangle 2

X(81) = perspector of Gemini triangle 24 and cross-triangle of ABC and Gemini triangle 24

X(81) = perspector of Gemini triangle 28 and cross-triangle of ABC and Gemini triangle 28

= sin A csc(A + ω) : sin B csc(B + ω) : sin C csc(C + ω)

Barycentrics a/(b^{2} + c^{2}) :
b/(c^{2} + a^{2}) : c/(a^{2} +
b^{2})

Let A'B'C' be the circummedial triangle. Let A" be the trilinear product B'*C', and define B", C" cyclically. A", B", C" are collinear on line X(798)X(812) (the trilinear polar of X(3112)). The lines AA", BB", CC" concur in X(82). (Randy Hutson, October 15, 2018)

X(82) lies on these lines: 1,560 10,83 31,75 37,251 58,596 689,715 759,827

X(82) = isogonal conjugate of X(38)

X(82) = isotomic conjugate of X(1930)

X(82) = cevapoint of X(1) and X(31)

X(82) = trilinear pole of line X(661)X(830)

X(82) = crossdifference of every pair of points on line X(2084)X(2530)

X(82) = perspector of ABC and extraversion triangle of X(82) (which is also the anticevian triangle of X(82))

X(82) = crosspoint of X(1) and X(31) wrt the excentral triangle

= csc(A + ω) : csc(B + ω) : csc(C + ω)

Barycentrics 1/(b^{2} + c^{2}) :
1/(c^{2} + a^{2}) : 1/(a^{2} +
b^{2})

Let K denote the symmedian point, X(6). Let A'B'C' be the cevian triangle of K. Let K_{A} be K of the triangle AB'C'; let K_{B} be K of A'BC' and let K_{C} be K of A'B'C. The lines AK_{A}, BK_{B}, CK_{C} concur in X(83). (Randy Hutson, 9/23/2011)

Let A'B'C' be the 1st Brocard triangle. Let A" be the reflection of A' in BC, and define B" and C" cyclically. AA", BB", CC" concur in X(83). (Randy Hutson, December 26, 2015)

Let Ba, Ca be the intersections of lines CA, AB, resp., and the antiparallel to BC through X(2). Define Cb, Ab, Ac, Bc cyclically. Triangles ABaCa, AbBCb, AcBcC are similar to each other and inversely similar to ABC. Let Sa be the similitude center of triangles AbBCb and AcBcC. Define Sb and Sc cyclically. The lines ASa, BSb, CSc concur in X(83). (Randy Hutson, December 26, 2015)

Let (Oa) be the circle whose diameter is the orthogonal projections of PU(1) on line BC. Define (Ob) and (Oc) cyclically. X(83) is the radical center of circles (Oa), (Ob), (Oc). (Randy Hutson, December 26, 2015)

Let A'B'C' be the circummedial triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(83). (Randy Hutson, December 26, 2015)

X(83) lies on these lines:

2,32 3,262 4,182 5,98
6,76 10,82 17,624
18,623 39,99 213,239
217,287 275,297 597,671 689,729

X(83) = isogonal conjugate of X(39)

X(83) = isotomic conjugate of X(141)

X(83) = complement of X(2896)

X(83) = cevapoint of X(2) and X(6)

X(83) = X(i)-cross conjugate of X(j) for these (i,j): (2,308), (6,251), (512,99)

X(83) = trilinear pole of line X(23)X(385) (line is the polar of X(2) wrt the circumcircle, and also the anticomplement of the de Longchamps line, and also the polar of X(5) wrt {circumcircle, nine-point circle}-inverter)

X(83) = crossdifference of every pair of points on line X(688)X(3005)

X(83) = pole wrt polar circle of trilinear polar of X(427)

X(83) = X(48)-isoconjugate (polar conjugate) of X(427)

X(83) = perspector of ABC and medial triangle of 1st Brocard triangle

X(83) = crosspoint of X(2) and X(6) wrt both the anticomplementary and tangential triangles

X(83) = trilinear product of vertices of circummedial triangle

X(83) = midpoint of PU(137)

X(83) = bicentric sum of PU(i) for these i: 137, 141

X(83) = homothetic center of 5th anti-Brocard triangle and medial triangle

X(83) = X(8290)-of-1st-Brocard-triangle

X(83) = perspector of ABC and 1st Brocard triangle of medial triangle

X(83) = perspector of ABC and 1st Brocard triangle of 5th anti-Brocard triangle

X(83) = homothetic center of ABC and cross-triangle of ABC and 5th anti-Brocard triangle

X(83) = Cundy-Parry Phi transform of X(262)

X(83) = Cundy-Parry Psi transform of X(182)

X(83) = barycentric product of circumcircle intercepts of line X(316)X(512)

Trilinears a^2[a^2 - (b - c)^2]^2 - (b - c)^2[a^2 - (b + c)^2]^2 : :

Let A',B',C' be the excenters. The perpendiculars from B' to AB and from C' to AC meet in a point A". Points B" and C" are determined cyclically. The hexyl triangle, A"B"C", is perspective to ABC, and X(84) is the perspector.

Let A'B'C' be the extouch triangle. Let A" be the orthocenter of AB'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(84). (Randy Hutson, September 14, 2016)

Let A1B1C1 be the 1st Conway triangle. Let A' be the crosspoint of B1 and C1, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(84). (Randy Hutson, December 2, 2017)

X(84) lies on the Darboux cubic, the circumellipse with center X(9), and these lines: 1,221 3,9 4,57 7,946 8,20 21,285 33,603 36,90 46,80 58,990 64,3353 171,989 256,988 294,580 309,314 581,941 944,1000 2130,3345 3346,3472 3347,3355

X(84) = reflection of X(i) in X(j) for these (i,j): (40,1158), (1490,3)

X(84) = isogonal conjugate of X(40)

X(84) = isotomic conjugate of X(322)

X(84) = X(i)-Ceva conjugate of X(j) for these (i,j): (189,282), (280,1)

X(84) = X(i)-cross conjugate of X(j) for these (i,j): (19,57), (56,1)

X(84) = X(280)-aleph conjugate of X(84)

X(84) = X(i)-beth conjugate of X(j) for these (i,j): (271,3), (280,280), (285,84)

X(84) = X(68)-of-the-hexyl-triangle.
X(84) = trilinear pole of line X(650)X(1459)

X(84) = perspector of ABC and the reflection in X(9) of the antipedal triangle of X(9)

X(84) = Danneels point of X(110)

X(84) = trilinear product of vertices of hexyl triangle (i.e., the extraversions of X(40))

X(84) = hexyl-isotomic conjugate of X(12717)

X(84) = perspector of ABC and cross-triangle of extouch and Hutson-extouch triangles

X(84) = Cundy-Parry Phi transform of X(9)

X(84) = Cundy-Parry Psi transform of X(57)

X(84) = intouch-to-excentral similarity image of X(4)

= tan(A/2) csc

Barycentrics bc/(b + c - a) : ca/(c + a - b) : ab/(a + b - c)

Barycentrics cot A' : :, where A'B'C' is the excentral triangle

Let A_{38}B_{38}C_{38} be Gemini triangle 38. Let A' be the perspector of conic {{A,B,C,B_{38},C_{38}}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(85). (Randy Hutson, January 15, 2019)

X(85) lies on these lines:

1,664 2,241 7,8 12,120
29,34 56,870 57,274
76,226 92,331 109,767
150,355 264,309

X(85) = isogonal conjugate of X(41)

X(85) = isotomic conjugate of X(9)

X(85) = complement of X(3177)

X(85) = anticomplement of X(1212)

X(85) = X(274)-Ceva conjugate of X(348)

X(85) = cevapoint of X(i) and X(j) for these (i,j): (1,169), (2,7), (57,77), (92,342)

X(85) = X(i)-cross conjugate of X(j) for these (i,j): (2,75), (57,273), (92,309), (142,2), (226,7)

X(85) = X(i)-beth conjugate of X(j) for these (i,j): (76,76), (85,279), (99,1), (274,85), (668,85), (789,85), (799,85), (811,85)

X(85) = trilinear pole of line X(522)X(693) (the isotomic conjugate of the circumconic centered at X(1), conic {A,B,C,X(100),X(664),X(1120),X(1320)}; also the polar of X(33) wrt polar circle)

X(85) = pole wrt polar circle of trilinear polar of X(33) (line X(657)X(4041))

X(85) = polar conjugate of X(33)

X(85) = {X(7),X(8)}-harmonic conjugate of X(6604)

X(85) = trilinear square of X(508)

X(85) = trilinear product of vertices of Gemini triangle 9

X(85) = trilinear product of vertices of Gemini triangle 10

Barycentrics 1/(b + c) : 1/(c + a) : 1/(a + b)

X(86) = 2(r^{2} + 2rR +
s^{2})*X(1) + 3(r^{2} + s^{2})*X(2) -
4r^{2}*X(3) (Peter Moses, April 2, 2013)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let La be the tangent to the circumcircle at A', and define Lb and Lc cyclically. Let A" be the point where La is tangent to the Steiner circumellipse, and define B" and C" cyclically. Let A* = BB"∩CC", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(86). (Randy Hutson, December 10, 2016)

Let A1B1C1 be the 1st Conway triangle. Let A' be the trilinear pole of line B1C1, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(86). (Randy Hutson, December 10, 2016)

Let A5B5C5 be the 5th Conway triangle. Let A' be the trilinear pole of line B5C5, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(86). (Randy Hutson, December 10, 2016)

Let A_{3}B_{3}C_{3} and A_{4}B_{4}C_{4} be Gemini triangles 3 and 4, resp. Let L_{A} be the tangent at A to conic {{A,B_{3},C_{3},B_{4},C_{4}}}, and define L_{B}, L_{C} cyclically. Let A' = L_{B}∩L_{C}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(86). (Randy Hutson, January 15, 2019)

Let A_{21}B_{21}C_{21} be Gemini triangle 21. Let A' be the perspector of conic {{A,B,C,B_{21},C_{21}}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(86). (Randy Hutson, January 15, 2019)

X(86) lies on these lines:

1,75 2,6 7,21 10,319
29,34 37,190 58,238
60,272 99,106 110,675
142,284 239,1100 269,1088
283,307 310,350 741,789 870,871

X(86) = isogonal conjugate of X(42)

X(86) = isotomic conjugate of X(10)

X(86) = complement of X(1654)

X(86) = anticomplement of X(1213)

X(86) = X(274)-Ceva conjugate of X(333)

X(86) = cevapoint of X(i) and X(j) for these (i,j): (1,2), (7,77), (21,81)

X(86) = crosssum of X(1) and X(1045)

X(86) = crossdifference of every pair of points on line X(512)X(798)

X(86) = X(i)-cross conjugate of X(j) for these (i,j): (1,81), (2,274), (7,286), (21,333), (58,27), (513,190)

X(86) = X(i)-beth conjugate of X(j) for these (i,j): (86,1014), (99,86), (261,86), (314,314), (645,86), (811,86)

X(86) = X(2)-Ceva conjugate of X(6626)

X(86) = intersection of tangents at X(1) and X(2) to the bianticevian conic of X(1) and X(2); see X(99)

X(86) = crosspoint of X(1) and X(2) wrt both the excentral and anticomplementary triangles

X(86) = trilinear pole of line X(239)X(514) (Lemoine axis of excentral triangle)

X(86) = pole wrt polar circle of trilinear polar of X(1826)

X(86) = X(48)-isoconjugate (polar conjugate)-of-X(1826)

X(86) = perspector of Gemini triangle 1 and cross-triangle of ABC and Gemini triangle 1

X(86) = perspector of ABC and cross-triangle of ABC and Gemini triangle 23

X(86) = perspector of ABC and cross-triangle of ABC and Gemini triangle 24

X(86) = perspector of ABC and cross-triangle of Gemini triangles 23 and 24

X(86) = perspector of ABC and Gemini triangle 25

X(86) = barycentric product of vertices of Gemini triangle 23

X(86) = barycentric product of vertices of Gemini triangle 24

X(86) = barycentric product of vertices of Gemini triangle 25

Barycentrics a/(ab + ac - bc) : b/(bc + ba - ca) : c/(ca + cb - ab)

X(87) lies on these lines: 1,192 6,43 9,292 10,979 34,242 56,238 58,978 106,932

X(87) = isogonal conjugate of X(43)

X(87) = isotomic conjugate of X(6376)

X(87) = perspector of ABC and extraversion triangle of X(43)

X(87) = trilinear product of extraversions of X(43)

X(87) = cevapoint of X(2) and X(330)

X(87) = X(2)-cross conjugate of X(1)

X(87) = X(932)-beth conjugate of X(87)

Barycentrics a/(b + c - 2a) : b/(c + a - 2b) : c/(a + b - 2c)

Let A_{9}B_{9}C_{9} be Gemini triangle 9. Let A' be the perspector of conic {{A,B,C,B_{9},C_{9}}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(88). (Randy Hutson, January 15, 2019)

X(88) lies on these lines: 1,100 2,45 6,89 28,162 44,679 57,651 81,662 105,901 274,799 278,653 279,658 291,660

X(88) = isogonal conjugate of X(44)

X(88) = isotomic conjugate of X(4358)

X(88) = complement of X(30578)

X(88) = cevapoint of X(i) and X(j) for these (i,j): (1,44), (6,36)

X(88) = X(i)-cross conjugate of X(j) for these (i,j): (44,1), (517,7)

X(88) = X(i)-aleph conjugate of X(j) for these (i,j): (88,1), (679,88), (903,63), (1022,1052)

X(88) = X(333)-beth conjugate of X(190)

X(88) = perspector of conic {{A,B,C,PU(50)}}

X(88) = trilinear pole of PU(55); the line X(1)X(513), the line through X(1) parallel to its trilinear polar; also normal to Feuerbach hyperbola at X(1)

X(88) = crossdifference of every pair of points on line X(678)X(1635)

X(88) = X(6)-isoconjugate of X(519)

X(88) = BSS(a^2->a) of X(111)

Barycentrics a/(2b + 2c - a) : b/(2c + 2a - b) : c/(2a + 2b - c)

Let A_{9}B_{9}C_{9} be the Gemini triangle 9. Let L_{A} be the line through A_{9} parallel to BC, and define L_{B} and L_{C} cyclically. Let A'_{9} = L_{B}∩L_{C}, and define B'_{9}, C'_{9} cyclically. Triangle A'_{9}B'_{9}C'_{9} is homothetic to ABC at X(89). (Randy Hutson, November 30, 2018)

X(89) lies on these lines: 1,902 2,44 6,88 649,1022

X(89) = isogonal conjugate of X(45)

X(89) = isotomic conjugate of X(4671)

Trilinears 1/(a^3 + a^2 (b + c) - a (b^2 + c^2) - (b - c)^2 (b + c)) : :

Barycentrics a/(cos B + cos C - cos A) : b/(cos C + cos A - cos B) : c/(cos A + cos B - cos C)

X(90) = (r + R)^{2}*X(1) - 6rR*X(2) - 2r(r - R)*X(3) (Peter Moses, April 2, 2013)

X(90) lies on these lines: 1,155 4,46 9,35 21,224 33,47 36,84 40,80 57,79

X(90) = isogonal conjugate of X(46)

X(90) = isotomic conjugate of X(20930)

X(90) = X(3)-cross conjugate of X(1)

X(90) = perspector of ABC and extraversion triangle of X(46)

X(90) = trilinear product of the extraversions of X(46), which is also the cross-triangle of the orthic and excentral triangles

X(90) = trilinear product of PU(125)

X(90) = Cundy-Parry Phi transform of X(90)

X(90) = Cundy-Parry Psi transform of X(46)

Barycentrics sin A sec 2A : sin B sec 2B : sin C sec 2C

Trilinears cot A tan 2A : :

X(91) lies on these lines: 19,920 31,1087 37,498 47,92 63,921 65,68 225,847 255,1109 759,925

X(91) = isogonal conjugate of X(47)

X(91) = X(48)-cross conjugate of X(92)

X(91) = trilinear product of X(485) and X(486)

X(91) = polar conjugate of X(1748)

X(91) = X(92)-isoconjugate of X(563)

X(91) = trilinear pole of line X(661)X(2618)

X(91) = perspector of ABC and extraversion triangle of X(91) (which is also the anticevian triangle of X(91))

X(91) = trilinear product of vertices of outer and inner Vecten triangles

Trilinears cot A + tan A : :

Barycentrics sec A : sec B : sec C

X(92) = (sec A)*[A] + (sec B)*[B] + (sec C)*[C], where A, B, C denote angles and [A], [B], [C] denote vertices

Let L_{A} be the line through X(4) parallel to the internal bisector of angle A, and let

A' = BC∩L_{A}. Define B' and C' cyclically.

**Alexei Myakishev,** "The M-Configuration of a Triangle," *Forum Geometricorum* 3 (2003) 135-144,

proves that the lines AA', BB', CC' concur in X(92). He notes that another construction follows from Proposition 2 of the article: let A_{1} be the midpoint of the arc BC of the circumcircle that passes through A, and let A_{2} be the point, other than A, in which the A-altitude meets the circumcircle. Let A" = A_{1}A_{2}∩BC. Define B" and C" cyclically. Then the lines AA", BB", CC" concur in X(92).

Suppose that T = A'B'C' is a central triangle. Let A'' be the pole with respect to the polar circle of the line B'C', and define B'' and C'' cyclically. The appearance of T in the following list means that the lines AA'', BB'', CC'' concur in X(92): Feurerbach, incentral, excentral, extangents, Apollonius, mixtilinear excentral. (Randy Hutson, December 26, 2015)

X(92) lies on these lines:

1,29 2,273 4,8 7,189
10,1838 19,27 25,242
28,2975 31,162 33,1897
34,1220 38,240 40,412
47,91 48,2167 53,4415
55,243 56,1940 57,653
81,2995 85,331 100,917
108,1311 171,1430 226,342
239,607 255,1087 257,297
264,306 304,561 345,3262
388,1118 394,1943 406,1068
427,2969 429,3948 429,3948
459,1446 497,1857 518,1859
608,894 651,2988 823,2349
938,3176 942,1148 960,1882
984,1860 994,1845 1146,1952
1172,2997 1211,1865 1309,2717
1435,3306 1585,1659 1621,4183
1707,1733 1726,1746 1731,1751
1785,4656 1842,1891 1844,3874
1870,5136 1947,2994 1954,1955
1956,2632 1973,3112 2064,3596
2331,5256 2399,4391
3064,4468 4198,4968

X(92) = isogonal conjugate of X(48)

X(92) = isotomic conjugate of X(63)

X(92) = anticomplement of X(1214)

X(92) = anticomplementary conjugate of X(2897)

X(92) = X(i)-Ceva conjugate of X(j) for these (i,j): (85, 342), (264,318), (286,4), (331,273)

X(92) = cevapoint of X(i) and X(j) for these (i,j): (1,19), (4,281), (47,48), (196,278)

X(92) = X(i)-cross conjugate of X(j) for these (i,j): (1,75), (4,273), (19,158), (48,91), (226,2), (281,318)

X(92) = crosspoint of X(i) and X(j) for these (i,j): (85,309), (264,331)

X(92) = crossdifference of every pair of points on line X(810)X(822)

X(92) = X(275)-aleph conjugate of X(47)

X(92) = X(i)-beth conjugate of X(j) for these (i,j): (92,278), (312,329), (648,57)

X(92) = {X(19),X(63)}-harmonic conjugate of X(1748)

X(92) = barycentric product of PU(20)

X(92) = trilinear product of PU(i) for these i: 21, 45

X(92) = bicentric sum of PU(130)

X(92) = midpoint of PU(130)

X(92) = trilinear product X(2)*X(4)

X(92) = trilinear pole of line X(240)X(522) (polar of X(1) wrt polar circle)

X(92) = pole of antiorthic axis wrt polar circle

X(92) = X(6)-isoconjugate of X(3)

X(92) = X(48)-isoconjugate (polar conjugate) of X(1)

X(92) = X(91)-isoconjugate of X(563)

X(92) = inverse-in-Fuhrmann-circle of X(5174)

X(92) = perspector of ABC and extraversion triangle of X(92) (which is also the anticevian triangle of X(92))

X(92) = crosspoint of X(1) and X(19) wrt excentral triangle

X(92) = crosspoint of X(47) and X(48) wrt excentral triangle

X(92) = perspector of ABC and cross-triangle of Gemini triangles 37 and 38

X(92) = perspector of ABC and cross-triangle of ABC and Gemini triangle 37

X(92) = perspector of ABC and cross-triangle of ABC and Gemini triangle 38

X(92) = barycentric product of vertices of Gemini triangle 37

X(92) = barycentric product of vertices of Gemini triangle 38

Barycentrics sin A sec 3A : sin B sec 3B : sin C sec 3C

Barycentrics 1/(a^2 (a^2 - b^2 - c^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - b^2 c^2)) : :

Let O_{A}O_{B}O_{C} be the Kosnita triangle. Let A' be the pole wrt polar circle of line O_{B}O_{C}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(93). (Randy Hutson, June 7, 2019)

X(93) lies on these lines: 4,562 49,94 186,252

X(93) = isogonal conjugate of X(49)

X(93) = X(50)-cross conjugate of X(94)

X(93) = polar conjugate of X(1994)

X(93) = X(2964)-isoconjugate of X(3)

Barycentrics sin A csc 3A : sin B csc 3B : sin C csc 3C

Barycentrics b^2 c^2 / ((a^2 - b^2 - c^2)^2 - b^2 c^2) : :

Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. Let A' be the trilinear pole of line A1A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(94). Let A1'B1'C1' and A2'B2'C2' be the 1st and 2nd Ehrmann inscribed triangles. Then X(94) is the radical center of nine-point circles of AA1'A2', BB1'B2', CC1'C2'. (Randy Hutson, June 27, 2018)

X(94) lies on the Kiepert hyperbola and these lines: 2,300 4,143 23,98 49,93 96,925 275,324

X(94) = isogonal conjugate of X(50)

X(94) = isotomic conjugate of X(323)

X(94) = cevapoint of X(49) and X(50)

X(94) = X(i)-cross conjugate of X(j) for these (i,j): (30,264), (50,93), (265,328)

X(94) = X(300)-Hirst inverse of X(301)

X(94) = trilinear pole of PU(5) (line X(5)X(523))

X(94) = pole wrt polar circle of trilinear polar of X(186)

X(94) = X(48)-isoconjugate (polar conjugate) of X(186)

X(94) = barycentric product X(476)*X(850)

X(94) = trilinear pole of PU(173)

X(94) = trilinear product X(74)*X(107) (circumcircle-X(4) antipodes)

X(94) = barycentric quotient X(476)/X(110)

X(94) = intersection of the tangent to hyperbola {{A,B,C,X(6),X(13),X(16)}} at X(13) and the tangent to hyperbola {{A,B,C,X(6),X(14),X(15)}} at X(14)

Let A'B'C' be the symmedial triangle. Let La be the reflection of line B'C' in line BC, and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(95). (Randy Hutson, August 19, 2015)

Let A' be the intersection, other than A, of the circumcircle and the branch of the Lucas cubic that contains A, and define B' and C' cyclically. The triangle A'B'C' is here introduced as the Lucas triangle (not to be confused with the Lucas central triangle). The vertices A', B', C' lie on the rectangular hyperbola {{X(2),X(20),X(54),X(69),X(110),X(2574),X(2575),X(2979)}}. (See https://bernard-gibert.pagesperso-orange.fr/Exemples/k007.html.) Also, X(95) is the trilinear product of the vertices of the Lucas triangle. (Randy Hutson, August 19, 2015)

X(95) lies on these lines:

2,97 3,264 54,69 76,96
99,311 140,340 141,287 160,327 183,305 216,648 307,320

X(95) = isogonal conjugate of X(51)

X(95) = isotomic conjugate of X(5)

X(95) = anticomplement of X(233)

X(95) = X(276)-Ceva conjugate of X(275)

X(95) = cevapoint of X(i) and X(j) for these (i,j): (2,3), (6,160), (54,97)

X(95) = X(i)-cross conjugate of X(j) for these (i,j): (2,276), (3,97), (54,275), (140,2), (340,1494)

X(95) = intersection of tangents at X(2) and X(3) to bianticevian conic of X(2) and X(3)

X(95) = crosspoint of X(2) and X(3) wrt both the anticomplementary triangle and anticevian triangle of X(3)

X(95) = trilinear pole of line X(323)X(401) (polar of X(53) wrt polar circle, and polar of X(69) wrt de Longchamps circle)

X(95) = pole wrt polar circle of trilinear polar of X(53)

X(95) = X(48)-isoconjugate (polar conjugate) of X(53)

X(95) = X(92)-isoconjugate of X(217)

Barycentrics a sec 2A sec(B - C) : b sec 2B sec(C - A) : c sec 2C sec(A - B)

Let A'B'C' be the reflection triangle. Let B_{A} and C_{A} be the orthogonal projections of B' and C' on line BC, resp. Let (O_{A}) be the circle with segment B_{A}C_{A} as diameter. Define (O_{B}) and (O_{C}) cyclically. X(96) is the radical center of circles (O_{A}), (O_{B}), (O_{C}). (Randy Hutson, June 7, 2019)

X(96) lies on these lines: 2,54 4,231 24,847 76,95 94,925

X(96) = isogonal conjugate of X(52)

X(96) = cevapoint of X(3) and X(68)

X(96) = X(3)-cross conjugate of X(54)

X(96) = polar conjugate of X(467)

X(96) = Cundy-Parry Phi transform of X(5392)

X(96) = Cundy-Parry Psi transform of X(571)

Barycentrics cos A sec(B - C) : cos B sec(C - A) : cos C sec(A - B)

X(97) lies on these lines: 2,95 3,54 110,418 216,288 276,401

X(97) = isogonal conjugate of X(53)

X(97) = isotomic conjugate of X(324)

X(97) = X(95)-Ceva conjugate of X(54)

X(97) = X(3)-cross conjugate of X(95)

X(97) = cevapoint of X(3) and X(577)

X(97) = X(51)-isoconjugate of X(92)

X(97) = Cundy-Parry Phi transform of X(7592)

Λ(P,X) = isogonal conjugate of the point where line PX meets the line at infinity.

Let Y = Λ(P,X), let Q = isogonal conjugate of P, and let Y
and Z be the points where line YQ meets the circumcircle;

then **Ψ(P,X) = Z.**

Trilinears bc/(b

Barycentrics 1/(b

Tripolars |cos(A + ω)| : :

Tripolars |a (b^4 + c^4 - a^2 b^2 - a^2 c^2)| : :

Let A' be the apex of the isosceles triangle BA'C constructed inward on BC such that ∠A'BC = ∠A'CB = ω. Define B' nad C' cyclically. Let Ha be the orthocenter of BA'C, and define Hb and Hc cyclically. The lines AHa, BHb, CHc concur in X(98). (Randy Hutson, July 20, 2016)

Let A' be the apex of the isosceles triangle BA'C constructed outward on BC such that ∠A'BC = ∠A'CB = ω/2. Define B' and C' cyclically. Let Oa be the circumcenter of BA'C, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X(98). (Randy Hutson, July 20, 2016)

If you have The Geometer's Sketchpad, you can view X(98).

**J. W. Clawson,** "Points on the circumcircle," *American Mathematical Monthly* 32 (1925) 169-174.

Let A'B'C' be the 1st Brocard triangle. Let A" be the reflection of A in B'C', and define B" and C" cyclically. X(98) is the radical center of the circumcircles of AA'A", BB'B", CC'C". (Randy Hutson, November 18, 2015)

Let A'B'C' be the orthic triangle. Let La be the Lemoine axis of triangle AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. Triangle A"B"C" is inversely similar to ABC, with similicenter X(2). The lines AA", BB", CC" concur in X(98), which is X(4)-of-A"B"C". (Randy Hutson, July 31 2018)

Let P be a point on the Steiner circumellipse. Let A' be the orthocenter of BCP, and define B' and C' cyclically. Let Q be the centroid of A'B'C'. The locus of Q as P varies is an ellipse similar and orthogonal to the Steiner circumellipse, and also centered at X(2). When P = X(671), Q = X(98). See also X(6054) and X(24808). (Randy Hutson, October 15, 2018)

X(98) lies on these lines:

2,110 3,76 4,32 5,83
6,262 10,101 13,1080
14,383 20,148 22,925
23,94 25,107 30,671
100,228 109,171 111,1637 186,935
275,427 376,543 381,598
385,511 468,685 523,842
620,631 804,878

X(98) is the {X(2),X(147)}-harmonic conjugate of X(114). For a list of harmonic conjugates, click **Tables** at the top of this page.

X(98) = midpoint between X(20) and X(148)

X(98) = reflection of X(i) in X(j) for these (i,j): (4,115), (99,3), (147,114), (1513,230)

X(98) = isogonal conjugate of X(511)

X(98) = isotomic conjugate of X(325)

X(98) = complement of X(147)

X(98) = anticomplement of X(114)

X(98) = X(290)-Ceva conjugate of X(287)

X(98) = cevapoint of X(i) and X(j) for these (i,j): (2,385), (6,237)

X(98) = X(i)-cross conjugate of X(j) for these (i,j): (230,2), (237,6), (248,287), (446,511)

X(98) = crosssum of X(385) and X(401)

X(98) = X(2)-Hirst inverse of X(287)

X(98) = perspector of ABC and triangle formed by Lemoine axis (or PU(1) or PU(2)) reflected in sides of ABC

X(98) = Λ(X(4), X(69)) (the line that is the isotomic conjugate of the Jerabek hyperbola)

X(98) = trilinear pole of line X(6)X(523) (polar of X(297) wrt polar circle, and the radical axis of circles with segments X(13)X(16) and X(14)X(15) as diameters)

X(98) = pole wrt polar circle of trilinear polar of X(297) (line X(114)X(132))

X(98) = pole wrt {circumcircle, nine-point circle}-inverter of line X(115)X(125)

X(98) = X(48)-isoconjugate (polar conjugate) of X(297)

X(98) = X(6)-isoconjugate of X(1959)

X(98) = inverse-in-polar-circle of X(132)

X(98) = inverse-in-{circumcircle, nine-point circle}-inverter of X(125)

X(98) = inverse-in-circle-{{X(1687),X(1688),PU(1),PU(2)}} of X(2715)

X(98) = Ψ(X(6), X(523))

X(98) = Ψ(X(190), X(71))

X(98) = Kiepert-hyperbola antipode of X(4)

X(98) = reflection of X(842) in the Euler line

X(98) = reflection of X(2698) in the Brocard axis

X(98) = reflection of X(2699) in line X(1)X(3)

X(98) = X(129)-of-excentral-triangle

X(98) = X(130)-of-hexyl-triangle

X(98) = X(3)-of-1st-anti-Brocard-triangle

X(98) = perspector of ABC and 1st Neuberg triangle

X(98) = trilinear product of vertices of 1st Neuberg triangle

X(98) = orthocenter of X(13)X(14)X(2394)

X(98) = 2nd-Parry-to-ABC similarity image of X(2)

X(98) = trilinear product of PU(88)

X(98) = X(2456) of 6th Brocard triangle

X(98) = midpoint of PU(135)

X(98) = bicentric sum of PU(135)

X(98) = perspector of ABC and circumsymmedial triangle of Artzt triangle

X(98) = McCay-to-Artzt similarity image of X(381)

X(98) = circumcircle-antipode of X(99)

X(98) = the point of intersection, other than A, B, and C, of the circumcircle and Kiepert hyperbola

X(98) = Ψ(X(101), X(100)

X(98) = Λ(X(3), X(6))

X(98) = homothetic center of 5th anti-Brocard triangle and Euler triangle

X(98) = Thomson-isogonal conjugate of X(512)

X(98) = Lucas-isogonal conjugate of X(512)

X(98) = X(1380)-of-circummedial-triangle

X(98) = X(6233)-of-circumsymmedial-triangle

X(98) = Cundy-Parry Phi transform of X(76)

X(98) = Cundy-Parry Psi transform of X(32)

X(98) = perspector of ABC and vertex-triangle of reflection triangles of PU(1)

X(98) = X(2)-of-2nd-anti-Parry-triangle

X(98) = excentral-to-ABC functional image of X(1282)

X(98) = orthic-to-ABC functional image of X(129)

X(98) = Ψ(X(1), X(810))

X(98) = Ψ(X(811), X(1))

X(98) = barycentric product of circumcircle intercepts of line X(2)X(647)

X(98) = trilinear pole, wrt Thomson triangle, of line X(2)X(1350)

X(98) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,X(3),X(25),X(32)}} (isogonal conjugate of line X(4)X(69))

X(98) = BSS(a->a^2) of X(673)

Trilinears b

Barycentrics 1/(b

Barycentrics d(A,L) : : , where d = directed distance and L = X(115)X(125)

Tripolars |a (b^2 - c^2)| : :

Let L_{A} be the reflection of the line X(3)X(6) in line BC,
and define L_{B} and L_{C} cyclically. Let A' =
L_{B}∩L_{C}, B' = L_{C}∩L_{A},
C' = L_{A}∩L_{B}. The lines AA', BB', CC' concur in
X(99). (Randy Hutson, 9/23/2011)

X(99) is the only point on the circumcircle whose isotomic conjugate lies on the line at infinity. (Randy Hutson, 9/23/2011)

X(99) is the center of the bianticevian conic of X(1) and X(2), which is the rectangular hyperbola H that passes through these points: X(1), X(2), X(20), X(63), X(147), X(194), X(487), X(488), X(616), X(617), X(627), X(628), X(1764), X(2896), the excenters, the vertices of the anticomplementary triangle, and the extraversions of X(63). Also, H is the anticomplementary conjugate of line X(4)X(69), the anticomplementary isotomic conjugate of line X(2)X(6), the excentral isogonal conjugate of line X(40)X(511), and the excentral isotomic conjugate of line X(1045)X(2951); also, H is tangent to line X(1)X(75) at X(1), to line X(2)X(6) at X(2), and meets the line at infinity (and the Kiepert hyperbola, other than at X(2)) at X(3413) and X(3414). (Randy Hutson, December 26, 2015)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let A"B"C" be the tangential triangle of A'B'C'. Let A* be the cevapoint of B" and C", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(99). (Randy Hutson, February 10, 2016)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Brocard axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to ABC with similarity ratio 3. Let A"B"C" be the reflection of A'B'C' in the Brocard axis. The triangle A"B"C" is homothetic to ABC, with center of homothety X(115) and centroid X(99). See Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, February 10, 2016)

Let A', B' and C' be the intersections of the de Longchamps line and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(99). (Randy Hutson, February 10, 2016)

Let A', B', C' be the intersections of the Brocard axis and lines BC, CA, AB, resp. Let Oa, Ob, Oc be the circumcenters of AB'C', BC'A', CA'B', resp. The lines AOa, BOb, COc concur in X(99). (Randy Hutson, February 10, 2016)

Let A'B'C' be the 1st Brocard triangle. Let La be the line through A parallel to B'C', and define Lb and Lc cyclically. La, Lb, Lc concur in X(99). (Randy Hutson, February 10, 2016)

Let A' be the trilinear pole of the perpendicular bisector of BC, and define B' and C' cyclically. A'B'C' is also the anticomplement of the anticomplement of the midheight triangle. X(99) is the trilinear product A'*B'*C'. (Randy Hutson, January 29, 2018)

Let A'B'C' be the 1st Brocard triangle. Let A" be the reflection of A in line B'C', and define B", C" cyclically. Let A"' be the reflection of A' in line BC, and define B"', C"' cyclically. Let A* = B"B"'∩\C"C"', and define B*, C* cyclically. Triangle A*B*C* is homothetic to A'B'C' at X(99). (Randy Hutson, June 27, 2018)

See a construction: Ercole Suppa, **Hyacinthos 29064**.

If you have The Geometer's Sketchpad, you can view the following dynamic sketches:

X(99) and Steiner Circum-ellipse (showing X(99) and an area-ratio property)

For more about the Steiner circumellipse, visit MathWorld.

X(99) lies on these lines:

1,741 2,111 3,76 4,114
6,729 13,303 14,302
20,147 21,105 22,305
30,316 31,715 32,194
36,350 38,745 39,83
58,727 69,74 75,261
81,739 86,106 95,311
100,668 101,190 102,332
103,1043 104,314 108,811
109,643 110,690 112,648
141,755 163,825 187,385
249,525 264,378 286,915
298,531 299,530 310,675
476,850 512,805 523,691
524,843 666,919 669,886
670,804 692,785 695,711
813,1016 889,898 935,3267

X(99) is the {X(39),X(384)}-harmonic conjugate of X(83). For a list of other harmonic conjugates of X(99), click **Tables** at the top of this page.

X(99) = midpoint of X(i) and X(j) for these (i,j): (20,147), (616,617)

X(99) = reflection of X(i) in X(j) for these (i,j): (4,114), (13,619), (14,618), (98,3), (115,620), (148,115), (316,325), (385,187), (671,2)

X(99) = isogonal conjugate of X(512)

X(99) = isotomic conjugate of X(523)

X(99) = complement of X(148)

X(99) = anticomplement of X(115)

X(99) = cevapoint of X(i) and X(j) for these (i,j): (2,523), (3,525), (39,512), (100,190)

X(99) = X(1019)-cross conjugate of X(1509)

X(99) = crossdifference of every pair of points on line X(351)X(865)

X(99) = X(i)-cross conjugate of X(j) for these (i,j): (3,249), (22,250), (512,83), (523,2), (525,76)

X(99) = X(21)-beth conjugate of X(741)

X(99) = X(6)-of-1st-anti-Brocard-triangle

X(99) = X(381)-of-anti-McCay-triangle

X(99) = circumcircle-antipode of X(98)

X(99) = point of intersection, other than A, B, and C, of the circumcircle and Steiner ellipse

X(99) = Ψ(X(i), X(j) for these (i,j): (1,75), (2,39), (3,69), (4,69), (37,2), (51,5), (351,690)

X(99) = point of intersection, other than A, B, C, of the circumcircle and hyperbola {{A,B,C,PU(1)}}

X(99) = point of intersection, other than A, B, C, of the circumcircle and hyperbola {{A,B,C,PU(37)}}

X(99) = trilinear product of PU(90)

X(99) = similitude center of (equilateral) antipedal triangles of X(13) and X(14)

X(99) = Steiner-circumellipse-antipode of X(671)

X(99) = projection from Steiner inellipse to Steiner circumellipse of X(2482)

X(99) = trilinear pole of line X(2)X(6)

X(99) = pole wrt polar circle of trilinear polar of X(2501) (line X(115)X(2971))

X(99) = X(48)-isoconjugate (polar conjugate) of X(2501)

X(99) = X(6)-isoconjugate of X(661)

X(99) = X(1577)-isoconjugate of X(32)

X(99) = concurrence of reflections in sides of ABC of line X(4)X(69)

X(99) = Λ(X(1), X(512))

X(99) = isotomic conjugate wrt 1st Brocard triangle of X(76)

X(99) = perspector of ABC and the tangential triangle, wrt the anticomplementary triangle, of the bianticevian conic of X(1) and X(2)

X(99) = perspector of ABC and the tangential triangle, wrt the tangential triangle, of the Stammler hyperbola

X(99) = reflection of X(691) in the Euler line

X(99) = reflection of X(805) in the Brocard axis

X(99) = reflection of X(2703) in line X(1)X(3)

X(99) = reflection of X(316) in the de Longchamps line

X(99) = X(130)-of-excentral-triangle

X(99) = X(129)-of-hexyl-triangle

X(99) = inverse-in-polar-circle of X(5139)

X(99) = inverse-in-{circumcircle, nine-point circle}-inverter of X(126)

X(99) = inverse-in-2nd-Brocard-circle of X(76)

X(99) = trilinear product of vertices of circumcircle antipode of circumorthic triangle

X(99) = 1st-Parry-to-ABC similarity image of X(2)

X(99) = crossdifference of PU(105)

X(99) = X(1691) of 6th Brocard triangle

X(99) = eigencenter of circummedial triangle

X(99) = eigencenter of circumsymmedial triangle

X(99) = perspector of ABC and cross-triangle of circumcevian triangles of PU(1)

X(99) = X(98)-of-anti-Artzt-triangle

X(99) = X(2)-of-1st-anti-Parry-triangle

X(99) = Thomson-isogonal conjugate of X(511)

X(99) = Lucas-isogonal conjugate of X(511)

X(99) = X(1379)-of-circummedial-triangle

X(99) = X(6323)-of-circumsymmedial-triangle

X(99) = Kiepert image of X(2)

X(99) = Cundy-Parry Phi transform of X(14265)

X(99) = intersection of antipedal lines of X(1379) and X(1380)

X(99) = homothetic center of anticomplementary triangle and mid-triangle of antipedal triangles of X(13) and X(14)

X(99) = barycentric square root of X(4590)

X(99) = barycentric product of circumcircle intercepts of line X(2)X(39)

X(99) = perspector of ABC and vertex triangle of 1st and 2nd isodynamic-Dao triangles

X(99) = orthic-to-ABC functional image of X(130)

X(99) = tangential-isogonal conjugate of X(33704)

X(99) = trilinear pole, wrt circumtangential triangle, of Brocard axis

Trilinears (a - b)(a - c) : (b - c)(b - a) : (c - a)(c - b)

Barycentrics a/(b - c) : b/(c - a) : c/(a - b)

Tripolars |b - c| : :

X(100) = 2R*X(1) - 3R*X(2) + 2r*X(3) (Peter Moses, April 2, 2013)

Let L_{A} be the reflection of the line X(1)X(3) in line BC, and define L_{B} and L_{C} cyclically. Let A' = L_{B}∩L_{C}, B' = L_{C}∩L_{A}, C' = L_{A}∩L_{B}. The lines AA', BB', CC' concur in X(100). (Randy Hutson, 9/23/2011)

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically; then X(100) = X(36)-of-IaIbIc. Also, let P be a point on line X(4)X(8) other than X(4). Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AB'C', BC'A', and CA'B' concur at X(100). (Randy Hutson, 9/5/2015)

Let IaIbIc be the excentral triangle. The Euler lines of triangles BCIa, CAIb, ABIc concur in X(100). (Randy Hutson, June 27, 2018)

X(100) lies on these lines:

1,88 2,11 3,8 4,119
6,739 7,1004 9,1005
10,21 20,153 22,197
25,1862 231,43 32,713 36,519
37,111 40,78 42,81
46,224 56,145 59,521
63,103 72,74 75,675
76,767 92,917 98,228
99,668 101,644 107,823
108,653 109,651 110,643
112,162 144,480 190,659
198,346 213,729 238,899
281,1013 329,972 442,943
484,758 513,765 516,908
517,953 518,840 522,655
560,697 594,1030 645,931
649,660 650,919 658,664
667,898 693,927 731,869
733,893 753,984 756,846
789,874 976,986 2859,3267

X(100) is the {X(10),X(35)}-harmonic conjugate of X(21). For a list of other harmonic conjugates of X(100), click **Tables** at the top of this page.

X(100) = midpoint of X(20) and X(153)

X(100) = reflection of X(i) in X(j) for these (i,j): (1,214), (4,119), (8,1145), (80,10), (104,3), (145,1317), (149,11), (962,1537), (1156,9), (1320,1), (1484,140)

X(100) = isogonal conjugate of X(513)

X(100) = isotomic conjugate of X(693)

X(100) = complement of X(149)

X(100) = anticomplement of X(11)

X(100) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,5375), (99,190)

X(100) = cevapoint of X(i) and X(j) for these (i,j): (1,513), (3,521), (10,522), (142,514), (442,523)

X(100) = X(i)-cross conjugate of X(j) for these (i,j): (3,59), (513,1), (521,8), (522,21)

X(100) = crosssum of X(i) and X(j) for these (i,j): (1,1054), (244,764), (512,661), (649,663)

X(100) = crossdifference of every pair of points on line X(244)X(665)

X(100) = circumcircle-antipode of X(104)

X(100) = Ψ(X(i),X(j)) for these (i,j): (1,2), 2,37), (3,63), (4,8), (6,1), (7,8), (48,3), (56,55), (68,72)

X(100) = X(1)-line conjugate of X(244)

X(100) = X(113)-of-the-hexyl-triangle

X(100) = concurrence of reflections in sides of ABC of line X(4)X(8)

X(100) = perspector of Hutson-Moses hyperbola

X(100) = trilinear pole of line X(1)X(6) (and PU(28)) (van Aubel line of excentral triangle)

X(100) = trilinear product of PU(33)

X(100) = trilinear product of intercepts of circumcircle and Nagel line

X(100) = the point of intersection, other than A, B, and C, of the circumcircle and the circumellipse centered at X(1) (viz., {{A,B,C,X(100),X(664),X(1120),X(1320)}})

X(100) = the point of intersection, other than A, B, and C, of the circumcircle and the circumellipse centered at X(9) (viz., {{A,B,C,X(100),X(658),X(662),X(799),X(1821),X(2580),X(2581),PU(34)}})

X(100) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,PU(8)}}

X(100) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,PU(32)}}

X(100) = Collings transform of X(1)

X(100) = Collings transform of X(9)

X(100) = center of hyperbola passing through X(1), X(9), and the excenters

X(100) = X(125)-of-excentral-triangle

X(100) = trilinear pole wrt 1st circumperp triangle of line X(3)X(142)

X(100) = X(110)-of=1st-circumperp-triangle

X(100) = reflection of X(1290) in the Euler line

X(100) = reflection of X(2703) in the Brocard axis

X(100) = reflection of X(901) in line X(1)X(3)

X(100) = cevapoint of X(59) and inverse-in-circumcircle-of-X(59)

X(100) = X(i)-isoconjugate of X(j) for these (i,j): (6,514), (63,6591), (1333,1577)

X(100) = inverse-in-{circumcircle, nine-point circle}-inverter of X(120)

X(100) = exsimilicenter of circumcircle and AC-incircle

X(100) = X(i)-aleph conjugate of X(j) for these (i,j) (1,1052), (100,1), (190,63), (643,411), (666,673), (765,100), (1016,190)

X(100) = X(i)-beth conjugate of X(j) for these (i,j): (8,80), (21,106), (100,109), (333,673), (643,100), (765,100)

X(100) = the point of intersection, other than A, B, and C, of the circumcircle and ellipse {{A,B,C,PU(75)}}

X(100) = crossdifference of PU(27)

X(100) = Thomson-isogonal conjugate of X(517)

X(100) = Lucas-isogonal conjugate of X(517)

X(100) = homothetic center of 2nd Schiffler triangle and excenters-midpoints triangle

X(100) = Feuerbach image of X(2)

X(100) = Cundy-Parry Phi transform of X(14266)

X(100) = perspector of anti-Mandart-incircle and anticomplementary triangles

X(100) = intersection of antipedal lines of X(1381) and X(1382)

X(100) = eigencenter of Gemini triangle 2

X(100) = barycentric product of vertices of Gemini triangle 5

X(100) = barycentric product of vertices of Gemini triangle 6

X(100) = perspector of ABC and side-triangle of Gemini triangles 29 and 30

X(100) = homothetic center of 2nd Schiffler triangle and excenters-midpoints triangle

X(100) = barycentric product of vertices of Gemini triangle 29

X(100) = barycentric product of vertices of Gemini triangle 30

X(100) = intersection, other than A, B, C, of {ABC, Gemini 29}-circumconic and {ABC, Gemini 30}-circumconic

X(100) = barycentric product of circumcircle intercepts of line X(2)X(37)

X(100) = excentral-to-ABC barycentric image of X(1768)

X(100) = intouch-to-ABC barycentric image of X(11)

X(100) = ABC-to-excentral barycentric image of X(11)

X(100) = inverse-in-Stevanovic-circle of X(919)

X(100) = trilinear pole, wrt circumtangential triangle, of line X(1)X(3)

X(100) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(20344)

X(100) = de-Longchamps-circle-inverse of X(34188)

X(100) = BSS(a^2->a) of X(110)

Trilinears a(a - b)(a - c) : b(b - c)(b - a) : c(c - a)(c - b)

Barycentrics a

Barycentrics a^2 (sin^2 A + sin B sin C - sin C sin A - sin A sin B) : :

Tripolars |b c (b - c)| : :

Let L_{A} be the reflection of the line X(1)X(7) in line BC,
and define L_{B} and L_{C} cyclically. Let A' =
L_{B}∩L_{C}, B' = L_{C}∩L_{A},
C' = L_{A}∩L_{B}. The lines AA', BB', CC' concur in
X(101). (Randy Hutson, 9/23/2011)

Let I_{a}I_{b}I_{c} be the excentral triangle. The Brocard axes of BCI_{a}, CAI_{b}, ABI_{c} concur in X(101). (Randy Hutson, February 10, 2016)

Let P be a point on line X(4)X(9) other than X(4). Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AA'B', BC'A', CA'B' concur in X(101). (Randy Hutson, February 10, 2016)

Let Q be a point on the Nagel line other than X(2). Let A'B'C' be the cevian triangle of Q. Let A" be the {B,C}-harmonic conjugate of A' (or equivalently, A" = BC∩B'C'), and define B" and C" cyclically. The circumcircles of AB"C", BC"A", CA"B" concur in X(101). (Randy Hutson, February 10, 2016)

Let A', B', C' be the intersections of the antiorthic axis and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(101). (Randy Hutson, February 10, 2016)

X(101) is the perspector of the anticevian triangle of X(109) and the unary cofactor triangle of the intangents triangle. (Randy Hutson, January 29, 2018)

X(101) lies on these lines:

1,41 2,116 3,103 4,118
6,106 9,48 10,98 19,913
20,152 31,609 32,595
36,672 37,284 40,972
42,111 56,218 58,172
59,657 71,74 75,767
78,205 99,190 100,644
102,198 109,654 110,163
514,664 517,910 522,929
560,713 643,931 649,901
651,934 663,919 667,813
668,789 692,926 733,904
743,869 761,984 765,898

X(101) = midpoint of X(20) and X(152)

X(101) = reflection of X(i) in X(j) for these (i,j): (4,118), (103,3), (150,116)

X(101) = isogonal conjugate of X(514)

X(101) = isotomic conjugate of X(3261)

X(101) = complement of X(150)

X(101) = anticomplement of X(116)

X(101) = X(59)-Ceva conjugate of X(55)

X(101) = cevapoint of X(354) and X(513)

X(101) = X(i)-cross conjugate of X(j) for these (i,j): (55,59), (199,250)

X(101) = crosssum of X(i) and X(j) for these (i,j): (513,650), (523,661), (649,1459)

X(101) = crossdifference of every pair of points on line X(11)X(244)

X(101) = X(i)-aleph conjugate of X(j) for these (i,j): (100,165), (509,1052), (662,572), (664,169)

X(101) = X(i)-beth conjugate of X(j) for these (i,j): (21,105), (644,644)

X(101) = circumcircle-antipode of X(103)

X(101) = Ψ(X(i),X(j)) for these (i,j): (1,6), (2,1), (3,48), (4,9), (6,31), (7,2), (8,9), (9,55), (57,56), (63,3), (69,63), (76,10)

X(101) = X(114)-of-the-hexyl-triangle

X(101) = trilinear product of PU(i) for these i: 26, 49

X(101) = barycentric product of PU(33)

X(101) = the point of intersection, other than A, B, C, of the circumcircle and hyperbola {{A,B,C,PU(9)}}

X(101) = the point of intersection, other than A, B, C, of conic {{A,B,C,X(1),PU(93)}}

X(101) = trilinear pole of line X(6)X(31) (the isogonal conjugate of the isotomic conjugate of the Nagel line)

X(101) = trilinear pole wrt 1st circumperp triangle of line X(9)X(165)

X(101) = X(99)-of -1st-circumperp-triangle

X(101) = crossdifference of PU(i) for these i: 121, 123

X(101) = concurrence of reflections of line X(4)X(9) in sides of ABC

X(101) = isogonal conjugate of isotomic conjugate of trilinear pole of Nagel line

X(101) = center of Kiepert hyperbola of excentral triangle (i.e. X(115) of excentral triangle)

X(101) = reflection of X(2690) in the Euler line

X(101) = reflection of X(2702) in the Brocard axis

X(101) = reflection of X(1308) in line X(1)X(3)

X(101) = reflection of X(5011) in antiorthic axis

X(101) = inverse-in-polar-circle of X(5190)

X(101) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5513)

X(101) = X(6)-isoconjugate of X(693)

X(101) = X(92)-isoconjugate of X(1459)

X(101) = X(1577)-isoconjugate of X(58)

X(101) = eigencenter of 2nd circumperp triangle

X(101) = perspector of 3rd mixtilinear triangle and unary cofactor triangle of 5th mixtilinear triangle

X(101) = trilinear product of vertices of 1st circumperp triangle

X(101) = Thomson-isogonal conjugate of X(516)

X(101) = Lucas-isogonal conjugate of X(516)

X(101) = intersection of Lemoine axes of 1st & 2nd Montesdeoca bisector triangles

X(101) = focus of Yff parabola

X(101) = polar conjugate of isogonal conjugate of X(32656)

X(101) = trilinear product X(6)*X(100)

X(101) = trilinear product of circumcircle intercepts of line X(1)X(6)

X(101) = barycentric product of circumcircle intercepts of the Nagel line

X(101) = inverse-in-Stevanovic-circle of X(2222)

Trilinears a/[2a

Barycentrics (sin A)/[sin B (sec A - sec B) + sin C (sec A - sec C)] : :

Tripolars |sin B (sec A - sec B) + sin C (sec A - sec C)| : :

X(102) lies on these lines:

1,108 2,117 3,109 4,124
19,282 29,107 40,78
73,947 77,934 99,332
101,198 103,928 110,283
112,284 226,1065 516,929

X(102) = midpoint of X(20) and X(153)

X(102) = reflection of X(i) in X(j) for these (i,j): (4,124), (109,3), (151,117)

X(102) = isogonal conjugate of X(515)

X(102) = complement of X(151)

X(102) = anticomplement of X(117)

X(102) = X(21)-beth conjugate of X(108)

X(102) = circumcircle-antipode of X(109)

X(102) = Λ(X(1), X(4))

X(102) = trilinear pole of line X(6)X(652)

X(102) = Ψ(X(6), X(652))

X(102) = Ψ(X(1), X(521))

X(102) = Ψ(X(108), X(1))

X(102) = Ψ(X(651), X(63))

X(102) = Ψ(X(653), X(2))

X(102) = trilinear product of circumcircle intercepts of line X(1)X(521)

X(102) = trilinear pole wrt 2nd circumperp triangle of line X(971)X(1001)

X(102) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,X(4),X(58)}}

X(102) = reflection of X(2695) in the Euler line

X(102) = reflection of X(2708) in the Brocard axis

X(102) = reflection of X(2716) in line X(1)X(3)

X(102) = X(131)-of-excentral-triangle

X(102) = X(136)-of-hexyl-triangle

X(102) = X(925)-of-2nd-circumperp-triangle

X(102) = Thomson-isogonal conjugate of X(522)

X(102) = Lucas-isogonal conjugate of X(522)

X(102) = Cundy-Parry Phi transform of X(10571)

X(102) = Cundy-Parry Psi transform of X(10570)

X(102) = Ψ(X(19), X(650))

Barycentrics a

Barycentrics a^2 / (2 a^3 - a^2 (b + c) - (b - c)^2 (b + c)) : :

Tripolars |b c ((a - b) cot C + (a - c) cot B)| : :

Let A'B'C' be the excentral triangle. The Lemoine axes of triangles A'BC, B'CA, C'AB bound a triangle perspective to ABC at X(103). (Randy Hutson, June 27, 2018)

X(103) lies on these lines:

1,934 2,118 3,101 4,116
20,150 27,107 33,57
55,109 58,112 63,100
99,1043 102,928 295,813
376,544 515,929 516,927
572,825 672,919 910,971

X(103) = midpoint of X(20) and X(150)

X(103) = reflection of X(i) in X(j) for these (i,j): (4,116), (101,3), (152,118)

X(103) = isogonal conjugate of X(516)

X(103) = complement of X(152)

X(103) = anticomplement of X(118)

X(103) = X(21)-beth conjugate of X(934)

X(103) = X(115)-of-the-hexyl-triangle

X(103) = perspector of ABC and the triangle formed by reflecting line PU(10) in the sidelines of ABC

X(103) = X(114)-of-excentral-triangle

X(103) = trilinear pole of line X(6)X(657)

X(103) = Ψ(X(i),X(j)) for these (i,j): (6,657), (101,3), (190,69)

X(103) = Λ(X(1), X(7))

X(103) = trilinear pole wrt 2nd circumperp triangle of line X(1001)X(1012)

X(103) = X(99)-of-2nd-circumperp-triangle

X(103) = reflection of X(2688) in the Euler line

X(103) = reflection of X(2700) in the Brocard axis

X(103) = reflection of X(2717) in line X(1)X(3)

X(103) = Thomson-isogonal conjugate of X(514)

X(103) = Lucas-isogonal conjugate of X(514)

X(103) = Cundy-Parry Phi transform of X(3730)

X(103) = Cundy-Parry Psi transform of X(14377)

X(103) = polar conjugate of isogonal conjugate of X(32657)

X(103) = SR(P,U), where P and U are the circumcircle intercepts of the Soddy line

Trilinears 1/[b^3 + c^3 - (a^2 + bc)(b + c) + 2abc] : :

Barycentrics a/(-1 + cos B + cos C) : b/(-1 + cos C + cos A) : c/(-1 + cos C + cos B)

Tripolars |-1 + cos B + cos C| : :

X(104) = 2R*X(1) - 3R*X(2) + (2R - 2r)*X(3) (Peter Moses, April 2, 2013)

Let L_{A} be the reflection of the line X(1)X(513) in line BC, and define L_{B} and L_{C} cyclically. Let A' = L_{B}∩L_{C}, B' = L_{C}∩L_{A}, C' = L_{A}∩L_{B}. The lines AA', BB', CC' concur in X(104). (Randy Hutson, 9/23/2011)

Let A', B', C' be the intersections of the antiorthic axis and lines BC, CA, AB, resp. Let Oa, Ob, Oc be the circumcenters of AB'C', BC'A', CA'B', resp. The lines AOa, BOb, COc conucr in X(104). (Randy Hutson, December 2, 2017)

Let A'B'C' be the excentral triangle (i.e. the antiorthic triangle). Let A"B"C" be the triangle bounded by the orthic axes of A'BC, B'CA, C'AB. Then A"B"C" is perspective to ABC at X(104); c.f. X(8068). (Randy Hutson, December 2, 2017)

Let A'B'C' be the excentral triangle. The de Longchamps lines of triangles A'BC, B'CA, C'AB bound a triangle perspective to ABC at X(104). (Randy Hutson, June 27, 2018)

Let A'B'C' be the excentral triangle. The Hatzipolakis axes of triangles A'BC, B'CA, C'AB bound a triangle perspective to ABC at X(104). (Randy Hutson, June 27, 2018)

X(104) lies on these lines:

1,109 2,119 3,8 4,11
7,934 9,48 20,149
21,110 28,107 36,80
55,1000 79,946 99,314
105,885 112,1108 256,1064
294,919 355,404 376,528
513,953 517,901 631,958

X(104) = midpoint of X(20) and X(149)

X(104) = reflection of X(i) in X(j) for these (i,j): (4,11), (100,3), (153,119), (1537,1387)

X(104) = isogonal conjugate of X(517)

X(104) = isotomic conjugate of X(3262)

X(104) = complement of X(153)

X(104) = anticomplement of X(119)

X(104) = cevapoint of X(i) and X(j) for these (i,j): (1,36), (44,55)

X(104) = X(21)-beth conjugate of X(109)

X(104) = circumcircle-antipode of X(100)

X(104) = point of intersection, other than A, B, and C, of the circumcircle and Feuerbach hyperbola

X(104) = Λ(X(1), X(3))

X(104) = Ψ(X(101), X(9))

X(104) = X(125)-of-the-hexyl-triangle

X(104) = X(113)-of-excentral-triangle

X(104) = X(110)-of-2nd-circumperp-triangle

X(104) = trilinear pole of line X(6)X(650)

X(104) = Ψ(X(6), X(650))

X(104) = Ψ(X(190), X(63))

X(104) = Ψ(X(1), X(522))

X(104) = Ψ(X(19), X(649))

X(104) = Ψ(X(109), X(1))

X(104) = Ψ(X(651), X(2))

X(104) = Feuerbach hyperbola antipode of X(4)

X(104) = trilinear pole wrt 2nd circumperp triangle of line X(3)X(142)

X(104) = reflection of X(2687) in the Euler line

X(104) = reflection of X(2699) in the Brocard axis

X(104) = reflection of X(953) in line X(1)X(3)

X(104) = crossdifference of every pair of points on line X(1769)X(3310)

X(104) = Thomson-isogonal conjugate of X(513)

X(104) = Lucas-isogonal conjugate of X(513)

X(104) = Cundy-Parry Phi transform of X(8)

X(104) = Cundy-Parry Psi transform of X(56)

X(104) = trilinear product of circumcircle intercepts of line X(1)X(522)

X(104) = trilinear pole, wrt Thomson triangle, of line X(40)X(392)

Barycentrics a/[b

Tripolars |b^2 + c^2 - a (b + c)| : :

X(105) is the perspector of ABC and the (degenerate) side-triangle of the circumcevian triangles of X(3513) and X(3514). (Randy Hutson, June 7, 2019)

X(105) lies on these lines:

1,41 2,11 3,277 6,1002
21,99 25,108 28,112
31,57 56,279 81,110
88,901 104,885 106,1022
165,1054 238,291 330,932
513,840 644,1083 659,884
666,898 825,985 910,919
961,1104

X(105) = reflection of X(i) in X(j) for these (i,j): (644,1083), (1292,3)

X(105) = isogonal conjugate of X(518)

X(105) = anticomplement of X(120)

X(105) = cevapoint of X(1) and X(238)

X(105) = X(1)-Hirst inverse of X(294)

X(105) = X(i)-beth conjugate of X(j) for these (i,j): (21,101), (927,105)

X(105) = Λ(X(1), X(6))

X(105) = isotomic conjugate of X(3263)

X(105) = crossdifference of every pair of points on line X(665)X(1642)

X(105) = Ψ(X(i), X(j)) for these (i,j): (1,514), (2,650), (6,513), (57,649), (59,651), (100,2), (101,1), (190,9), (650,11), (651,6)

X(105) = reflection of X(2752) in the Euler line

X(105) = reflection of X(2711) in the Brocard axis

X(105) = reflection of X(840) in line X(1)X(3)

X(105) = X(132)-of-excentral-triangle

X(105) = X(127)-of-hexyl-triangle

X(105) = X(6)-isoconjugate of X(3912)

X(105) = inverse-in-{circumcircle, nine-point circle}-inverter of X(11)

X(105) = trilinear pole of PU(i) for these i: 46, 54

X(105) = trilinear product of PU(96)

X(105) = bicentric sum of PU(142)

X(105) = Thomson-isogonal conjugate of X(3309)

X(105) = Lucas-isogonal conjugate of X(3309)

X(105) = polar conjugate of isogonal conjugate of X(32658)

X(105) = trilinear product of circumcircle intercepts of line X(1)X(514)

X(105) = trilinear pole, wrt Thomson triangle, of line X(354)X(612)

Barycentrics a

Tripolars |b c (2 a - b - c)| : :

X(106) lies on these lines:

1,88 2,121 3,1293 6,101 34,108
36,901 56,109 58,110
86,99 87,932 105,1022
238,898 269,934 292,813
614,998 663,840 789,870
833,977 919,1055

X(106) = reflection of X(1293) in X(3)

X(106) = isogonal conjugate of X(519)

X(106) = isotomic conjugate of X(3264)

X(106) = complement of X(21290)

X(106) = anticomplement of X(121)

X(106) = X(36)-cross conjugate of X(58)

X(106) = X(i)-beth conjugate of X(j) for these (i,j): (21,100), (901,106)

X(106) = Λ(X(1), X(2))

X(106) = X(122)-of-hexyl-triangle

X(106) = trilinear pole of line X(6)X(649)

X(106) = Ψ(X(i), X(j)) for these (i,j): (1,513), (2,514), (6,649), (9,650), (100,1), (101,6), (190,2), (651,57)

X(106) = trilinear pole wrt 2nd circumperp triangle of line X(1)X(6)

X(106) = X(107) of 2nd circumperp triangle

X(106) = trilinear pole wrt circumsymmedial triangle of line X(6)X(31)

X(106) = reflection of X(2758) in the Euler line

X(106) = reflection of X(2712) in the Brocard axis

X(106) = reflection of X(2718) in line X(1)X(3)

X(106) = X(6)-isoconjugate of X(4358)

X(106) = X(133)-of-excentral triangle

X(106) = barycentric product of PU(50)

X(106) = trilinear product of PU(98)

X(106) = eigencenter of 1st circumperp triangle

X(106) = Thomson-isogonal conjugate of X(3667)

X(106) = Lucas-isogonal conjugate of X(3667)

X(106) = polar conjugate of isogonal conjugate of X(32659)

X(106) = trilinear product of circumcircle intercepts of line X(1)X(513)

X(106) = trilinear pole, wrt Thomson triangle, of line X(1201)X(2177)

Trilinears (sec A)/(tan B - tan C) : :

Trilinears bc/[(b

Barycentrics 1/[(b

Tripolars |(cos A)(tan B - tan C)| : :

X(107) = center of the bianticevian conic of X(1) and X(4), the rectangular hyperbola passing through X(1), X(4), X(19), and the vertices of their anticevian triangles. This hyperbola is the excentral isogonal conjugate of line X(40)X(2939), the anticomplementary conjugate of line X(20)X(1330), and the anticomplementary isotomic conjugate of line X(1654)X(3164). (Randy Hutson, April 9, 2016)

X(107) lies on these lines:

2,122 3,1294 4,74 19,2249 20,3184 21,1295 23,2697 24,1093 25,98 27,103 28,104 29,102 51,275 100,823 109,162 110,648 111,393 158,759 186,477 250,687 450,511 468,842 741,1096 20,3184 21,1295 23,2697

X(107) = reflection of X(i) in X(j) for these (i,j): (4,133), (1294,3)

X(107) = isogonal conjugate of X(520)

X(107) = isotomic conjugate of X(3265)

X(107) = complement of X(34186)

X(107) = anticomplement of X(122)

X(107) = cevapoint of X(4) and X(523)

X(107) = cevapoint of X(24007) and X(24008) (the Kiepert hyperbola intercepts of the orthic axis)

X(107) = X(i)-cross conjugate of X(j) for these (i,j): (24,250), (108,162), (523,4)

X(107) = trilinear pole of line X(4)X(6)

X(107) = Ψ(X(i),X(j)) for these (i,j): (1,29), (3,2), (6,4), (4,51), (54,4), (64,4), (65,4), (67,4), (69,4)

X(107) = intersection of reflections in sides of ABC of line X(4)X(51)

X(107) = reflection of X(1304) in the Euler line

X(107) = reflection of X(2713) in the Brocard axis

X(107) = reflection of X(2719) in line X(1)X(3)

X(107) = inverse-in-polar-circle of X(125)

X(107) = inverse-in-{circumcircle, nine-point circle}-inverter of X(132)

X(107) = pole wrt polar circle of trilinear polar of X(525) (line X(122)X(125))

X(107) = X(48)-isoconjugate (polar conjugate) of X(525)

X(107) = X(1577)-isoconjugate of X(577)

X(107) = crossdifference of every pair of points on line X(1636)X(2972)

X(107) = X(134)-of-excentral-triangle

X(107) = circumcircle intercept, other than A, B, C, of conic {{A,B,C,PU(157)}}

X(107) = Thomson-isogonal conjugate of X(6000)

X(107) = Lucas-isogonal conjugate of X(6000)

X(107) = barycentric product of Steiner circumellipse intercepts of van Aubel line

X(107) = trilinear product of circumcircle intercepts of line X(1)X(29)

= g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = 1/[(b - c)(b + c - a)(b

Barycentrics a

Tripolars |b c (sec B - sec C)| : :

X(108) lies on these lines:

1,102 2,123 3,1295 4,11 7,1013
12,451 24,915 25,105
28,225 33,57 34,106
40,207 55,196 65,74
99,811 100,653 109,1020
110,162 204,223 273,675
318,404 331,767 388,406
429,961 608,739 648,931

X(108) = reflection of X(1295) in X(3)

X(108) = isogonal conjugate of X(521)

X(108) = complement of X(34188)

X(108) = anticomplement of X(123)

X(108) = X(162)-Ceva conjugate of X(109)

X(108) = cevapoint of X(i) and X(j) for these (i,j): (56,513), (429,523)

X(108) = X(513)-cross conjugate of X(4)

X(108) = crosspoint of X(107) and X(162)

X(108) = crosssum of X(520) and X(656)

X(108) = X(i)-beth conjugate of X(j) for these (i,j): (21,102), (162,108)

X(108) = point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,PU(18)}}

X(108) = trilinear pole of line X(6)X(19) (the polar of X(4391) wrt polar circle)

X(108) = pole wrt polar circle of trilinear polar of X(4391) (line X(11)X(123))

X(108) = X(48)-isoconjugate (polar conjugate) of X(4391)

X(108) = X(1577)-isoconjugate of X(2193)

X(108) = concurrence of the reflections of line X(4)X(65) in the sidelines of ABC

X(108) = Ψ(X(i),Xj)) for these (i,j): (1,4), (3,1), (4,65), (6,19), (7,4), (8,4), (9,4), (29,1), (69,7), (80,4)

X(108) = reflection of X(2766) in the Euler line

X(108) = reflection of X(2714) in the Brocard axis

X(108) = reflection of X(2720) in line X(1)X(3)

X(108) = inverse-in-polar-circle of X(11)

X(108) = X(135)-of-excentral-triangle

X(108) = barycentric product of PU(76)

X(108) = trilinear product of PU(100)

X(108) = Thomson-isogonal conjugate of X(6001)

X(108) = Lucas-isogonal conjugate of X(6001)

X(108) = trilinear product of circumcircle intercepts of line X(1)X(4)

X(108) = barycentric product of circumcircle intercepts of line X(2)X(92)

Trilinears a/[(b - c)(b + c - a)] : :

Barycentrics a

Barycentrics a^2 (cos^2 A + cos B cos C - cos C cos A - cos A cos B) : :

Tripolars |b c (cos B - cos C)| : :

If the line X(1)X(4) is reflected in every side of triangle ABC, then the reflections concur in X(109). (Randy Hutson, 9/23/2011)

Let P be a point on line X(1)X(4) other than X(4). Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of CA'B', BC'A', and CA'B' concur in X(109). (Randy Hutson, December 26, 2011)

Let Q be a point on line X(2)X(7) other than X(2). Let A'B'C' be the cevian triangle of Q. Let A" be the {B,C}-harmonic conjugate of A' (or equivalently, A" = BC∩B'C'), and define B" and C" cyclically. The circumcircles of AB"C", BC"A", CA"B" concur in X(109). (Randy Hutson, December 26, 2011)

Let A', B', C' be the Fuhrmann triangle. The circumcircles of AB'C', BC'A', CA'B' concur in X(109). (Randy Hutson, December 26, 2011)

Let A', B', C' be the intersections of the Gergonne line and lines BC, CA, AB, respectively. The circumcircles of AB'C', BC'A', CA'B' concur in X(109). (Randy Hutson, December 26, 2011)

Let P and Q be the points where the line tangent to the incircle at X(11) intersects the circumcircle. Let L(P) be the line through P, other than PQ, that is tangent to the incircle; let L(Q) be the line through Q, other than PQ, that is tangent to the incircle. Then X(109) = L(P)∩L(Q). (Piotr Ambroszczyk, December 21, 2016)

X(109) lies on these lines:

1,104 2,124 3,102 4,117
7,675 20,151 31,57
34,46 35,73 36,953
40,255 55,103 56,106
58,65 59,901 85,767
98,171 99,643 100,651
101,654 107,162 108,1020
112,163 165,212 191,201
278,917 284,296 478,573
579,608 604,739 649,919
658,927 662,931 840,902

X(109) = midpoint of X(20) and X(151)

X(109) = reflection of X(i) in X(j) for these (i,j): (4,117), (102,3)

X(109) = isogonal conjugate of X(522)

X(109) = complement of X(33650)

X(109) = anticomplement of X(124)

X(109) = X(i)-Ceva conjugate of X(j) for these (i,j): (59,56), (162,108)

X(109) = cevapoint of X(65) and X(513)

X(109) = X(i)-cross conjugate of X(j) for these (i,j): (56,59), (513,58)

X(109) = crosspoint of X(110) and X(162)

X(109) = crosssum of X(i) and X(j) for these (i,j): (523,656), (652,663)

X(109) = crossdifference of every pair of points on line X(11)X(1146)

X(109) = X(i)-aleph conjugate of X(j) for these (i,j): (100,1079), (162,580), (651,223)

X(109) = X(i)-beth conjugate of X(j) for these (i,j): (21,104), (59,109), (100,100), (110,109), (765,109), (901,109)

X(109) = circumcircle-antipode of X(102)

X(109) = trilinear product X(1381)*X(1382)

X(109) = isogonal conjugate of isotomic conjugate of trilinear pole of line X(2)X(7)

X(109) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,PU(19)}

X(109) = trilinear pole of line X(6)X(41)

X(109) = trilinear pole wrt 1st circumperp triangle of line X(971)X(1158)

X(109) = trilinear pole, wrt circumtangential triangle, of line X(3)X(10)

X(109) = X(925)-of-1st-circumperp-triangle

X(109) = X(136)-of-excentral-triangle

X(109) = X(131)-of-hexyl-triangle

X(109) = Ψ(X(i),X(j)) for these (i,j): (1,3), (2,7), (3,73), (4,1), (6,41), (7,20), (21,2), (69,73), (77,3)

X(109) = reflection of X(2689) in the Euler line

X(109) = reflection of X(2701) in the Brocard axis

X(109) = reflection of X(2222) in line X(1)X(3)

X(109) = X(6)-isoconjugate of X(4391)

X(109) = X(92)-isoconjugate of X(652)

X(109) = X(1577)-isoconjugate of X(284)

X(109) = barycentric product of PU(57)

X(109) = trilinear product of PU(102)

X(109) = Thomson isogonal conjugate of X(515)

X(109) = Lucas isogonal conjugate of X(515)

X(109) = isotomic conjugate of polar conjugate of X(32674)

X(109) = polar conjugate of isogonal conjugate of X(32660)

X(109) = intersection of antipedal lines of circumcircle intercepts of line X(3)X(10)

X(109) = barycentric product of circumcircle intercepts of line X(2)X(7)

Trilinears cos(B - C) - cos(C - A)cos(A - B) : :

Trilinears a/(b

Barycentrics a

Tripolars |sin(B - C)| : :

Tripolars |b c (b^2 - c^2)| : :

X(110) is the center of the Stammler hyperbola, SH, which is the rectangular hyperbola that passes through X(1), X(3), X(6), X(155), X(159), X(195), X(399), X(1498), X(2916), X(2917), X(2918), X(2929), X(2930), X(2931), X(2935), X(2948), X(3511), the excenters, and the vertices of the tangential triangle. SH is the bianticevian conic of X(1) and X(6) and the antipedal-anticevian conic of X(3). SH is the tangential isogonal conjugate of the Euler line, the tangential isotomic conjugate of the van Aubel line, the excentral isogonal conjugate of line X(30)X(40), and the excentral isotomic conjugate of line X(191)X(2938). SH is tangent to the Euler line at X(3) and meets the line at infinity (and the Jerabek hyperbola, other than at X(3) and X(6)) at X(2574) and X(2575). SH is the locus of a point P for which the P-Brocard triangle is perspective to ABC. (Randy Hutson, 9/23/2011, 1/29/2015)

**J. W. Clawson,** "Points on the circumcircle," *American Mathematical Monthly* 32 (1925) 169-174.

**Roland H. Eddy and R. Fritsch,** "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," *Mathematics Magazine* 67 (1994) 188-205.

**Benedetto Scimemi,** "Paper-folding and Euler's Theorem Revisited," *Forum Geometricorum*.

Scimemi proves that if the Euler line is reflected in every side of triangle ABC, then the three reflections concur in X(110).

Seven constructions from Randy Hutson, January 29, 2015:

(1) Let P be a point, other than X(4), on Euler line. Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AA'B', BC'A', and CA'B' concur in X(110).

(2) Let Q be a point on line X(2)X(6) other than X(2). Let A'B'C' be the cevian triangle of Q. Let A" be the {B,C}-harmonic conjugate of A' (or equivalently, A" = BC∩B'C'), and define B", C" cyclically. The circumcircles of AB"C", BC"A", CA"B" concur in X(110).

(3) Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Euler line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B', C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Euler line. The triangle A"B"C" is homothetic to ABC, with center of homothety X(125) and centroid X(110). (See Hyacinthos #16741/16782, Sep 2008.)

(4) Let A', B', C' be the intersections of the Lemoine axis and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(110).

(5) Let A', B', C' be the intersections of line X(36)X(238) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(110).

(6) Let A', B', C' be the intersections of the Euler line and lines BC, CA, AB, resp. Let Oa, Ob, Oc be the circumcenters of AB'C', BC'A', CA'B', resp. The lines AOa, BOb, COc concur in X(110).

(7) Let Na be the reflection of X(5) in the perpendicular bisector of BC, and define Nb, Nc cyclically. X(110) = X(2070) of NaNbNc.

Let A2B2C2 and A3B3C3 be the 2nd and 3rd Parry triangles. Let A' be the barycentric product A2*A3, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(110). (Randy Hutson, February 10, 2016)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the anticomplementary triangle at X(110). The nine-point circle of A'B'C' touches the circumcircle of ABC at X(110). Also, X(110) = X(125)-of-A'B'C'. (Randy Hutson, Novermber 2, 2016)

X(110): Let A' be the trilinear pole of the perpendicular bisector of BC, and define B', C' cyclically. A'B'C' is homothetic to the circumanticevian triangle of X(4) at X(110). X(110) is also the barycentric product A'*B'*C'. (Randy Hutson, January 29, 2018)

X(110) lies on the circumcircle, the Parry circle, and these lines:

1,60 2,98 3,74 4,113 5,49 6,111 11,215
20,146 21,104 22,154
23,323 24,155 27,917
28,915 30,477 31,593
32,729 39,755 58,106
65,229 67,141 69,206
81,105 86,675 97,418
99,690 100,643 101,163
102,283 107,648 108,162
143,195 187,352 190,835
249,512 250,520 251,694
274,767 324,436 351,526
353,574 373,575 376,541
476,523 525,935 560,715
595,849 668,839 669,805
670,689 681,823 685,850
789,799 859,953 2868,3266

X(110) is the {X(5),X(49)}-harmonic conjugate of X(54). For a list of other harmonic conjugates of X(110), click **Tables** at the top of this page.

X(110) = midpoint of X(i) and X(j) for these (i,j): (3,399), (20,146), (23,323), (1495,3292)

X(110) = reflection of X(i) in X(j) for these (i,j): (3,1511), (4,113), (23,1495), (67,141), (74,3), (265,5), (382,1539), (895,6), (1177,206)

X(110) = circumcircle-antipode of X(74)

X(110) = isogonal conjugate of X(523)

X(110) = isotomic conjugate of X(850)

X(110) = isogonal conjugate of the isotomic conjugate of X(99)

X(110) = inverse of X(2) in the Brocard circle

X(110) = complement of X(3448)

X(110) = anticomplement of X(125)

X(110) = X(i)-Ceva conjugate of X(j) for these (i,j): (249,6), (250,3)

X(110) = cevapoint of X(i) and X(j) for these (i,j): (3,520), (5,523), (6,512), (141,525)

X(110) = crossdifference of every pair of points on line X(115)X(125)

X(110) = X(i)-Hirst inverse of X(j) for these (i,j): (1,245), (2,125), (3,246), (4,247)

X(110) = X(i)-beth conjugate of X(j) for these (i,j): (21,759), (643,643)

X(110) = X(23)-of-1st-Brocard triangle

X(110) = X(111)-of-circumsymmedial-triangle

X(110) = X(323)-of-orthocentroidal-triangle

X(110) = X(137)-of-excentral-triangle

X(110) = X(128)-of-hexyl-triangle

X(110) = trilinear pole of the Brocard axis

X(110) = trilinear pole of PU(29) (see ETC->Tables->Bicentric Pairs)

X(110) = perspector of ABC and vertex-triangle of anticevian triangles of X(3) and X(6)

X(110) = Johnson-circumconic antipode of X(265)

X(110) = MacBeath-circumconic antipode of X(895)

X(110) = perspector of conic {A,B,C,PU(2)}

X(110) = intersection of trilinear polars of P(2) and U(2)

X(110) = intersection of tangents to Steiner circumellipse at X(99) and X(648)

X(110) = crosspoint of X(99) and X(648)

X(110) = reflection of X(476) in the Euler line

X(110) = reflection of X(691) in the Brocard axis

X(110) = reflection of X(23) in the Lemoine axis

X(110) = reflection of X(1290) in line X(1)X(3)

X(110) = reflection of X(111) in line X(3)X(351)

X(110) = inverse-in-polar-circle of X(136)

X(110) = inverse-in-{circumcircle, nine-point circle}-inverter of X(114)

X(110) = inverse-in-Moses-radical-circle of X(2715)

X(110) = inverse-in-O(15,16) of X(843), where O(15,16) is the circle having segment X(15)X(16) as diameter

X(110) = X(i)-isoconjugate of X(j) for these (i,j): (6,1577), (92,647), (1577,6)

X(110) = perspector of circumorthic triangle and Johnson triangle (the reflection triangles of X(4) and X(3), resp.)

X(110) = trilinear product of vertices of circumtangential triangle

X(110) = {X(3),X(156)}-harmonic conjugate of X(1614)

X(110) = orthocentroidal-to-ABC similarity image of X(2)

X(110) = 4th-Brocard-to-circumsymmedial similarity image of X(2)

X(110) = isogonal conjugate of isotomic conjugate of trilinear pole of line X(2)X(6)

X(110) = the point of intersection, other than A, B, C, of the circumcircle and Johnson circumconic

X(110) = the point of intersection, other than A, B, C, of the circumcircle and MacBeath circumconic

X(110) = the point of intersection, other than A, B, C, of the circumcircle and circumconic {{A,B,C,PU(5)}}

X(110) = Collings transform of X(5)

X(110) = Collings transform of X(6)

X(110) = intersection of tangents at X(61) and X(62) to the Napoleon-Feuerbach cubic K005

X(110) = SR(PU(4))

X(110) = insimilicenter of nine-point circle and sine-triple-angle circle

X(110) = insimilicenter of circumcircle and nine-point circle of tangential triangle; the exsimilicenter is X(1614)

X(110) = X(7972)-of-Trinh-triangle

X(110) = Ψ(X(i), X(j)) for these (i,j): (1,21), (2,6), (3,49), (4,2), (5,51), (6,3), (7,21), (8,21), (9,21), (19,1), (53,5), (54,3), (64,3), (65,1), (66,3), (67,3), (68,3), (69,3), (73,3), (74,3), (75,1), (76,2), (93,4), (95,54), (115,125), (190,99), (250,186)

X(110) = X(110)-of-1st-Parry-triangle

X(110) = X(74)-of-2nd-Parry-triangle

X(110) = center of similitude of ABC and 1st Parry triangle

X(110) = inverse-in-Parry-isodynamic-circle of X(111); see X(2)

X(110) = barycentric product of PU(i) for these i: 78, 145

X(110) = perspector of unary cofactor triangles of outer and inner Napoleon triangles

X(110) = X(6792)-of-4th-anti-Brocard-triangle

X(110) = X(111)-of-anti-Artzt-triangle

X(110) = perspector of circumcevian triangle of X(36) and cross-triangle of ABC and 2nd circumperp triangle

X(110) = perspector of circumcevian triangle of X(187) and cross-triangle of ABC and circumsymmedial triangle

X(110) = Kiepert image of X(3)

X(110) = Jerabek image of X(2)

X(110) = Cundy-Parry Phi transform of X(14264)

X(110) = endo-homothetic center of X(2)-altimedial and X(2)-anti-altimedial triangles

X(110) = endo-homothetic center of X(20)-altimedial and X(3)-anti-altimedial triangles

X(110) = Thomson isogonal conjugate of X(30)

X(110) = Lucas isogonal conjugate of X(30)

X(110) = center of the perspeconic of these triangles: inner and outer Napoleon

X(110) = intersection of antipedal lines of X(1113) and X(1114)

X(110) = X(104)-of-circumorthic-triangle if ABC is acute

X(110) = perspector, wrt circumorthic triangle, of polar circle

X(110) = trilinear product of circumcircle intercepts of line X(1)X(21)

X(110) = barycentric product of circumcircle intercepts of line X(2)X(6)

X(110) = barycentric product X(6)*X(99)

X(110) = barycentric product of Steiner circumellipse intercepts of Brocard axis

X(110) = Feuerbach point of the tangential triangle if ABC is acute; otherwise, a vertex of the Feuerbach triangle of the tangential triangle

X(110) = barycentric product of MacBeath circumconic intercepts of Euler line

X(110) = excentral-to-ABC functional image of X(1768)

X(110) = intouch-to-ABC functional image of X(11)

X(110) = trilinear pole, wrt circumtangential triangle, of Euler line

X(110) = trilinear pole, wrt 1st Parry triangle, of Lemoine axis

X(110) = trilinear product of circumcircle intercepts of line X(1)X(21)

X(110) = de-Longchamps-circle-inverse of X(34186)

Barycentrics a

Tripolars |b c (2 a^2 - b^2 - c^2)| : :

Let L be a line tangent to the Brocard circle. Let P be the trilinear pole of L, and let P' be the isogonal conjugate of P. As L varies, P' traces a parabola with focus at X(111). The parabola meets the line at infinity at X(524). Also, X(111) is the QA-P4 center (Isogonal Center) of quadrangle X(13)X(14)X(15)X(16) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/25-qa-p4.html) Also, let A' = BC∩X(115)X(125), and define B' and C' cyclically. The circumcircles of AB'C', BC'A', CA'B' concur in X(111). (Randy Hutson, October 13, 2015)

Let A"B"C" be the 2nd Ehrmann triangle. Let Pa be the pole of line B"C" wrt the A-Ehrmann circle, and define Pb and Pc cyclically. Let Pa' be the pole of line BC wrt the A-Ehrmann circle, and define Pb' and Pc' cyclically. The lines APaPa', BPbPb', CPcPc' concur in X(111). Also, let A* be the trilinear pole of line B"C", and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(111). (Randy Hutson, November 18, 2015)

Let A1B1C1 and A3B3C3 be the 1st and 3rd Parry triangles. Let A' be the barycentric product A1*A3, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(111). (Randy Hutson, February 10, 2016)

Let A'B'C' and A"B"C" be the 4th Brocard and 4th anti-Brocard triangles, resp. Let A* be the barycentric product A'*A'', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(111). (Randy Hutson, December 2, 2017)

X(111) lies on the Parry circle and these lines:

2,99 3,1296 6,110 23,187
25,112 37,100 42,101 98,1637
107,393 182,353 230,476
251,827 308,689 352,511
385,892 468,935 512,843
647,842 694,805 931,941

X(111) = reflection of X(1296) in X(3)

X(111) = isogonal conjugate of X(524)

X(111) = isotomic conjugate of X(3266)

X(111) = inverse-in-Brocard-circle of X(353)

X(111) = anticomplement of X(126)

X(111) = cevapoint of X(6) and X(187)

X(111) = X(i)-cross conjugate of X(j) for these (i,j): (23,251), (187,6), (351,110)

X(111) = crossdifference of every pair of points on line X(351)X(690)

X(111) = point of intersection, other than A, B, C, of circumcircle and hyperbola {{A,B,C,X(2),X(6)}}

X(111) = trilinear pole of line X(6)X(512)

X(111) = Λ(X(2), X(6))

X(111) = Ψ(X(i),X(j)) for these (i,j): (6,512), (21,650), (190,10)

X(111) = trilinear pole wrt circumsymmedial triangle of Brocard axis

X(111) = trilinear pole wrt circummedial triangle of line X(2)X(6)

X(111) = X(110)-of-circumsymmedial-triangle

X(111) = X(23)-of-4th-Brocard-triangle

X(111) = X(352)-of-orthocentroidal-triangle

X(111) = X(138)-of-excentral-triangle

X(111) = reflection of X(2770) in the Euler line

X(111) = reflection of X(843) in the Brocard axis

X(111) = reflection of X(2721) in line X(1)X(3)

X(111) = reflection of X(110) in line X(3)X(351)

X(111) = inverse-in-polar-circle of X(1560)

X(111) = inverse-in-{circumcircle, nine-point circle}-inverter of X(115)

X(111) = inverse-in-Moses-radical-circle of X(842)

X(111) = inverse-in-circle-O(15,16) of X(691)

X(111) = X(1577)-isoconjugate of X(5467)

X(111) = SR(P,U), where P and U are the circumcircle intercepts of line X(2)X(6)

X(111) = one of two harmonic traces of the McCay circles; X(2) is the other

X(111) = X(1296)-of-1st-Parry-triangle

X(111) = X(111)-of-2nd-Parry-triangle

X(111) = X(691)-of-3rd-Parry-triangle

X(111) = center of similitude of ABC and 2nd Parry triangle

X(111) = inverse-in-Parry-isodynamic-circle of X(110); see X(2)

X(111) = 3rd-Parry-to-circumsymmedial similarity image of X(352)

X(111) = center of similitude of ABC and circumsymmedial triangle of Artzt triangle

X(111) = eigencenter of circumtangential triangle

X(111) = perspector of ABC and unary cofactor triangle of 2nd Brocard triangle

X(111) = X(2)-of-4th-anti-Brocard-triangle

X(111) = anti-Artzt-to-4th-anti-Brocard similarity image of X(2)

X(111) = Thomson-isogonal conjugate of X(1499)

X(111) = Lucas-isogonal conjugate of X(1499)

X(111) = Cundy-Parry Phi transform of X(14262)

X(111) = Cundy-Parry Psi transform of X(13608)

X(111) = X(6)-isoconjugate of X(14210)

X(111) = inverse-in-circle-{{X(1687),X(1688),PU(1),PU(2)}} of X(32694)

X(111) = barycentric product of circumcircle intercepts of line X(2)X(523)

X(111) = inverse-in-Stevanovic-circle of X(2758)

X(111) = trilinear pole, wrt Thomson triangle, of line X(6)X(5646)

X(111) = trilinear pole, wrt 2nd Parry triangle, of line X(511)X(3569)

Trilinears a/[(b

Trilinears tan A csc(B - C) : :

Barycentrics a

Tripolars |cot A sin(B - C)| : :

If the line X(4)X(6) is reflected in every side of triangle ABC, then the reflections concur in X(112). (Randy Hutson, 9/23/2011)

Let P be a point on the van Aubel line other than X(4). Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AA'B', BC'A', and CA'B' concur at X(112). (Randy Hutson, December 26, 2015)

Let Q be a point on the Euler line other than X(2). Let A'B'C' be the cevian triangle of Q. Let A" be the {B,C}-harmonic conjugate of A' (or equivalently, A" = BC∩B'C'), and define B" and C" cyclically. The circumcircles of AB"C", BC"A", CA"B" concur in X(112). (Randy Hutson, December 26, 2015)

Let A', B', C' be the intersections of the orthic axis and lines BC, CA, AB, respectively. The circumcircles of AB'C', BC'A', CA'B' concur in X(112). (Randy Hutson, December 26, 2015)

Let A' be the reflection of X(6) in BC, and define B' and C' cyclically. The circumcircles of AB'C', BC'A', CA'B' concur in X(112). (Randy Hutson, December 26, 2015)

Let A'B'C' be the circummedial triangle. Let A" be the pole, wrt the polar circle, of line B'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(112). (Randy Hutson, December 26, 2015)

Let L_{1} be the line that is the barycentric product of the Euler line and P(4). Let L_{2} be the line that is the barycentric product of the Euler line and U(4). Then X(112) = L_{1}∩L_{2}. (Randy Hutson, July 11, 2019)

X(112) lies on these lines:

2,127 3,1297 4,32 6,74 19,759
25,111 27,675 28,105
33,609 50,477 54,217
58,103 99,648 100,162
102,284 104,1108 109,163
186,187 230,403 250,691
251,427 286,767 376,577
393,571 523,935 647,1304 789,811

X(112) = reflection of X(i) in X(j) for these (i,j): (4,132), (1297,3)

X(112) = isogonal conjugate of X(525)

X(112) = isotomic conjugate of X(3267)

X(112) = anticomplement of X(127)

X(112) = X(i)-Ceva conjugate of X(j) for these (i,j): (249,24), (250,25)

X(112) = cevapoint of X(i) and X(j) for these (i,j): (32,512), (427,523)

X(112) = X(i)-cross conjugate of X(j) for these (i,j): (25,250), (512,4), (523,251)

X(112) = crossdifference of every pair of points on line X(122)X(125)

X(112) = barycentric product of X(1113) and X(1114)

X(112) = isogonal conjugate of isotomic conjugate of trilinear pole of Euler line

X(112) = point of intersection, other than A, B, C, of circumcircle and hyperbola {{A,B,C,X(4),PU(39)}}

X(112) = trilinear pole of line X(6)X(25)

X(112) = X(647)-cross conjugate of X(6)

X(112) = pole wrt polar circle of trilinear polar of X(850) (line X(115)X(127))

X(112) = X(48)-isoconjugate (polar conjugate) of X(850)

X(112) = X(92)-isoconjugate of X(520)

X(112) = X(1577)-isoconjugate of X(3)

X(112) = trilinear pole wrt circumsymmedial triangle of line X(6)X(647)

X(112) = reflection of X(935) in the Euler line

X(112) = reflection of X(2715) in the Brocard axis

X(112) = reflection of X(2722) in line X(1)X(3)

X(112) = inverse-in-polar-circle of X(115)

X(112) = inverse-in-{circumcircle, nine-point circle}-inverter of X(1560)

X(112) = inverse-in-Moses-radical-circle of X(1304)

X(112) = inverse-in-[circle with diameter X(15)X(16) and center X(187)] of X(842)

X(112) = inverse-in-circle-{X(1687),X(1688),PU(1),PU(2)} of X(2698)

X(112) = inverse-in-Stevanovic-circle of X(2766)

X(112) = X(139)-of-excentral-triangle

X(112) = barycentric product of PU(74)

X(112) = trilinear product of PU(108)

X(112) = eigencenter of circumnormal triangle

X(112) = Thomson-isogonal conjugate of X(1503)

X(112) = Lucas-isogonal conjugate of X(1503)

X(112) = perspector of circumcevian triangle of X(468) and cross-triangle of ABC and circumcevian triangle of X(25)

X(112) = trilinear product of circumcircle intercepts of line X(1)X(19)

X(112) = barycentric product X(3)*X(107)

X(112) = barycentric product X(4)*X(110)

X(112) = X(6)-of-1st-anti-orthosymmedial-triangle

X(112) = cevapoint of Jerabek hyperbola intercepts of Lemoine axis

X(112) = Ψ(X(i),X(j)) for these (i,j): (1,19), (2,3), (3,6), (4,6), (5,53), (6,25), (69,2), (76,4), (125,115)

lie on the nine-point circle.

Suppose that X is a point on the nine-point circle, and let X' be
the reflection of X in the orthocenter, H. Then X is the anticenter of
the cyclic quadrilateral ABCX'. Let H_{A} be the orthocenter of
triangle BCX, Let H_{B} be the orthocenter of CAX, and let
H_{C} be the orthocenter of triangle ABX. Then the
quadrilateral HH_{A}H_{B}H_{C} is homothetic to
and congruent to the cyclic quadrilateral ABCX', and X is the center of
homothety. (Randy Hutson, 9/23/2011)

Barycentrics (sin C)/(cos C - 2 cos A cos B) + (sin B)/(cos B - 2 cos A cos C) : :

Barycentrics b

Barycentrics (2a^4 - b^4 - c^4 - a^2b^2 - a^2c^2 + 2b^2c^2)[a^4(b^2 + c^2) - 2a^2(b^4 - b^2c^2 + c^4) + (b^2 - c^2)^2(b^2 + c^2)] : :

Let A'B'C' be the orthic triangle. Let L be the line through A' parallel to the Euler line, and define M and N cyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L', M', N' concur in X(113). (Randy Hutson, August 26, 2014)

Let A'B'C' be the orthic triangle. Let M_{A} be the
reflection of the orthic axis in line B'C', and define Let
M_{B} and Let M_{C} cyclically. Let A'' = Let
M_{B}∩M_{C}, and define B'' and C'' cyclically. The
lines A'A'', B'B'' C'C'' concur in X(113). (Randy Hutson, August 26,
2014)

Let A'B'C' be the orthic triangle. Let N_{A} be the orthic
axis of AB'C', and define N_{B} and N_{C} cyclically.
Let A'' = N_{B}∩N_{C}, B'' =
N_{C}∩A_{C}, C'' = N_{A}∩B_{C}.
Then triangle A''B''C'' is inversely similar to ABC, with similitude
center X(6), and the lines A'A'', B'B'', C'C'' concur in X(113). Also,
X(113) = X(3)-of-A''B''C''. (Randy Hutson, August 26, 2014)

X(113) lies on the bicevian conic of X(2) and X(110) and on these lines:

2,74 3,122 4,110 5,125
6,13 11,942 52,135 114,690 123,960 127,141 137,546

X(113) = midpoint of X(i) and X(j) for these (i,j): (4,110), (74,146), (265,399), (1553,3258)

X(113) = reflection of X(i) in X(j) for these (i,j): (52,1112), (125,5)

X(113) = complementary conjugate of X(30)

X(113) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,30), (2,3003)

X(113) = crosspoint of X(4) and X(403)

X(113) = crossdifference of every pair of points on line X(526)X(686)

X(113) = nine-point-circle-antipode of X(125)

X(113) = X(74)-of-medial-triangle

X(113) = X(104)-of-orthic-triangle if ABC is acute

X(113) = X(186)-of-X(4)-Brocard-triangle

X(113) = center of rectangular circumhyperbola that passes through X(110)

X(113) = center of rectangular hyperbola {{X(3),X(4),X(110),X(155),X(1351),X(1352),X(2574,X(2575)}}

X(113) = perpsector of circumconic centered at X(3003)

X(113) = inverse-in-polar-circle-of X(1300)

X(113) = inverse-in-{circumcircle, nine-point circle}-inverter of X(1302)

X(113) = anticenter of cyclic quadrilateral ABCX(110)

X(113) = Λ(X(2),X(3))-with-respect-to-orthic-triangle

X(113) = barycentric product X(30)*X(3580)

X(113) = complement of X(74)

X(113) = reflection of X(3) in X(5972)

X(113) = antipode of X(3) in the bicevian conic of X(2) and X(110)

X(113) = antipode of X(52) in the Hatzipolakis-Lozada hyperbola

X(113) = orthopole of line X(3)X(523)

X(113) = perspector of Ehrmann mid-triangle and orthic triangle

X(113) = excentral-to-ABC functional image of X(104)

Trilinears cos(B - C) cos 2ω - sin ω sin(A + ω) (Peter J. C. Moses, 9/12/03)

Barycentrics b sec(B + ω) + c sec(C + ω) : :

Barycentrics (b^4 + c^4 - a^2b^2 - a^2c^2)(2a^4 + b^4 + c^4 - a^2b^2 - a^2c^2 - 2b^2c^2) : :

X(114) is the QA-P30 center (Reflection of QA-P2 in QA-P11) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/58-qa-p30.html)

Let A'B'C' be the orthic triangle. Let La be the Lemoine axis of triangle AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. Triangle A"B"C" is inversely similar to ABC, with similicenter X(2). The lines A'A", B'B", C'C" concur in X(114), which is X(3)-of-A"B"C".

X(114) lies on the bicevian conic of X(2) and X(99), and on these lines:

2,98 3,127 4,99 5,39
25,135 52,211 113,690
132,684 136,427 325,511 381,543

X(114) = isogonal conjugate of X(2065)

X(114) = midpoint of X(i) and X(j) for these (i,j): (4,99), (98,147)

X(114) = reflection of X(i) in X(j) for these (i,j): (3,620), (115,5)

X(114) = complementary conjugate of X(511)

X(114) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,230), (4,511)

X(114) = crosspoint of X(2) and X(325)

X(114) = orthojoin of X(230)

X(114) = nine-point-circle-antipode of X(115)

X(114) = X(98)-of-medial triangle

X(114) = X(103)-of-orthic triangle if ABC is acute

X(114) = perspector of circumconic centered at X(230)

X(114) = center of circumconic that is locus of trilinear poles of lines passing through X(230)

X(114) = center of rectangular hyperbola {{X(4),X(76),X(99),X(376),X(487),X(488)}}

X(114) = X(1513)-of-1st-Brocard-triangle

X(114) = inverse-in-polar-circle of X(3563)

X(114) = inverse-in-{circumcircle, nine-point circle}-inverter of X(110)

X(114) = center of inverse-in-{circumcircle, nine-point circle}-inverter of Brocard circle

X(114) = X(5)=of=1st=antiBrocard=triangle

X(114) = anticenter of cyclic quadrilateral ABCX(99)

X(114) = Λ(X(3), X(6)), wrt orthic triangle

X(114) = centroid of mid-triangle of X(15)- and X(16)-Fuhrmann triangles

Trilinears cos A - 2 cos(B - C) + cot ω sin A : : (Peter J. C. Moses, 9/12/03)

Trilinears sin A sin

Barycentrics (b

Barycentrics (SB - SC)^2 : :

Barycentrics (SA - SW) (SB + SC) + 4 SB SC : :

X(115) = X(13) + X(14) (Randy Hutson, July 23, 2015)

The circumcircle of the incentral triangle intersects the nine-point circle at 2 points, X(11) and X(115), and X(115) lies on the incentral circle and the cevian circle of every point on the Kiepert hyperbola. Let A'B'C' be the orthic triangle. The Brocard axes of AB'C', BC'A', CA'B' concur in X(115). Let P be a point on the Brocard circle, and let L be the line tangent to the Brocard circle at P. Let P' be the trilinear pole of L, and let P" be the isotomic conjugate of P'. As P varies, P" traces an ellipse with center at X(115). (Randy Hutson, July 23, 2015)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Brocard axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B', C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Brocard axis. The triangle A"B"C" is homothetic to ABC, with center of homothety X(115); see Hyacinthos #16741/16782, Sep 2008.

X(115) is the QA-P2 center (Euler-Poncelet Point) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/10-mathematics/quadrangle-objects/12-qa-p2.html)

Let F be the Feuerbach point, X(11), and FaFbFc be the Feuerbach triangle (the extraversion triangle of X(11)). Let A' be the barycentric product F*Fa, and define B', C' cyclically. The lines AA', BB', CC' concur in X(115). (Randy Hutson, January 29, 2018)

If you have The Geometer's Sketchpad, you can view Kiepert Hyperbola, showing X(115).

**Roland H. Eddy and R. Fritsch,** "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," *Mathematics Magazine* 67 (1994) 188-205.

X(115) lies on the nine-point circle, the Steiner inellipse, the bicevian conic of X(2) and X(98), and on these lines:

2,99 4,32 5,39 6,13
11,1015 30,187 50,231
53,133 76,626 116,1086
120,442 125,245 127,338
128,233 129,389 131,216
232,403 316,385 325,538
395,530 396,531 427,1560 593,1029
804,1084

X(115) = midpoint of X(i) and X(j) for these (i,j): (4,98), (13,14), (99,148), (316,385), (2009,2010)

X(115) = midpoint of PU(40)

X(115) = reflection of X(i) in X(j) for these (i,j): (99,620), (114,5), (187,230), (325,625)

X(115) = isogonal conjugate of X(249)

X(115) = isotomic conjugate of X(4590)

X(115) = complement of X(99)

X(115) = anticomplement of X(620)

X(115) = inverse-in-orthocentroidal-circle of X(6)

X(115) = complementary conjugate of X(512)

X(115) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,523), (4,512), (338,125)

X(115) = crosspoint of X(i) and X(j) for these (i,j): (2,523), (68,525), (3413,3414)

X(115) = crosssum of X(i) and X(j) for these (i,j): (6,110), (24,112), (163,849), (1379,1380)

X(115) = crossdifference of every pair of points on line X(110)X(351)

X(115) = X(2)-Hirst inverse of X(148)

X(115) = X(99)-of-medial triangle

X(115) = X(101)-of-orthic triangle if ABC is acute

X(115) = X(325)-of-1st-Brocard-triangle

X(115) = X(187)-of-4th-Brocard-triangle

X(115) = X(187)-of-orthocentroidal-triangle

X(115) = X(141)-of-1st-antiBrocard-triangle

X(115) = barycentric product X(11)*X(12)

X(115) = {X(5),X(39)}-harmonic conjugate of X(1506)

X(115) = projection from Steiner circumellipse to Steiner inellipse of X(671)

X(115) = center of similitude of incentral and Feuerbach triangles

X(115) = center of circumconic that is locus of trilinear poles of lines parallel to the orthic axis (i.e. lines that pass through X(523))

X(115) = perspector of circumconic centered at X(523) (parabola {{A,B,C,X(476),X(523),X(685)})

X(115) = trilinear pole wrt medial triangle of line X(2)X(6)

X(115) = inverse-in-circumcircle of X(2079)

X(115) = inverse-in-polar-circle of X(112)

X(115) = inverse-in-{circumcircle, nine-point circle}-inverter of X(111)

X(115) = inverse-in-Moses-radical-circle of X(3258)

X(115) = inverse-in-Steiner-circumellipse of X(148)

X(115) = inverse-in-excircles-radical-circle of X(5213)

X(115) = {X(99),X(671)}-harmonic conjugate of X(148)

X(115) = {X(6108),X(6109)}-harmonic conjugate of X(6055)

X(115) = inverse-in-circle-{{X(2),X(13),X(14),X(111),X(476)}} of X(1648)

X(115) = orthopole of Brocard axis

X(115) = orthic isogonal conjugate of X(512)

X(115) = incentral isogonal conjugate of X(512)

X(115) = perspector of orthic triangle and tangential triangle of hyperbola {{A,B,C,X(2),X(6)}}

X(115) = similitude center of (equilateral) pedal triangles of X(15) and X(16)

X(115) = exsimilicenter of Moses circle and the nine-point circle

X(115) = anticenter of cyclic quadrilateral ABCX(98)

X(115) = Λ(X(187), X(237))-wrt-orthic-triangle

X(115) = X(1101)-isoconjugate of X(2)

X(115) = harmonic center of nine-point circle and Gallatly circle

X(115) = perspector of medial triangle and Schroeter triangle

X(115) = trilinear pole of line X(1648)X(8029)

X(115) = barycentric square of X(523)

X(115) = inverse-in-Hutson-Parry-circle of X(1648)

X(115) = {X(13636),X(13722)}-harmonic conjugate of X(1648)

X(115) = homothetic center of medial triangle and mid-triangle of antipedal triangles of X(13) and X(14)

X(115) = homothetic center of Ehrmann vertex-triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles

X(115) = centroid of reflection triangle of X(187)

X(115) = centroid of mid-triangle of 1st and 2nd isodynamic-Dao triangles

X(115) = centroid of mid-triangle of 3rd and 4th isodynamic-Dao triangles

X(115) = point of concurrence of cevian circles of the excenters

X(115) = inverse-in-circle-{{X(3102),X(3103),PU(1)}} of X(32448)

X(115) = inverse-in-Stevanovic-circle of X(5520)

X(115) = intersection of Fermat axes of outer and inner Vecten triangles

Barycentrics (b - c)

Let A'B'C' be the orthic triangle. Let La be the Soddy line of triangle AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(116), which is X(3)-of-A"B"C". (Randy Hutson, July 31 2018)

X(116) lies on the nine-point circle and these lines: 2,101 4,103 5,118 10,120 115,1086 119,142 121,141 124,928

X(116) = midpoint of X(i) and X(j) for these (i,j): (4,103), (101,150)

X(116) = reflection of X(118) in X(5)

X(116) = isotomic conjugate of isogonal conjugate of X(20974)

X(116) = complementary conjugate of X(514)

X(116) = X(4)-Ceva conjugate of X(514)

X(116) = complement of X(101)

X(116) = inverse-in-excircles-radical-circle of X(3034)

X(116) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(675)

X(116) = anticenter of cyclic quadrilateral ABCX(103)

X(116) = Λ(Gergonne line), wrt orthic triangle

X(116) = X(2)-Ceva conjugate of X(6586)

X(116) = X(101)-of-medial triangle.

g(b,c,a) = b

Barycentrics h(a,b,c) : h(b,c,a) : h(c,a,b) where h(a,b,c) = af(a,b,c)

X(117) lies on the nine-point circle

X(117) = X(102)-of-medial triangle.

X(117) lies on these lines: 2,102 4,109 5,124 10,123 11,65 118,928 136,407

X(117) = midpoint of X(i) and X(j) for these (i,j): (4,109), (102,151)

X(117) = reflection of X(124) in X(5)

X(117) = complement of X(102)

X(117) = complementary conjugate of X(515)

X(117) = X(4)-Ceva conjugate of X(515)

X(117) = inverse-in-polar-circle of X(32706)

X(117) = anticenter of cyclic quadrilateral ABCX(109)

X(117) = Λ(X(1), X(4)), wrt orthic triangle

Let A'B'C' be the orthic triangle. Let La be the Gergonne line of triangle AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(118), which is X(3)-of-A"B"C". (Randy Hutson, July 31 2018)

X(118) lies on the nine-point circle and these lines: 2,103 4,101 5,116 11,226 117,928 122,440 136,430 381,544 516,910

X(118) = midpoint of X(i) and X(j) for these (i,j): (4,101), (103,152)

X(118) = reflection of X(116) in X(5)

X(118) = complementary conjugate of X(516)

X(118) = X(4)-Ceva conjugate of X(516)

X(118) = complement of X(103)

X(118) = inverse-in-polar-circle of X(917)

X(118) = anticenter of cyclic quadrilateral ABCX(101)

X(118) = X(103)-of-medial triangle.

X(118) = Λ(X(1), X(7)), wrt orthic triangle

X(118) = Λ(X(4), X(9)), wrt orthic triangle

Barycentrics (-1 + cos B + cos C)[sin 2B + sin 2C + 2(-1 + cos A)(sin B + sin C)] : :

Let Na = X(5)-BCX(1), Nb = X(5)-of-CAX(1), Nc = X(5)-of-ABX(1). Then X(119) = X(2071)-of-NaNbNc. (Randy Hutson, January 29, 2018)

Let A'B'C' be the orthic triangle. Let La be the antiorthic axis of AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. Then triangle A"B"C" is inversely similar to ABC, with similicenter X(9). The lines A'A", B'B", C'C" concur in X(119). Also, X(119) = X(3) of A"B"C". (Randy Hutson, January 29, 2018)

X(119) lies on these lines:

1,5 2,104 3,123 4,100
10,124 116,142 125,442
135,431 136,429 214,515
381,528 517,908

X(119) = midpoint of X(i) and X(j) for these (i,j): (4,100), (104,153)

X(119) = reflection of X(i) in X(j) for these (i,j): (11,5), (3,3035)

X(119) = isogonal conjugate of X(15381)

X(119) = complement of X(104)

X(119) = complementary conjugate of X(517)

X(119) = X(4)-Ceva conjugate of X(517)

X(119) = nine-point-circle-antipode of X(11)

X(119) = X(104)-of-medial triangle

X(119) = X(2072)-of-Fuhrmann-triangle

X(119) = inverse-in-polar-circle of X(915)

X(119) = anticenter of cyclic quadrilateral ABCX(100)

X(119) = Λ(X(1), X(3)), wrt orthic triangle

X(119) = Λ(X(4), X(8)), wrt orthic triangle

X(119) = center of rectangular circumhyperbola passing through isogonal and isotomic conjugates of X(3657)

Barycentrics [2abc - (b + c)(a

X(120) lies on the nine-point circle and these lines: 2,11 10,116 12,85 115,442

X(120) = complementary conjugate of X(518)

X(120) = X(4)-Ceva conjugate of X(518)

X(120) = perspector of circumconic centered at X(3290)

X(120) = center of circumconic that is locus of trilinear poles of lines passing through X(3290)

X(120) = X(2)-Ceva conjugate of X(3290)

X(120) = inverse-in-excircles-radical-circle of X(3033)

X(120) = inverse-in-{circumcircle, nine-point circle}-inverter of X(100)

X(120) = X(1292)-of-Euler-triangle

X(120) = midpoint of X(4) and X(1292)

X(120) = Λ(X(1), X(6)), wrt orthic triangle

X(120) = orthopole of PU(44) (line X(3)X(667))

f(a,b,c) = bc(b + c - 2a)[b

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where
g(a,b,c) = (b + c - 2a)[b^{3} + c^{3} + a(b^{2}
+ c^{2}) - 2bc(b + c)]

X(121) lies on the nine-point circle

X(121) = X(106)-of-medial triangle.

X(121) lies on these lines: 2,106 10,11 116,141

X(121) = complementary conjugate of X(519)

X(121) = X(4)-Ceva conjugate of X(519)

X(121) = complement of X(106)

X(121) = inverse-in-excircles-radical-circle of X(3032)

X(121) = Λ(X(1), X(2)), wrt orthic triangle

Barycentrics a(b

Barycentrics (b^2 - c^2)^2 (a^2 - b^2 - c^2)^2 (3 a^4 - 2 a^2 (b^2 + c^2) - (b^2 - c^2)^2) : :

X(122) lies on the nine-point circle, the cevian circle of X(20), and these lines: 2,107 3,113 5,133 118,440 125,684 138,233

X(122) = reflection of X(133) in X(5)

X(122) = isogonal conjugate of X(15384)

X(122) = complementary conjugate of X(520)

X(122) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,520), (253,525)

X(122) = crosssum of X(i) and X(j) for these (i,j): (64,1301), (112,154)

X(122) = crosspoint of X(253) and X(525)

X(122) = crossdifference of every pair of points on line X(112)X(1301)

X(122) = X(107)-of-medial triangle

X(122) = center of the rectangular hyperbola that passes through A, B, C, and X(20)

X(122) = X(1293)-of-orthic-triangle if ABC is acute

X(122) = complement of X(107)

X(122) = perspector of orthic triangle and tangential triangle of hyperbola {{A,B,C,X(2),X(3)}}

X(122) = inverse-in-polar-circle of X(1301)

X(122) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(1297)

X(122) = inverse-in-Moses-radical-circle of X(33504)

X(122) = X(2)-Ceva conjugate of X(6587)

X(122) = barycentric product X(20)*X(15526) = X(20)*X(525)^2

Barycentrics (sec B - sec C)[(sec A)(sin

Barycentrics (a - b - c)(b - c)^2(a^2 - b^2 - c^2)(a^4 - b^4 - c^4 + 2a^2bc - 2ab^2c - 2abc^2 + 2b^2c^2) : :

X(123) lies on the nine-point circle and hese lines: 2,108 3,119 10,117 113,960

X(123) = complement of X(108)

X(123) = complementary conjugate of X(521)

X(123) = X(2)-Ceva conjugate of X(6588)

X(123) = X(4)-Ceva conjugate of X(521)

X(123) = X(108)-of-medial triangle

f(a,b,c) = bc(b + c - a)(b - c)

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where

g(a,b,c)
= (b + c - a)(b - c)^{2} [(b + c)(b^{2} + c^{2}
- a^{2} - bc) + abc]

X(124) lies on the nine-point circle

X(124) = X(109)-of-medial triangle

X(124) = center of the rectangular hyperbola that passes through A, B,
C, and X(58)

X(124) lies on these lines: 2,109 4,102 5,117 10,119 116,928

X(124) = midpoint of X(4) and X(102)

X(124) = reflection of X(117) in X(5)

X(124) = complementary conjugate of X(522)

X(124) = X(4)-Ceva conjugate of X(522)

X(124) = complement of X(109)

X(124) = crosssum of circumcircle intercepts of line X(3)X(10)

X(124) = orthopole of line X(3)X(10)

X(124) = focus of Mandart parabola

X(124) = anticenter of cyclic quadrilateral ABCX(102)

X(124) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(1311)

X(124) = X(2)-Ceva conjugate of X(6589)

X(124) = center of rectangular circumhyperbola that is isogonal conjugate of line X(3)X(10)

Trilinears (sec A)(c cos C - b cos B)

Trilinears bc(b

Barycentrics (sin 2A)[sin(B - C)]

**Roland H. Eddy and R. Fritsch,** "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," *Mathematics Magazine* 67 (1994) 188-205.

X(125) is the pole of the Fermat axis with respect to the Dao-Moses-Telv circle. (Randy Hutson, December 14, 2014)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Euler line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B', C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Euler line. The triangle A"B"C" is homothetic to ABC, with center of homothety X(125); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

Let A'B'C' be the orthic triangle. The Euler lines of AB'C', BC'A', CA'B' concur in X(125). (Randy Hutson, March 25, 2016)

X(125) lies on these curves: nine-point circle, orthic inconic, symmedial circle, Johnson circumconic of the medial triangle, cevian circle of every point on the Jerabek hyperbola, and bicevian conic of X(2) and X(72). X(125) also lies on these lines:

2,98 3,131 4,74 5,113
6,67 51,132 68,1092
69,895 115,245 119,442
122,684 126,141 128,140
136,338 381,541 511,858

X(125) = midpoint of X(i) and X(j) for these (i,j): (3,265), (4,74), (6,67), (110,3448)

X(125) = reflection of X(i) in X(j) for these (i,j): (113,5), (185,974), (1495,468), (1511,140), (1539,546)

X(125) = isogonal conjugate of X(250)

X(125) = isotomic conjugate of X(18020)

X(125) = inverse-in-Brocard-circle of X(184)

X(125) = complement of X(110)

X(125) = complementary conjugate of X(523)

X(125) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,523), (66,512), (68,520), (69,525), (338,115)

X(125) = crosspoint of X(i) and X(j) for these (i,j): (4,523), (69,525), (338,339)

X(125) = crosssum of X(i) and X(j) for these (i,j): (3,110), (25,112), (162,270), (1113,1114)

X(125) = crossdifference of every pair of points on the line X(110)X(112)

X(125) = X(115)-Hirst inverse of X(868)

X(125) = X(2)-line conjugate of X(110)

X(125) = orthopole of the Euler line

X(125) = perspector of orthic triangle and Schroeter triangle

X(125) = X(110)-of-medial triangle

X(125) = X(100)-of-orthic triangle, if ABC is acute

X(125) = X(858)-of-1st-Brocard triangle

X(125) = anticenter of cyclic quadrilateral ABCX(74)

X(125) = Λ(X(230), X(231)), wrt orthic triangle

X(125) = anticomplement of X(5972)

X(125) = pole of Fermat axis wrt Dao-Moses-Telv circle

X(125) = orthic-isogonal conjugate of X(523)

X(125) = perspector of circumconic centered at X(647)

X(125) = center of circumconic that is locus of trilinear poles of lines passing through X(647)

X(125) = X(2)-Ceva conjugate of X(647)

X(125) = trilinear pole wrt orthic triangle of van Aubel line

X(125) = inverse-in-polar-circle of X(107)

X(125) = inverse-in-{circumcircle, nine-point circle}-inverter of X(98)

X(125) = inverse-in-orthosymmedial-circle of X(51)

X(125) = centroid of (degenerate) pedal triangle of X(74)

X(125) = X(i)-isoconjugate of X(j) for these {i,j}: {4,1101}, {92,249}

X(125) = inverse-in-Hutson-Parry-circle of X(868)

X(125) = {X(13636),X(13722)}-harmonic conjugate of X(868)

X(125) = crosssum of MacBeath circumconic intercepts of Brocard axis

X(125) = excentral-to-ABC functional image of X(100)

f(a,b,c) = bc(2a

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where
g(a,b,c) = (2a^{2} - b^{2} -
c^{2})[b^{4} + c^{4} +
a^{2}(b^{2} + c^{2}) -
4b^{2}c^{2}]

X(126) lies on the nine-point circle

X(126) = X(111)-of-medial triangle.

X(126) lies on these lines: 2,99 125,141 625,858

X(126) = complement of X(111)

X(126) = complementary conjugate of X(524)

X(126) = X(4)-Ceva conjugate of X(524)

X(126) = perspector of circumconic centered at X(3291)

X(126) = center of circumconic that is locus of trilinear poles of lines passing through X(3291)

X(126) = X(2)-Ceva conjugate of X(3291)

X(126) = one of two intersections (X(3258) is the other) of the nine-point circle of ABC and the Parry circle of the X(2)-Brocard triangle

X(126) = inverse-in-polar-circle of X(2374)

X(126) = inverse-in-{circumcircle, nine-point circle}-inverter of X(99)

X(126) = Λ(X(2), X(6)), wrt orthic triangle

Barycentrics (sin 2B - sin 2C)[(b

Barycentrics (b^2 - c^2)^2(b^2 + c^2 - a^2)(b^4 + c^4 - a^4) : :

Let A'B'C' be the orthic triangle. Let La be the van Aubel line of triangle AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(127), which is X(3)-of-A"B"C". (Randy Hutson, July 31 2018)

X(127) lies on the nine-point circle, the cevian circle of X(22), and on these lines: 2,112 3,114 5,132 113,141 115,338 133,381 125,140

X(127) = reflection of X(132) in X(5)

X(127) = complement of X(112)

X(127) = anticomplementary conjugate of X(525)

X(127) = X(4)-Ceva conjugate of X(525)

X(127) = X(1292)-of-orthic-triangle if ABC is acute

X(127) = perspector of circumconic centered at X(2485)

X(127) = center of circumconic that is locus of trilinear poles of lines passing through X(2485)

X(127) = X(2)-Ceva conjugate of X(2485)

X(127) = inverse-in-polar-circle of X(1289)

X(127) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(2373)

X(127) = X(112)-of-medial triangle

X(127) = center of the rectangular hyperbola that passes through A, B, C, and X(22)

f(A,B,C) = (sec A)(cos 2B + cos 2C)(1 + 2 cos 2A)(cos 2A + 2 cos 2B cos 2C)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B) where g(A,B,C) = (sin A) f(A,B,C)

X(128) lies on the nine-point circle

X(128) = X(74)-of-orthic triangle.

X(128) lies on these lines: 5,137 52,134 53,139 115,233 125,140

X(128) = reflection of X(137) in X(5)

X(128) = isogonal conjugate of X(15401)

X(128) = complement of X(1141)

X(128) = perspector of circumconic centered at X(231)

X(128) = center of circumconic that is locus of trilinear poles of lines passing through X(231)

X(128) = inverse-in-polar-circle of X(2383)

X(128) = X(2)-Ceva conjugate of X(231)

X(128) = orthojoin of X(231)

X(128) = excentral-to-ABC functional image of X(74)

f(A,B,C) = (sec A)(sin 2A)(sin 2B + sin 2C) s(A,B,C) t(A,B,C),

s(A,B,C) = sin

t(A,B,C) = sin

u(A,B,C) = sin 2B sin 2C - sin

v(A,B,C) = (sin 2B sin 2C)(sin 2B - sin 2C)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B) where g(A,B,C) = (sin A) f(A,B,C)

X(129) lies on the nine-point circle

X(129) = X(98)-of-orthic triangle.

X(129) lies on these lines: 5,130 51,137 52,139 115,389

X(129) = reflection of X(130) in X(5)

X(129) = complement of X(1298)

X(129) = complementary conjugate of X(32428)

X(129) = Λ(X(5), X(53)), wrt orthic triangle

X(129) = excentral-to-ABC functional image of X(98)

f(A,B,C) = (sin A)(sin 2B + sin 2C)[(sin 2B - sin 2C)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(130 lies on the nine-point circle

X(130) = X(99)-of-orthic triangle

X(130) = center of the rectangular hyperbola that passes through A, B,
C, and X(51)

X(130) lies on these lines: 5,129 51,138

X(130) = reflection of X(129) in X(5)

X(130) = complement of X(1303)

X(130) = excentral-to-ABC functional image of X(99)

X(130) = trilinear pole wrt orthic triangle of line X(51)X(53)

f(A,B,C) = (sec A)[2T - S(sec 2B + sec 2C)](T - S sec 2A),

S = sin 2A + sin 2B + sin 2C, T = tan 2A + tan 2B + tan 2C

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(131) lies on the nine-point circle

X(131) = X(102)-of-orthic triangle if ABC is acute.

X(131) lies on these lines: 3,125 4,135 5,136 115,216

X(131) = reflection of X(136) in X(5)

X(131) = complement of X(1300)

X(131) = complementary conjugate of X(13754)

X(131) = inverse-in-polar-circle of X(1299)

X(131) = Λ(X(4), X(52)), wrt orthic triangle

f(A,B,C) = (sec A) u(A,B,C) v(A,B,C),

u(A,B,C) = [sin

v(A,B,C) = [sin

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(132) lies on the nine-point circle and these lines: 2,107 4,32 5,127 25,136 51,125 114,684 137,428 147,648

X(132) = midpoint of X(4) and X(112)

X(132) = reflection of X(127) in X(5)

X(132) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,232), (4,1503)

X(132) = X(4)-line conjugate of X(248)

X(132) = crossdifference of every pair of points on line X(248)X(684)

X(132) = X(105)-of-orthic triangle if ABC is acute

X(132) = complement of X(1297)

X(132) = perspector of circumconic centered at X(232)

X(132) = center of rectangular hyperbola {{A,B,C,X(4),X(112),PU(39)}} that is locus of trilinear poles of lines passing through X(232)

X(132) = center of rectangular hyperbola {X(4),X(112),X(371),X(372),X(378),X(1064)}

X(132) = inverse-in-polar-circle of X(98)

X(132) = circumcircle-inverse of X(34131)

X(132) = inverse-in-{circumcircle, nine-point circle}-inverter of X(107)

X(132) = anticenter of cyclic quadrilateral ABCX(112)

X(132) = Λ(X(4), X(6)), wrt orthic triangle

X(132) = orthopole of PU(37)

X(132) = isogonal conjugate of X(15407)

Barycentrics (tan A)[(sin 2B - sin 2C)

X(133) lies on the nine-point circle

X(133) = X(106)-of-orthic triangle is ABC is acute.

X(133) lies on these lines: 2,1294 4,74 5,122 53,115 127,381 136,235

X(133) = midpoint of X(4) and X(107)

X(133) = reflection of X(122) in X(5)

X(133) = isogonal conjugate of X(15404)

X(133) = complement of X(1294)

X(133) = perspector of circumconic centered at X(1990)

X(133) = center of circumconic that is locus of trilinear poles of lines passing through X(1990)

X(133) = X(2)-Ceva conjugate of X(1990)

X(133) = trilinear pole wrt Euler triangle of van Aubel line

X(133) = inverse-in-polar-circle of X(74)

X(133) = anticenter of cyclic quadrilateral ABCX(107)

X(133) = Λ(X(4), X(51)), wrt orthic triangle

f(A,B,C) = (sec A) u(A,B,C) [v(B,C,A) - v(C,B,A)],

u(A,B,C) = (sin 2A)[sin

v(B,C,A) = sin 2C [sin

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(134) lies on the nine-point circle

X(134) = X(107)-of-orthic triangle

X(134) = center of the rectangular hyperbola that passes through A, B, C, and X(52)

X(134) lies on this line: 52,128

X(134) = trilinear pole wrt orthic triangle of line X(52)X(53)

f(A,B,C) = (sec A)[(sin 2B)/(sec 2C - sec 2A) + (sin 2C)/(sec 2A - sec 2B)]

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where

&nbnbsp; g(A,B,C)
= (tan A)[(sin 2B)/(sec 2C - sec 2A) + (sin 2C)/(sec 2A - sec 2B)]

X(135) lies on the nine-point circle

X(135) = X(108)-of-orthic-triangle if ABC is acute

X(135) = center of the rectangular hyperbola that passes through A, B,
C, and X(24)

X(135) lies on these lines: 4,131 25,114 52,113 119,431

X(135) = inverse-in-polar-circle of X(925)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where

f(A,B,C) = (sec A)[(sin 2B - sin 2C)

u(A,B,C) = [sin

Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(136) lies on the nine-point circle

X(136) =X(109)-of-orthic triangle if ABC is acute

X(136) = center of the rectangular hyperbola that passes through A, B,
C, and X(93)

X(136) lies on these lines:

2,925 4,110 5,131
25,132 68,254 114,427
117,407 118,430 119,429
125,338 127,868 133,235

X(136) = reflection of X(131) in X(5)

X(136) = complement of X(925)

X(136) = complementary conjugate of X(924)

X(136) = X(254)-Ceva conjugate of X(523)

X(136) = perspector of circumconic centered at X(2501)

X(136) = center of circumconic that is locus of trilinear poles of lines passing through X(2501) (hyperbola {{A,B,C,X(4),X(93)}})

X(136) = X(2)-Ceva conjugate of X(2501)

X(136) = inverse-in-polar-circle of X(110)

X(136) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(3563)

f(A,B,C) = (sec A)(sin 2B + sin 2C)[(sin 2B - sin 2C)

u(A,B,C) = [sin

Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(137) lies on the nine-point circle

X(137) = X(110)-of-orthic triangle

X(137) = center of the rectangular hyperbola that passes through A, B,
C, X(5), and X(53)

X(137) lies on the cevian circle of X(5) and these lines: 5,128 51,129 53,138 113,546 132,428

X(137) = reflection of X(128) in X(5)

X(137) = complement of X(930)

X(137) = X(4)-Ceva conjugate of X(1510)

X(137) = crosssum of X(252) and X(930)

X(137) = trilinear pole wrt orthic triangle of line X(5)X(53)

X(137) = inverse-in-polar-circle of X(933)

X(137) = excentral-to-ABC functional image of X(110)

u = u(A,B,C) = (sin 2A)/(2 sin

Barycentrics (v + w) tan A : (w + u) tan B : (u + v) tan C

X(138) lies on the nine-point circle

X(138) = X(111)-of-orthic triangle

X(138) lies on these lines: 51,130 53,137 122,233

X(138) = Λ(X(51), X(53)), wrt orthic triangle

f(A,B,C) = (sec A)(sin 2B + sin 2C)[(sin 2B - sin 2C)

u(A,B,C) = (sin 2B)

X(139) lies on the nine-point circle and these lines: 52,129 53,128

X(139) = inverse-in-polar-circle of X(32692)

X(139) = X(112)-of-orthic-triangle

113- 127, 140- 143: centers of the medial triangle

128- 139: centers of the orthic triangle

144- 153: centers of the anticomplementary triangle

154- 157, 159- 163: centers of the tangential triangle

164- 170: centers of the excentral triangle

Trilinears cos A + 2 sin B sin C : cos B + 2 sin C sin A : cos C + 2 sin A sin B

Trilinears 3 cos A + 2 cos B cos C : 3 cos B + 2 cos C cos A : 3 cos C + 2 cos A cos B

Trilinears 2 sec A + 3 sec B sec C : 2 sec B + 3 sec C sec A : 2 sec C + 3 sec A sec B

Trilinears bc[b cos(C - A) + c cos(B - A)] : :

Barycentrics b cos(C - A) + c cos(B - A) : :

Barycentrics 3 - cot B cot C : :

Barycentrics 2a^4 - 3a^2(b^2 + c^2) + (b^2 - c^2)^2 : :

Barycentrics 3 S^2 - SB SC : :

As a point on the Euler line, X(140) has Shinagawa coefficients (3, -1).

Let A' be the midpoint between A and X(3), and define B' and C' cyclically; the triangle A'B'C' is homothetic to the medial triangle, and the center of homothety is X(140). Let A'' be the centroid of the triangle BCX(3), and define B'' and C'' cyclically; then A''B''C'' is homothetic to ABC, and the center of homothety is X(140). Also, X(140) is the center of the conic consisting of the centers of all the conics which pass through A, B, C, and X(3). (Randy Hutson, 9/23/2011) This conic is also the locus of crosssums of the intersections of the circumcircle and lines through X(4). Furthmore, this conic is the bicevian conic of X(2) and X(3). (Randy Hutson, 9/14/2016)

Let Oa be the circle centered at A and passing through the A-vertex of the Euler triangle; define Ob and Oc cyclically. The radical center of Oa, Ob, Oc is X(140). (Randy Hutson, September 14, 2016)

Let Oa be the circle centered at A with radius 1/2*sqrt(b^2 + c^2), and define Ob and Oc cyclically. The radical center of Oa, Ob, Oc is X(140). (Randy Hutson, September 14, 2016)

Let P be a point on the circumcircle. The bicevian conic of X(2) and P is a rectangular hyperbola, H. Let X be the center of H. As P varies, X traces a circle centered at X(140). (Randy Hutson, November 2, 2017)

X(140) is the centroid of the six circumcenters in the construction of the van Lamoen circle. (Randy Hutson, October 15, 2018)

X(140) lies on these lines:

{1,5432}, {2,3}, {6,5418}, {8,1483}, {9,5843}, {10,214}, {11,35}, {12,36}, {13,5237}, {14,5238}, {15,18}, {16,17}, {32,3815}, {39,230}, {40,3624}, {46,11375}, {49,5012}, {51,10263}, {52,3917}, {53,10979}, {54,252}, {55,496}, {56,495}, {57,6147}, {61,395}, {62,396}, {69,1353}, {72,10202}, {76,6390}, {79,5131}, {83,2080}, {95,340}, {98,7832}, {100,1484}, {104,5260}, {110,10264}, {113,10990}, {114,6292}, {115,10992}, {119,5251}, {125,128}, {141,182}, {142,5762}, {143,511}, {156,9306}, {165,8227}, {183,3933}, {184,13336}, {185,5876}, {187,1506}, {195,323}, {216,1990}, {231,570}, {233,6748}, {236,8129}, {262,7846}, {298,628}, {299,627}, {302,633}, {303,634}, {325,1078}, {343,569}, {355,1698}, {371,615}, {372,590}, {385,13571}, {389,1154}, {392,11729}, {394,12161}, {484,5443}, {485,1152}, {486,1151}, {497,10386}, {515,3634}, {516,9955}, {517,1125}, {518,13373}, {523,1116}, {524,575}, {539,11264}, {542,6698}, {551,10222}, {567,3580}, {568,7998}, {572,1213}, {574,3054}, {576,597}, {577,6749}, {578,13567}, {601,748}, {602,750}, {618,630}, {619,629}, {620,2782}, {623,13350}, {624,13349}, {625,7830}, {626,13335}, {671,10185}, {758,5885}, {908,3916}, {912,5044}, {930,1263}, {936,5791}, {942,3911}, {944,5790}, {946,3579}, {956,5552}, {958,3820}, {970,6703}, {971,6666}, {993,1329}, {999,3085}, {1001,10200}, {1007,3785}, {1040,8144}, {1056,5265}, {1058,5281}, {1071,12691}, {1141,11016}, {1145,4861}, {1155,12047}, {1173,12834}, {1210,12433}, {1319,10039}, {1351,3618}, {1352,3763}, {1376,10267}, {1387,3057}, {1388,12647}, {1478,5204}, {1479,5217}, {1482,3616}, {1493,13366}, {1503,5092}, {1587,6398}, {1588,6221}, {1621,11849}, {1697,11373}, {1737,2646}, {1768,3652}, {1834,4256}, {1837,3612}, {1853,9833}, {2077,5259}, {2095,5761}, {2548,3053}, {2777,5893}, {2794,6721}, {2800,13145}, {2808,6710}, {2818,6711}, {2831,11259}, {2883,3357}, {2888,7666}, {2896,7925}, {2979,3567}, {3019,13329}, {3068,3312}, {3069,3311}, {3070,6396}, {3071,6200}, {3086,3295}, {3095,7786}, {3096,7940}, {3098,5480}, {3167,11411}, {3216,5396}, {3303,10072}, {3304,10056}, {3316,3590}, {3317,3591}, {3336,3649}, {3337,5557}, {3419,4855}, {3487,5435}, {3488,5704}, {3532,4846}, {3581,5888}, {3582,3746}, {3583,7173}, {3584,5298}, {3585,3614}, {3600,8164}, {3601,5722}, {3617,7967}, {3619,6776}, {3620,11898}, {3622,10247}, {3626,13607}, {3630,7916}, {3631,5965}, {3653,3679}, {3654,7982}, {3655,5881}, {3656,7991}, {3678,12005}, {3740,12675}, {3767,5013}, {3793,7762}, {3813,8715}, {3814,5267}, {3816,5248}, {3822,5841}, {3824,12436}, {3825,5840}, {3826,6796}, {3841,5842}, {3898,10284}, {3925,10902}, {3927,5744}, {4045,7886}, {4255,5292}, {4292,5122}, {4293,9654}, {4294,9669}, {4297,10175}, {4299,10895}, {4302,10896}, {4309,11238}, {4317,11237}, {4413,11499}, {4423,10310}, {5007,9300}, {5010,6284}, {5023,7737}, {5024,5286}, {5045,13405}, {5080,5303}, {5086,10609}, {5097,6329}, {5119,11376}, {5126,10106}, {5157,13562}, {5171,7808}, {5188,7889}, {5206,5475}, {5306,7772}, {5309,9607}, {5318,10646}, {5321,10645}, {5339,11480}, {5340,11481}, {5398,5718}, {5403,8160}, {5404,8161}, {5414,9661}, {5437,5709}, {5438,5705}, {5440,6734}, {5446,5943}, {5449,12038}, {5486,8548}, {5489,5664}, {5534,8580}, {5550,5603}, {5562,5650}, {5569,7775}, {5587,7987}, {5590,5874}, {5591,5875}, {5609,5642}, {5640,11465}, {5646,5654}, {5651,10539}, {5656,13093}, {5658,12684}, {5663,5907}, {5687,10527}, {5694,5884}, {5720,8726}, {5731,5818}, {5743,13323}, {5777,11227}, {5878,10606}, {5883,11281}, {5889,7999}, {5890,11444}, {5894,11204}, {5944,12134}, {6000,6696}, {6033,7944}, {6055,9167}, {6130,8552}, {6153,13433}, {6194,7875}, {6199,7582}, {6247,6759}, {6248,7820}, {6287,9751}, {6321,7847}, {6368,10213}, {6395,7581}, {6409,6561}, {6410,6560}, {6417,7586}, {6418,7585}, {6425,9680}, {6449,6459}, {6450,6460}, {6455,9541}, {6502,9646}, {6592,8902}, {6669,6673}, {6670,6674}, {6688,10110}, {6722,7861}, {6723,11801}, {6746,12363}, {7028,8130}, {7280,7354}, {7308,7330}, {7603,7747}, {7610,12040}, {7616,8859}, {7622,7781}, {7697,7835}, {7709,7891}, {7735,9605}, {7743,10624}, {7750,7752}, {7751,13468}, {7756,8589}, {7758,8667}, {7761,7862}, {7778,7800}, {7801,11168}, {7806,12251}, {7810,7821}, {7811,7814}, {7831,7899}, {7834,9737}, {7844,9734}, {7854,7888}, {7863,9466}, {7868,9744}, {7881,9755}, {7904,7912}, {7914,9996}, {7956,8167}, {8071,10320}, {8125,8128}, {8126,8127}, {8141,10319}, {8148,10595}, {8151,10190}, {8666,12607}, {8721,9756}, {8722,10358}, {9301,10357}, {9655,10590}, {9668,10591}, {9704,11003}, {9707,11457}, {9729,9820}, {9781,11451}, {9826,13416}, {10006,11247}, {10163,12506}, {10171,12512}, {10189,10280}, {10198,11249}, {10225,11813}, {10277,13582}, {10517,11916}, {10518,11917}, {10541,11179}, {10572,12019}, {10574,11459}, {10584,11928}, {10585,11929}, {10586,12000}, {10587,12001}, {10628,11561}, {10915,11260}, {10950,11545}, {11015,12690}, {11017,13474}, {11176,11615}, {11219,12738}, {11246,11544}, {11255,11511}, {11265,11513}, {11266,11514}, {11267,11515}, {11268,11516}, {11426,11433}, {11427,11432}, {11430,12241}, {11438,12233}, {11485,11489}, {11486,11488}, {11703,11792}, {11808,13365}, {12162,13491}, {12325,13432}, {12358,13148}, {13142,13352}, {13470,13565}

X(140) is the {X(2),X(3)}-harmonic conjugate of X(5). For a list of
other harmonic conjugates of X(140), click **Tables** at the top of
this page.

X(140) = midpoint of X(i) and X(j) for these (i,j): (3,5), (141,182), (389,1216), (2883, 3357)

X(140) = reflection of X(i) in X(j) for these (i,j): (546,5), (547,2), (548,3)

X(140) = isogonal conjugate of X(1173)

X(140) = complement of X(5)

X(140) = complementary conjugate of X(1209)

X(140) = X(2)-Ceva conjugate of X(233)

X(140) = crosspoint of X(i) and X(j) for these (i,j): (2,95), (17,18)

X(140) = crosssum of X(i) and X(j) for these (i,j): (6,51), (61,62)

X(140) = crosspoint of the two Napoleon points, X(17) and X(18)

X(140) = inverse-in-orthocentroidal-circle of X(1656)

X(140) = X(5)-of-medial triangle

X(140) = centroid of the quadrangle ABCX(3)

X(140) = perspector of circumconic centered at X(233)

X(140) = center of circumconic that is locus of trilinear poles of lines passing through X(233)

X(140) = intersection of tangents to Evans conic at X(3070) and X(3071)

X(140) = centroid of X(2)X(3)X(115)X(2482)

X(140) = anticomplement of X(3628)

X(140) = X(3) of polar triangle of complement of polar circle

X(140) = inverse-in-complement-of-polar-circle of X(2072)

X(140) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5189)

X(140) = center of inverse-in-{circumcircle, nine-point circle}-inverter of anticomplementary circle

X(140) = centroid of the six circumcenters in the construction of the van Lamoen circle.

X(140) = centroid of ABCX(3)

X(140) = Kosnita(X(3),X(2)) point

X(140) = center of circle that is locus of crosssums of antipodes on the 2nd Lemoine circle

X(140) = {X(2),X(5)}-harmonic conjugate of X(3628)

X(140) = {X(3),X(4)}-harmonic conjugate of X(550)

X(140) = {X(4),X(5)}-harmonic conjugate of X(3850)

X(140) = homothetic center of X(4)-altimedial and X(140)-anti-altimedial triangles

X(140) = X(3579)-of-orthic-triangle if ABC is acute

Trilinears csc

Barycentrics b

Barycentrics cot A + cot ω : :

Barycentrics SA + SW : :

X(141) = 3*X(2) + X(69) = 3*X(2) - X(6)

Let P be a point on the circumcircle, and let L be the line tangent to the circumcircle at P. Let P' be the trilinear pole of L, and let P" be the isotomic conjugate of P'. As P traces the circumcircle, P" traces an ellipse inscribed in ABC with center at X(141). (Randy Hutson, December 26, 2015)

Let A'B'C' be the 2nd Brocard triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(141). (Randy Hutson, December 26, 2015)

X(141) lies on the bicevian conic of X(2) and X(110) and on these lines:

2,6 3,66 5,211 10,142
37,742 39,732 45,344
53,264 67,110 75,334
76,698 95,287 99,755
113,127 116,121 125,126
140,182 239,319 241,307
308,670 311,338 317,458
320,894 384,1031 441,577
498,611 499,613 523,882
542,549 575,629 997,1060

X(141) is the {X(2),X(69)}-harmonic conjugate of X(6). For a list of
other harmonic conjugates of X(141), click **Tables** at the top of
this page.

X(141) = midpoint of X(i) and X(j) for these (i,j): (1,3416), (6,69), (8,3242), (66,159), (67,110), (69,3313), (1843,3313) (2930, 3448)

X(141) = reflection of X(i) in X(j) for these (i,j): (182,140), (597,2), (1353,575), (1386,1125)

X(141) = isogonal conjugate of X(251)

X(141) = isotomic conjugate of X(83)

X(141) = inverse-in-nine-point-circle of X(625)

X(141) = complement of X(6)

X(141) = complementary conjugate of X(2)

X(141) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,39), (67,524), (110,525)

X(141) = X(39)-cross conjugate of X(427)

X(141) = crosspoint of X(2) and X(76)

X(141) = crosssum of X(6) and X(32)

X(141) = X(39)-Hirst inverse of X(732)

X(141) = X(645)-beth conjugate of X(141)

X(141) = X(6)-of-medial triangle

X(141) = anticomplement of X(3589)

X(141) = centroid of ABCX(69)

X(141) = Kosnita(X(69),X(2)) point

X(141) = perspector of circumconic centered at X(39)

X(141) = center of circumconic that is locus of trilinear poles of lines passing through X(39)

X(141) = bicentric sum of PU(11)

X(141) = midpoint of PU(11)

X(141) = polar conjugate of X(32085)

X(141) = X(6)-of-X(2)-Brocard-triangle

X(141) = X(115)-of-1st-Brocard-triangle

X(141) = crosspoint of X(2) and X(2896) wrt excentral triangle

X(141) = crosspoint of X(2) and X(2896) wrt anticomplementary triangle

X(141) = crosspoint of X(6) and X(2916) wrt excentral triangle

X(141) = crosspoint of X(6) and X(2916) wrt tangential triangle

X(141) = {X(2),X(6)}-harmonic conjugate of X(3589)

X(141) = {X(395),X(396)}-harmonic conjugate of X(5306)

X(141) = trilinear cube root of X(14125)

X(141) = perspector of 2nd Brocard triangle and cross-triangle of ABC and 2nd Brocard triangle

X(141) = intersection, other than X(3), of the orthosymmedial circles of the 1st and 2nd Ehrmann inscribed triangles

Barycentrics ab + ac - (b - c)

Let A' be the midpoint between A and X(7), and define B' and C' cyclically; the triangle A'B'C' is homothetic to the medial triangle, and the center of homothety is X(142). (Randy Hutson, 9/23/2011)

Let A' be the intersection of these three lines:

(1) through midpoint of CA perpendicular to BX(1)

(2) through midpoint of AB perpendicular to CX(1)

(3) through midpoint of AX(1) perpendicular to BC.

Define B' and C' cyclically. Then X(142) = X(6)-of-A'B'C'. The triangle A'B'C' is the complement of the excentral triangle, and also the extraversion triangle of X(10). (Randy Hutson, September 14, 2016)

X(142) lies on these lines: 1,277 2,7 3,516 5,971 10,141 37,1086 86,284 116,119 214,528 269,948 354,3059 377,950 474,954

X(142) is the {X(2),X(7)}-harmonic conjugate of X(9). For a list of other harmonic conjugates, click **Tables** at the top of this page.

X(142) = midpoint of X(i) and X(j) for these (i,j): (7,9), (8,3243), (100,3254)

X(142) = reflection of X(1001) in X(1125)

X(142) = isogonal conjugate of X(1174)

X(142) = isotomic conjugate of X(32008)

X(142) = complement of X(9)

X(142) = X(100)-Ceva conjugate of X(514)

X(142) = crosspoint of X(2) and X(85)

X(142) = crosssum of X(6) and X(41)

X(142) = X(190)-beth conjugate of X(142)

X(142) = X(9)-of- medial triangle

X(142) = centroid of the set {X(1), X(4), X(7), X(40)}

X(142) = perspector of circumconic centered at X(1212)

X(142) = X(2)-Ceva conjugate of X(1212)

X(142) = centroid of ABCX(7)

X(142) = center of circumconic that is locus of trilinear poles of lines passing through X(1212)

X(142) = X(9969)-of-excentral-triangle

Trilinears (1 - 2 cos 2A)cos(B - C)]: :

Trilinears sec A cos(3A) cos(B - C) : :

Barycentrics (tan A)[cos(2C - 2A) + cos(2A - 2B)] : :

Barycentrics a^2 (a^2 b^2 + a^2 c^2 + 2 b^2 c^2 - b^4 - c^4) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - b^2 c^2) : :

X(143) is the third of three Spanish Points developed by Antreas P. Hatzipolakis and Javier Garcia Capitan in 2009; see X(3567).

Let A'B'C' be the cevian triangle of X(5). Let A", B", C" be the inverse-in-circumcircle of A', B', C'. The lines AA", BB", CC" concur in X(143). Also, X(143) = intersection of the tangent to hyperbola {{A,B,C,X(4),X(15)}} at X(61) and the tangent to the hyperbola {{A,B,C,X(4),X(16)}} at X(62). (Randy Hutson, July 23, 2015)

X(143) is the QA-P13 center (Nine-point Center of the QA-Diagonal Triangle) of quadrangle ABCX(4) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/41-qa-p13.html). Also, X(143) is the QA-P22 center (Midpoint QA-P1 and QA-P20) of quadrangle ABCX(4) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/50-qa-p22.html) /p>

X(143) lies on the curves K054, K416, K464, Q106, and these lines:

{2,6101}, {3,1173}, {4,94}, {5,51}, {6,26}, {23,1199}, {25,156}, {30,389}, {49,1493}, {54,2070}, {61,2912}, {62,2913}, {110,195}, {140,511}, {182,7525}, {185,3627}, {324,565}, {381,5876}, {382,5890}, {567,7488}, {569,7502}, {575,7555}, {576,1147}, {578,1658}, {632,3917}, {970,7508}, {1181,7530}, {1216,3628}, {1351,6642}, {1353,1843}, {1656,5640}, {1993,7506}, {2392,5885}, {2937,5012}, {2979,3526}, {3517,5093}, {3530,5892}, {3580,5576}, {3830,6241}, {3850,5907}, {3853,6000}, {5070,7999}, {5609,7545}, {6515,7528}, {7517,7592}

X(143) = midpoint of X(i) and X(j) for these {i,j}: {4, 6102}, {5, 52}, {185, 3627}, {389, 5446}, {1353, 1843}, {1493, 6152}, {3060, 5946}, {5876, 5889}, {6101, 6243}

X(143) = reflection of X(i) in X(j) for these {i,j}: {140, 5462}, {1216, 3628}, {5907,3850}

X(143) = isogonal conjugate of X(252)

X(143) = anticomplement of X(32142)

X(143) = X(137)-cross conjugate of X(1510)

X(143) = X(5)-of-orthic triangle

X(143) = X(249)-Ceva conjugate of X(1625)

X(143) = X(137)-cross conjugate of X(1510)

X(143) = excentral-to-ABC functional image of X(5)

X(143) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6243,6101), (3,3567,5946), (4,568,6102), (49,1994,1493), (51,52,5), (54,2070,5944), (381,5889,5876), (1112,6746,4), (1216,5943,3628), (1994,3518,49), (3060,3567,3)

X(143) = X(i)-isoconjugate of X(j) for these {i,j}: {1,252}, {54,2962}, {93,2169}, {930,2616}, {2167,2963}, {2190,3519}

Barycentrics tan B/2 + tan C/2 - tan A/2 : :

Barycentrics 1/(a - b - c) + 1/(a - b + c) + 1/(a + b - c) : :

Barycentrics 3a^2 - 2a(b + c) - (b - c)^2 : :

Barycentrics 3*a^2 - 2*a*b - b^2 - 2*a*c + 2*b*c - c^2 : :

X(144) = 3 X[2] - 4 X[9],9 X[2] - 8 X[142],5 X[2] - 4 X[6173],15 X[2] - 16 X[6666],9 X[2] - 10 X[18230],21 X[2] - 20 X[20195],2 X[3] - 3 X[21168],3 X[4] - 2 X[31671],3 X[7] - 4 X[142],X[7] - 3 X[6172],5 X[7] - 6 X[6173],5 X[7] - 8 X[6666],3 X[7] - 5 X[18230],7 X[7] - 10 X[20195],3 X[9] - 2 X[142],2 X[9] - 3 X[6172],5 X[9] - 3 X[6173],5 X[9] - 4 X[6666],6 X[9] - 5 X[18230],4 X[9] - X[20059],7 X[9] - 5 X[20195],2 X[75] - 3 X[27484],4 X[142] - 9 X[6172],10 X[142] - 9 X[6173],5 X[142] - 6 X[6666],4 X[142] - 5 X[18230],8 X[142] - 3 X[20059],14 X[142] - 15 X[20195],X[145] - 4 X[5698],3 X[210] - 2 X[15587],3 X[210] - X[31391]

X(144) lies on the Feuerbach circumhyperbola of the anticomplementary triangle, the Mandart hyperbola, the cubics K200, K202, K308, K710, K1044, K1084, and on these lines:: {2, 7}, {3, 5843}, {4, 2894}, {6, 3672}, {8, 516}, {10, 4312}, {12, 18231}, {20, 72}, {21, 954}, {37, 3945}, {40, 5815}, {42, 4335}, {44, 4000}, {45, 4648}, {65, 27288}, {69, 190}, {71, 27544}, {75, 391}, {77, 2324}, {78, 3522}, {85, 25001}, {86, 4747}, {92, 6994}, {100, 480}, {145, 192}, {149, 1156}, {153, 1145}, {165, 21060}, {175, 30557}, {176, 30556}, {191, 3085}, {194, 20036}, {200, 2951}, {210, 3474}, {213, 4352}, {218, 17691}, {219, 347}, {220, 279}, {238, 4310}, {239, 4452}, {241, 26669}, {269, 25930}, {281, 7282}, {306, 25734}, {320, 344}, {321, 14552}, {345, 33066}, {348, 10004}, {376, 3940}, {405, 11036}, {443, 15650}, {452, 3868}, {524, 17262}, {528, 12531}, {536, 5839}, {545, 4361}, {573, 21362}, {597, 17323}, {599, 17340}, {631, 31657}, {658, 17113}, {666, 30228}, {673, 4373}, {758, 18412}, {912, 6987}, {920, 15518}, {938, 10398}, {942, 5129}, {950, 20008}, {956, 6912}, {960, 3600}, {962, 6766}, {966, 4363}, {984, 4307}, {1001, 2975}, {1086, 16885}, {1119, 26003}, {1212, 24554}, {1213, 4470}, {1260, 7411}, {1266, 4402}, {1278, 20248}, {1418, 25067}, {1419, 3160}, {1434, 24557}, {1441, 4047}, {1633, 12329}, {1654, 2475}, {1721, 28043}, {1742, 2340}, {1743, 3663}, {1757, 24248}, {1761, 14543}, {1766, 7291}, {1788, 8165}, {1901, 31043}, {1959, 4704}, {1992, 4360}, {1999, 10889}, {2095, 6939}, {2245, 27039}, {2267, 18162}, {2287, 5781}, {2293, 24708}, {2321, 32099}, {2325, 17296}, {2345, 4643}, {2478, 5729}, {2664, 25570}, {2801, 6224}, {2895, 2897}, {3008, 3973}, {3059, 3681}, {3086, 6763}, {3091, 5805}, {3161, 3912}, {3174, 3935}, {3241, 30331}, {3243, 3623}, {3247, 4667}, {3339, 18250}, {3419, 3543}, {3434, 16112}, {3475, 3683}, {3476, 31165}, {3487, 17558}, {3523, 3916}, {3589, 17255}, {3616, 5542}, {3618, 4389}, {3619, 17273}, {3620, 4741}, {3621, 5853}, {3629, 17318}, {3630, 17309}, {3631, 17269}, {3640, 30334}, {3641, 30333}, {3648, 5696}, {3650, 5687}, {3664, 3731}, {3671, 5234}, {3686, 4659}, {3687, 10443}, {3715, 11246}, {3739, 7222}, {3758, 17258}, {3812, 28646}, {3826, 11681}, {3832, 5735}, {3839, 18482}, {3870, 4326}, {3873, 5572}, {3876, 6904}, {3946, 16670}, {3949, 24683}, {3952, 4019}, {3958, 8680}, {4001, 34255}, {4021, 16667}, {4073, 4712}, {4098, 29602}, {4110, 25278}, {4182, 20534}, {4190, 5784}, {4192, 22149}, {4292, 5785}, {4293, 5692}, {4294, 5904}, {4301, 24644}, {4308, 15829}, {4313, 11523}, {4321, 19861}, {4329, 5227}, {4343, 17018}, {4344, 7174}, {4353, 16469}, {4359, 20921}, {4370, 17267}, {4371, 4686}, {4384, 31995}, {4422, 7232}, {4430, 7671}, {4462, 6008}, {4468, 6006}, {4473, 16593}, {4512, 10578}, {4552, 20082}, {4640, 5281}, {4645, 27549}, {4652, 15717}, {4661, 7674}, {4670, 28640}, {4675, 16814}, {4678, 24393}, {4683, 33163}, {4699, 24633}, {4715, 4851}, {4748, 17303}, {4758, 28626}, {4788, 20016}, {4795, 28639}, {4847, 9812}, {4855, 21734}, {4859, 4887}, {4882, 5493}, {4888, 29571}, {4896, 25072}, {4902, 31183}, {4912, 17348}, {4915, 28228}, {5044, 17580}, {5175, 17578}, {5176, 17488}, {5177, 11662}, {5231, 9779}, {5274, 24477}, {5290, 18249}, {5302, 28629}, {5440, 10304}, {5587, 5775}, {5703, 31424}, {5708, 17559}, {5709, 5811}, {5714, 5791}, {5738, 31049}, {5739, 32933}, {5758, 7330}, {5766, 7675}, {5809, 12649}, {5832, 6871}, {5833, 9612}, {5857, 20060}, {5880, 15481}, {6007, 20012}, {6067, 11680}, {6144, 17388}, {6147, 16845}, {6244, 10307}, {6350, 32849}, {6360, 20211}, {6542, 20080}, {6600, 7676}, {6604, 32024}, {6605, 10509}, {6762, 9785}, {6764, 10624}, {6885, 31835}, {6908, 26921}, {6926, 24467}, {6995, 7717}, {7155, 17794}, {7172, 32937}, {7175, 9310}, {7201, 17451}, {7227, 17251}, {7238, 17265}, {7262, 33144}, {7277, 16777}, {7321, 17335}, {7613, 32857}, {7670, 16019}, {7672, 13601}, {7677, 24558}, {8055, 30567}, {8163, 12513}, {9533, 31627}, {9780, 30424}, {9797, 12575}, {10005, 32850}, {10177, 11025}, {10392, 24391}, {10442, 11679}, {10446, 21061}, {10580, 30330}, {10590, 17057}, {11008, 17377}, {11037, 31435}, {11160, 17373}, {12125, 12632}, {12514, 15298}, {12560, 19860}, {12630, 20014}, {12670, 14872}, {14555, 32939}, {14986, 15299}, {15254, 30340}, {15374, 28071}, {15492, 17278}, {15680, 20013}, {15913, 31527}, {16020, 24231}, {16284, 21872}, {16435, 23089}, {16552, 17753}, {16566, 28795}, {16669, 17301}, {16675, 17392}, {16713, 17139}, {17002, 26245}, {17007, 26032}, {17045, 24441}, {17051, 26105}, {17116, 17331}, {17118, 17330}, {17120, 17247}, {17132, 17151}, {17137, 27523}, {17147, 20043}, {17170, 17742}, {17183, 18206}, {17234, 31333}, {17243, 28333}, {17253, 17369}, {17261, 17316}, {17264, 17361}, {17272, 17355}, {17279, 17345}, {17281, 17344}, {17285, 21356}, {17288, 17339}, {17289, 17329}, {17298, 25101}, {17300, 29621}, {17327, 26039}, {17343, 21286}, {17375, 29583}, {17496, 20296}, {17582, 24470}, {17615, 17668}, {17776, 32859}, {18161, 21801}, {18600, 27644}, {18661, 24435}, {19742, 19789}, {19993, 20068}, {19998, 22312}, {20009, 20077}, {20018, 25264}, {20020, 20064}, {20101, 31087}, {20905, 30854}, {20930, 28974}, {20992, 21320}, {21039, 24341}, {21075, 27525}, {21084, 28124}, {21219, 30662}, {22003, 24048}, {24597, 33151}, {25000, 27541}, {26034, 32938}, {26768, 27136}, {26871, 32863}, {27268, 27475}, {27543, 28420}, {28628, 28645}, {30225, 32028}, {32007, 32100}, {33099, 33137}

X(144) = reflection of X(i) in X(j) for these {i,j}: {2, 6172}, {4, 5779}, {7, 9}, {8, 5223}, {20, 5759}, {100, 6068}, {145, 390}, {149, 1156}, {390, 5698}, {962, 11372}, {2550, 5220}, {3868, 5728}, {4312, 10}, {4430, 7671}, {4440, 673}, {4452, 5838}, {5880, 15481}, {5905, 8545}, {8581, 960}, {9965, 12848}, {17314, 17262}, {20014, 12630}, {20059, 7}, {20533, 190}, {25722, 3059}, {30628, 14100}, {31391, 15587}

X(144) = isogonal conjugate of X(11051)

X(144) = isotomic conjugate of X(10405)

X(144) = complement of X(20059)

X(144) = anticomplement of X(7)

X(144) = anticomplementary conjugate of X(3434)

X(144) = anticomplementary-isogonal conjugate of X(3434)

X(144) = polar conjugate of the isogonal of X(22117)

X(144) = perspector of anticomplementary triangle and **its** intouch triangle (inner Conway triangle)

X(144) = perspector of anticomplementary triangle and the extouch triangle of ABC

X(144) = perspector of extouch triangle and inner Conway triangle (Gemini triangle 30)

X(144) = X(7)-of-anticomplementary triangle

X(144) = X(i)-beth conjugate of X(j) for these (i,j): (190,144), (645,346)

X(144) = Conway-triangle-to-inner-Conway-triangle similarity image of X(7)

X(144) = X(i)-Ceva conjugate of X(j) for these (i,j): {8, 2}, {516, 20533}, {3729, 192}, {4416, 1654}, {4480, 17487}, {5223, 27484}, {31627, 3160}

X(144) = X(i)-cross conjugate of X(j) for these (i,j): {165, 3160}, {3160, 2}, {21060, 16284}, {21872, 165}

X(144) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11051}, {6, 3062}, {31, 10405}, {56, 19605}

X(144) = cevapoint of X(i) and X(j) for these (i,j): {7, 15913}, {9, 2951}, {57, 7955}, {220, 6244}, {3207, 22117}, {21060, 21872}

X(144) = crosspoint of X(i) and X(j) for these (i,j): {190, 1275}, {16284, 31627}

X(144) = crosssum of X(649) and X(14936)

X(144) = crossdifference of every pair of points on line {663, 20980}

X(144) = barycentric product X(i)*X(j) for these {i,j}: {1, 16284}, {8, 3160}, {9, 31627}, {75, 165}, {76, 3207}, {86, 21060}, {190, 7658}, {264, 22117}, {274, 21872}, {312, 1419}, {341, 17106}, {346, 9533}, {1275, 13609}

X(144) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3062}, {2, 10405}, {6, 11051}, {9, 19605}, {165, 1}, {1419, 57}, {3160, 7}, {3207, 6}, {7658, 514}, {9533, 279}, {13609, 1146}, {15856, 30330}, {16284, 75}, {17106, 269}, {21060, 10}, {21872, 37}, {22117, 3}, {23058, 24856}, {31627, 85}

X(144) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 3434}, {2, 21285}, {6, 7}, {8, 6327}, {9, 69}, {21, 17135}, {25, 12649}, {29, 20242}, {31, 145}, {32, 3210}, {33, 4}, {37, 2893}, {41, 2}, {42, 2475}, {43, 20350}, {48, 347}, {55, 8}, {57, 6604}, {58, 3873}, {60, 17140}, {71, 2897}, {75, 21280}, {78, 1370}, {81, 20244}, {100, 21302}, {101, 693}, {109, 3900}, {163, 4467}, {192, 20559}, {198, 5932}, {200, 3436}, {210, 1330}, {212, 20}, {213, 17778}, {219, 4329}, {220, 329}, {228, 3152}, {251, 20247}, {281, 21270}, {282, 21279}, {283, 20243}, {284, 75}, {294, 20347}, {312, 315}, {314, 17138}, {318, 11442}, {333, 17137}, {346, 21286}, {522, 21293}, {604, 4452}, {607, 5905}, {643, 512}, {644, 20295}, {645, 17217}, {646, 21304}, {650, 150}, {662, 4374}, {663, 149}, {692, 522}, {765, 3888}, {904, 29840}, {911, 9436}, {923, 4442}, {983, 25304}, {1110, 100}, {1126, 20292}, {1172, 17220}, {1174, 85}, {1212, 2890}, {1252, 21272}, {1253, 144}, {1320, 21282}, {1333, 3875}, {1334, 2895}, {1395, 11851}, {1397, 17480}, {1412, 17158}, {1415, 4025}, {1812, 18659}, {1857, 5906}, {1973, 30699}, {2053, 10453}, {2149, 664}, {2150, 4360}, {2175, 192}, {2176, 20537}, {2185, 17143}, {2188, 280}, {2191, 6601}, {2192, 962}, {2193, 17134}, {2194, 1}, {2195, 518}, {2200, 18667}, {2204, 3187}, {2212, 193}, {2251, 30577}, {2258, 388}, {2259, 1441}, {2287, 20245}, {2289, 6527}, {2299, 3868}, {2311, 30941}, {2316, 320}, {2319, 21281}, {2320, 21283}, {2321, 21287}, {2328, 3869}, {2329, 30660}, {2332, 92}, {2338, 4872}, {2340, 20344}, {2341, 17139}, {2342, 517}, {2344, 4441}, {2360, 20221}, {2361, 6224}, {3063, 4440}, {3207, 31527}, {3596, 21275}, {3683, 2891}, {3684, 20345}, {3685, 20554}, {3688, 21289}, {3689, 21290}, {3693, 20552}, {3699, 21301}, {3700, 21294}, {3709, 21221}, {3711, 21291}, {3939, 513}, {4041, 3448}, {4166, 20346}, {4182, 20555}, {4548, 21215}, {4612, 17159}, {4636, 17166}, {4845, 5057}, {4876, 20553}, {5546, 7192}, {5547, 17491}, {5548, 21297}, {6065, 3952}, {6602, 30695}, {7037, 9799}, {7054, 21273}, {7069, 2888}, {7070, 6225}, {7071, 5942}, {7072, 11415}, {7074, 6223}, {7075, 32548}, {7077, 4645}, {7084, 17784}, {7110, 21276}, {7115, 4566}, {7118, 9965}, {7156, 14361}, {7367, 189}, {8611, 13219}, {8750, 521}, {8851, 20352}, {9439, 497}, {9447, 194}, {9448, 17486}, {9456, 1266}, {10482, 3681}, {11051, 32003}, {13455, 637}, {14547, 2894}, {14827, 3177}, {14942, 20556}, {15374, 9801}, {18265, 17759}, {18889, 527}, {20967, 5484}, {21059, 7674}, {23990, 4552}, {24019, 23683}, {28615, 3879}, {32635, 20290}, {32652, 8058}, {32665, 4453}, {32674, 17896}, {32677, 22464}, {32739, 17496}, {33299, 1369}

X(144) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 9965, 21454}, {2, 20059, 7}, {2, 20078, 9965}, {2, 20214, 5905}, {6, 3672, 17014}, {6, 4419, 3672}, {6, 17334, 4419}, {7, 9, 2}, {7, 6172, 9}, {7, 18230, 142}, {7, 29007, 8232}, {8, 3729, 4461}, {8, 4488, 3729}, {8, 30625, 30695}, {8, 30695, 10405}, {9, 142, 18230}, {9, 6173, 6666}, {20, 72, 20007}, {37, 3945, 29624}, {37, 4644, 3945}, {44, 17276, 4000}, {45, 17365, 4648}, {57, 18228, 2}, {63, 329, 2}, {63, 908, 5744}, {63, 17781, 329}, {69, 190, 346}, {69, 346, 29616}, {142, 18230, 2}, {190, 17347, 69}, {192, 193, 145}, {192, 20072, 193}, {193, 20073, 192}, {193, 20111, 20110}, {210, 31391, 15587}, {226, 3929, 5273}, {226, 5273, 2}, {307, 27382, 2}, {320, 344, 4869}, {320, 17336, 344}, {329, 5744, 908}, {329, 20348, 20245}, {391, 4454, 75}, {480, 11495, 100}, {672, 1423, 27624}, {672, 30946, 2}, {894, 17257, 2}, {894, 17333, 17257}, {908, 5744, 2}, {984, 24695, 4307}, {1001, 11038, 3622}, {1743, 3663, 5222}, {2287, 8822, 14953}, {2345, 4643, 5232}, {2550, 5220, 5686}, {2550, 5686, 3617}, {3161, 21296, 3912}, {3177, 3869, 20535}, {3177, 20111, 145}, {3218, 31018, 2}, {3219, 5905, 2}, {3243, 8236, 3623}, {3305, 9776, 2}, {3452, 3928, 5435}, {3452, 5435, 2}, {3487, 31445, 17558}, {3662, 26685, 2}, {3664, 3731, 5308}, {3681, 25722, 3059}, {3686, 4659, 32087}, {3715, 11246, 26040}, {3729, 4416, 8}, {3729, 4480, 4488}, {3758, 17258, 17321}, {3869, 30616, 20111}, {3911, 5328, 2}, {3911, 31142, 5328}, {3912, 25728, 3161}, {3973, 4862, 3008}, {4000, 17276, 4346}, {4357, 5749, 2}, {4363, 17332, 966}, {4416, 4480, 3729}, {4416, 4488, 4461}, {4640, 25568, 5281}, {4643, 17351, 2345}, {4652, 27383, 15717}, {4661, 20075, 20015}, {4704, 20090, 29585}, {4741, 17280, 3620}, {5226, 5745, 2}, {5296, 10436, 2}, {5698, 10394, 6872}, {5745, 28609, 5226}, {5805, 5817, 3091}, {6646, 17350, 2}, {12246, 31793, 20}, {12526, 12527, 8}, {16552, 17753, 27304}, {17120, 17247, 26626}, {17183, 18206, 26818}, {17261, 17364, 17316}, {17272, 17355, 29611}, {17273, 17354, 3619}, {17288, 17339, 29579}, {17298, 25101, 29627}, {18228, 28610, 57}, {20072, 20073, 145}, {21151, 31658, 3523}, {24477, 24703, 5274}, {24909, 25679, 2}, {24952, 25461, 2}, {26059, 26125, 2}, {26065, 27184, 2}, {27058, 27170, 2}, {27282, 27334, 2}, {27509, 28739, 2}, {29621, 32093, 17300}, {31547, 31548, 8}

Trilinears -1 + csc A/2 sin B/2 sin C/2 : :

Trilinears 2(r/R) - sin B sin C : :

Barycentrics 3a - b - c : :

X(145) = 4 X(1) + 3 X(2)

Let A' be the reflection of the midpoint of segment BC in X(1), and define B' and C' cyclically. The lines AA', BB', CC' concur in X(145). Let A'' be the reflection of the A in X(1), and define B'' and C'' cyclically. Let A'''B'''C''' be the intouch triangle. The lines A''A''', B''B''', C''C''' concur in X(145). (Randy Hutson, 9/23/2011)

Let O_{A} be the circle tangent to side BC at its midpoint
and to the circumcircle on the same side of BC as A. Define
O_{B}, O_{C} cyclically. Then X(145) is the trilinear
pole of the line of the exsimilicenters (the Monge line) of
O_{A}, O_{B}, O_{C}. See the reference at
X(1001).

Let Ha be the hyperbola passing through A, with foci at B and C. This is called the A-Soddy hyperbola in (Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143). Let La be the polar of X(2) with respect to Ha. Define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(145). (Randy Hutson, September 5, 2015)

Let A'B'C' and A"B"C" be the intouch and extouch triangles. X(145) is the radical center of the circumcircles of AA'A", BB'B", CC'C". (Randy Hutson, September 14, 2016)

X(145) is the {X(1),X(8)}-harmonic conjugate of X(2). For a list of
other harmonic conjugates of X(145), click **Tables** at the top of
this page.

X(145) lies on these lines: 1,2 4,149 6,346 20,517 21,956 37,391 56,100 72,452 81,1043 144,192 218,644 279,664 329,950 330,1002 377,1056 404,999 515,962

X(145) = midpoint of X(2) and X(3241)

X(145) = reflection of X(i) in X(j) for these (i,j): (3,1483), (4,1482), (8,1), (20,944), (100,1317), (144,390), (149,1320)

X(145) = isogonal conjugate of X(3445)

X(145) = isotomic conjugate of X(4373)

X(145) = anticomplement of X(8)

X(145) = anticomplementary conjugate of X(3436)

X(145) = X(7)-Ceva conjugate of X(2)

X(145) = crosssum of X(663) and X(1015)

X(145) = X(643)-beth conjugate of X(56)

X(145) = exsimilicenter of incircle and AC-incircle

X(145) = X(64)-of-intouch-triangle

X(145) = trilinear pole of line X(2976)X(3667) (radical axis of incircle and AC-incircle, and the pole of X(2) wrt the Spieker circle)

X(145) = inverse-in-Steiner-circumellipse of X(3008)

X(145) = {X(i), X(j)-harmonic conjugate of X(k) for these (i,j,k): (1,2,3622), (1,10,3616), (2,8,3617), (8,10,4678)

X(145) = X(8)-of-anticomplementary-triangle

X(145) = crossdifference of every pair of points on line X(649)X(6363)

X(145) = X(11381)-of-excentral-triangle

X(145) = perspector of intouch triangle and Gemini triangle 29

Barycentrics -avw + bwu + cuv : -bwu + cuv + avw : -cuv + avw + bwu

Barycentrics a^10 + a^8 (b^2 + c^2) - a^6 (8 b^4 - 9 b^2 c^2 + 8 c^4) + 2 a^4 (b^2 + c^2) (4 b^4 - 7 b^2 c^2 + 4 c^4) - a^2 (b^2 - c^2)^2 (b^4 + 9 b^2 c^2 + c^4) - (b^2 - c^2)^4 (b^2 + c^2) : : X(146) lies on the anticomplementary circle and these lines: 2,74 4,94 20,110 30,323 147,690 148,193

X(146) is the {X(74),X(113)}-harmonic conjugate of X(2). For a list of harmonic conjugates, click **Tables** at the top of this page.

X(146) = reflection of X(i) in X(j) for these (i,j): (20,110), (74,113), (265,1539)

X(146) = isogonal conjugate of X(34178)

X(146) = anticomplementary conjugate of X(30)

X(146) = X(74)-of-anticomplementary triangle

X(146) = crosspoint of X(399) and X(2935) wrt both the excentral and tangential triangles

Barycentrics a sec(A + ω) - b sec(B + ω) - c sec(C + ω) : :

X(147) lies on the anticomplementary circle and these lines: 1,150 2,98 4,148 20,99 132,648 146,690 684,804

X(147) is the {X(98),X(114)}-harmonic conjugate of X(2). For a list of harmonic conjugates, click **Tables** at the top of this page.

X(147) = reflection of X(i) in X(j) for these (i,j): (20,99), (98,114), (148,4), (385,1513)

X(147) = anticomplementary conjugate of X(511)

X(147) = X(325)-Ceva conjugate of X(2)

X(147) = anticomplementary isotomic conjugate of X(385)

X(147) = X(4) of 1st anti-Brocard triangle

X(147) = perspector of anticomplementary and 2nd Neuberg triangles

X(147) = perspector of 1st anti-Brocard and 2nd Neuberg triangles

X(147) = perspector of 2nd Neuberg triangle and cross-triangle of ABC and 1st Neuberg triangle

X(147) = X(98)-of-anticomplementary triangle

X(147) = isogonal conjugate of X(34130)

X(147) = isotomic conjugate of X(9473)

X(147) = complement of X(5984)

X(147) = anticomplement of X(98)

f(a,b,c) = bc[a

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = a^{4} - (b^{2} - c^{2})^{2}
+ b^{2}c^{2} - a^{2}b^{2} -
a^{2}c^{2}

X(148) = X(99)-of-anticomplementary triangle

X(148) lies on the anticomplementary circle and these lines: 2,99 4,147 13,617 20,98 30,385 146,193 316,538

X(148) = reflection of X(i) in X(j) for these (i,j): (2,671), (20,98), (99,115), (147,4), (616,14), (617,13)

X(148) = anticomplementary conjugate of X(512)

X(148) = X(523)-Ceva conjugate of X(2)

X(148) = X(2)-Hirst inverse of X(115)

X(148) = crosssum of PU(2)

X(148) = crosspoint of PU(40)

X(148) = intersection of tangents at PU(40) to conic {{A,B,C,PU(40)}} (i.e., the Steiner circumellipse)

X(148) = trilinear pole wrt anticomplementary triangle of line X(2)X(6)

X(148) = inverse-in-Steiner-circumellipse of X(115)

X(148) = {X(99),X(671)}-harmonic conjugate of X(115)

X(148) = X(69)-of-1st-anti-Brocard-triangle

X(148) = center of conic through X(2), X(8), and the extraversions of X(8)

X(148) = pole of line X(115)X(125) wrt Steiner circumellipse

f(a,b,c) = bc[b

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where

g(a,b,c)
= b^{3} + c^{3} - a^{3} + (a^{2} -
bc)(b + c) + a(bc - b^{2} - c^{2})

Let A' be the reflection of the midpoint of segment BC in X(11), and define B' and C' cyclically. The lines AA', BB', CC' concur in X(149). (Randy Hutson, 9/23/2011)

X(149) lies on the anticomplementary circle and these lines: 2,11 4,145 8,80 20,104 151,962 377,1058 404,496

X(149) is the {X(11),X(100)}-harmonic conjugate of X(2). For a list
of harmonic conjugates, click **Tables** at the top of this
page.

X(149) = reflection of X(i) in X(j) for these (i,j): (3,1484), (8,80), (20,104), (100,11), (144,1156), (145,1320), (153,4)

X(149) = isogonal conjugate of X(3446)

X(149) = isotomic conjugate of X(8047)

X(149) = anticomplementary conjugate of X(513)

X(149) = trilinear pole wrt anticomplementary triangle of line X(7)X(8)

X(149) = center of conic through X(7), X(8), and the extraversions of X(8)

X(149) = X(100)-of-anticomplementary-triangle

X(149) = homothetic center of 2nd Schiffler triangle and polar triangle of AC-incircle

f(a,b,c) = bc[b

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where

g(a,b,c)
= b^{4} + c^{4} - a^{4} + a(bc^{2}
+cb^{2} - b^{3} - c^{3}) - bc(a^{2} +
b^{2} + c^{2}) + (b + c)a^{3}

X(150) = X(101)-of-anticomplementary triangle

X(150) lies on the anticomplementary circle and these lines: 1,147 2,101 4,152 7,80 20,103 69,668 85,355 295,334 348,944 664,952

X(150) = reflection of X(i) in X(j) for these (i,j): (20,103), (101,116), (152,4), (664,1565)

X(150) = isogonal conjugate of X(34179)

X(150) = anticomplementary conjugate of X(514)

f(a,b,c) = (by + cz - ax)/a, where x : y : z = X(102)

Barycentrics a^12 + a^11 (b + c) + a^10 b c + a^9 (3 b^3 - b^2 c - b c^2 + 3 c^3) - a^8 (3 b^4 + b^3 c + b c^3 + 3 c^4) - 2 a^7 (7 b^5 + b^4 c - 4 b^3 c^2 - 4 b^2 c^3 + b c^4 + 7 c^5) - 2 a^6 b c (b^4 - 2 b^3 c - 6 b^2 c^2 - 2 b c^3 + c^4) + 2 a^5 (7 b^7 + b^6 c - 5 b^5 c^2 - 3 b^4 c^3 - 3 b^3 c^4 - 5 b^2 c^5 + b c^6 + 7 c^7) + a^4 (3 b^8 + 2 b^7 c - 4 b^6 c^2 - 2 b^5 c^3 + 2 b^4 c^4 - 2 b^3 c^5 - 4 b^2 c^6 + 2 b c^7 + 3 c^8) - a^3 (3 b^9 - b^8 c + 8 b^6 c^3 - 10 b^5 c^4 - 10 b^4 c^5 + 8 b^3 c^6 - b c^8 + 3 c^9) + a^2 b c (b^8 - 4 b^7 c - 12 b^6 c^2 + 4 b^5 c^3 + 22 b^4 c^4 + 4 b^3 c^5 - 12 b^2 c^6 - 4 b c^7 + c^8) - a (b^11 + b^10 c - 3 b^9 c^2 - 3 b^8 c^3 + 2 b^7 c^4 + 2 b^6 c^5 + 2 b^5 c^6 + 2 b^4 c^7 - 3 b^3 c^8 - 3 b^2 c^9 + b c^10 + c^11) - b^12 - b^11 c + 4 b^10 c^2 + 3 b^9 c^3 - 7 b^8 c^4 - 2 b^7 c^5 + 8 b^6 c^6 - 2 b^5 c^7 - 7 b^4 c^8 + 3 b^3 c^9 + 4 b^2 c^10 - b c^11 - c^12 : :

X(151) lies on the anticomplementary circle and these lines: 2,102 20,109 49,962 152,928

X(151) = reflection of X(i) in X(j) for these (i,j): (20,109), (102,117)

X(151) = isogonal conjugate of X(34180)

X(151) = anticomplementary conjugate of X(515)

f(a,b,c) = (by + cz - ax)/a, where x : y : z = X(103)

Barycentrics a^8 - a^7 (b + c) + a^6 (2 b^2 + b c + 2 c^2) - a^5 (5 b^3 - b^2 c - b c^2 + 5 c^3) - a^4 b c (b^2 - 6 b c + c^2) + a^3 (5 b^5 + b^4 c - 6 b^3 c^2 - 6 b^2 c^3 + b c^4 + 5 c^5) - a^2 (2 b^6 + b^5 c + 2 b^4 c^2 - 10 b^3 c^3 + 2 b^2 c^4 + b c^5 + 2 c^6) + a (b^7 - b^6 c - 3 b^5 c^2 + 3 b^4 c^3 + 3 b^3 c^4 - 3 b^2 c^5 - b c^6 + c^7) - b^8 + b^7 c + 2 b^6 c^2 - b^5 c^3 - 2 b^4 c^4 - b^3 c^5 + 2 b^2 c^6 + b c^7 - c^8 : :

X(152) lies on the anticomplementary circle and these lines: 2,103 4,150 20,101 151,928

X(152) = reflection of X(i) in X(j) for these (i,j): (20,101), (103,118), (150,4)

X(152) = isogonal conjugate of X(34181)

X(152) = anticomplementary conjugate of X(516)

X(152) = X(103)-of-anticomplementary triangle

f(a,b,c) = (by + cz - ax)/a, where x : y : z = X(104)

Barycentrics a^7 - a^6 (b + c) - a^5 (b^2 - 7 b c + c^2) + a^4 (b + c) (b^2 - 6 b c + c^2) - a^3 (b^4 + 2 b^3 c - 10 b^2 c^2 + 2 b c^3 + c^4) + a^2 (b - c)^2 (b + c) (b^2 + 6 b c + c^2) + a (b^2 - c^2)^2 (b^2 - 5 b c + c^2) - (b - c)^4 (b + c)^3 : :

X(153) lies on the anticomplementary circle and these lines: 2,104 4,145 7,80 11,388 20,100 515,908

X(153) = reflection of X(i) in X(j) for these (i,j): (20,100), (104,119), (149,4), (1320,1537)

X(153) = isogonal conjugate of X(34182)

X(153) = anticomplementary conjugate of X(517)

X(153) = X(104)-of-anticomplementary triangle

Trilinears a(tan B + tan C - tan A) : b(tan C + tan A - tan B) : c(tan A + tan B - tan C)

Trilinears (sec A - sec B sec C)a^2 : :

Barycentrics a^2(3a^4 - b^4 - c^4 - 2a^2b^2 - 2a^2c^2 + 2b^2c^2) : :

Barycentrics (sin

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the Ara triangle at X(154).

X(154) lies on these lines:

3,64 6,25 22,110 26,155 31,56 48,55 160,418 197,692 198,212 205,220 237,682

X(154) is the {X(26),X(156)}-harmonic conjugate of X(155). For a
list of harmonic conjugates of X(154), click **Tables** at the top
of this page.

X(154) = isogonal conjugate of X(253)

X(154) = complement of X(32064)

X(154) = X(3)-Ceva conjugate of X(6)

X(154) = crosssum of X(i) and X(j) for these (i,j): (64,1073), (122,525)

X(154) = X(109)-beth conjugate of X(154)

X(154) = X(2)-of-tangential triangle

X(154) = centroid of cevian triangle of X(20)

X(154) = pole wrt circumcircle of the trilinear polar of X(3)

X(154) = Thomson isogonal conjugate of X(4)

X(154) = intouch-to-ABC functional image of X(2)

Barycentrics a^2 SA [a^2 (SA^2 - SB SC) - SA (SB^2 + SC^2)] : :

Barycentrics a^2 (a^2 - b^2 - c^2) (a^6 - 3 a^4 (b^2 + c^2) + a^2 (3 b^4 - 2 b^2 c^2 + 3 c^4) - (b^2 - c^2)^2 (b^2 + c^2)) : :

X(155) = 4 R^2 X[5] - SW X[6], (J^2 - 3) X[3] - 2(J^2 - 2) X[49], (J^2 + 1) X[24] - 2 J^2 X[110], (J^2 + 3) X[25] - 4 X[52]

3 X[2] - 4 X[9820],X[3] - 3 X[3167],3 X[3] - 2 X[7689],3 X[3] - 4 X[12038],X[3] + 2 X[15083],2 X[5] - 3 X[5654],2 X[5] + X[9936],4 X[5] - 3 X[14852],3 X[6] - 2 X[8548],2 X[26] - 3 X[154],3 X[51] - 2 X[12235],3 X[51] - X[21651],X[68] - 3 X[5654],2 X[68] - 3 X[14852],4 X[110] - X[15085],3 X[154] - 4 X[156],3 X[154] - X[17834],4 X[156] - X[17834],3 X[381] - 4 X[5448],3 X[381] - 2 X[9927],3 X[381] - X[12429],2 X[1147] - 3 X[3167],3 X[1147] - X[7689],3 X[1147] - 2 X[12038],4 X[1147] - X[12163],2 X[1147] + X[12164],X[1498] + 2 X[16266],5 X[1656] - 4 X[5449],4 X[1658] - 5 X[17821],3 X[1853] - 4 X[13371],3 X[1853] - 2 X[32140],3 X[3060] + X[12271],9 X[3167] - 2 X[7689],9 X[3167] - 4 X[12038],6 X[3167] - X[12163],3 X[3167] + X[12164],3 X[3167] + 2 X[15083],5 X[3567] - X[12282],9 X[5054] - 8 X[20191],3 X[5093] - X[6391],4 X[5448] - X[12429],3 X[5587] - X[9896],3 X[5654] + X[9936],3 X[5886] - 2 X[12259],2 X[6193] + X[12293],X[6193] + 2 X[22660],4 X[7689] - 3 X[12163],2 X[7689] + 3 X[12164],X[7689] + 3 X[15083],X[8548] - 3 X[19139],5 X[8567] - 4 X[32138],4 X[9820] - X[11411],3 X[9909] - 5 X[14530],2 X[9925] + X[11477],2 X[9936] + 3 X[14852],4 X[10282] - 3 X[14070],3 X[10606] - 4 X[11250],3 X[11206] - X[31305],4 X[11255] - 3 X[17813],8 X[12038] - 3 X[12163],4 X[12038] + 3 X[12164],2 X[12038] + 3 X[15083],X[12163] + 2 X[12164],X[12163] + 4 X[15083],X[12293] - 4 X[22660],2 X[12893] - 3 X[32609],4 X[13561] - 5 X[31283],3 X[18324] - 4 X[32171],4 X[18377] - 3 X[18405],5 X[19132] - 4 X[19154]

Let (A) be the pedal circle of A wrt the tangential triangle, and define (B), (C) cyclically. The radical center of (A), (B), (C) = X(155). (Randy Hutson, December 10, 2016)

X(155) = X(4)-of-tangential-triangle. This point is also the center of the circle which cuts (extended) lines BC, CA, AB in pairs of points A' and A", B' and B", C' and C", respectively, such that angles A'AA", B'BB", C'CC" are all right angles. This is the **Dou circle,** described in **Jordi Dou,** Problem 1140, *Crux Mathematicorum,* 28 (2002) 461-462.

Let A' be the isogonal conjugate of A with respect to the triangle BCX(4), and define B' and C' cyclically. Let A''B''C'' be the orthic triangle. Then the lines A'A'', B'B'', C'C'' concur in X(155). (Randy Hutson, 9/23/2011)

X(155) lies on the following curves: Feuerbach hyperbola of T, Jerabek hyperbola of T, Kiepert circumhyperbola of T, where T = tangential triangle; the Feuerbach circumhyperbola of the orthic triangle; the cubics K006, K044, K176, K339, K364, K389, K500, K742; and on these lines: {1, 90}, {2, 7592}, {3, 49}, {4, 254}, {5, 6}, {11, 10071}, {12, 10055}, {19, 12417}, {20, 323}, {22, 1614}, {24, 110}, {25, 52}, {26, 154}, {30, 1498}, {33, 9931}, {34, 19471}, {51, 7529}, {54, 7503}, {55, 6237}, {56, 7352}, {64, 2935}, {69, 3547}, {81, 6824}, {113, 19504}, {125, 19360}, {136, 20422}, {140, 17811}, {143, 13861}, {159, 511}, {182, 7393}, {193, 3089}, {195, 381}, {215, 9672}, {219, 26921}, {221, 14988}, {222, 24467}, {225, 8757}, {265, 15317}, {275, 19196}, {287, 28695}, {343, 3549}, {355, 12422}, {371, 8909}, {372, 8943}, {378, 9938}, {382, 399}, {389, 6642}, {403, 2904}, {427, 23307}, {450, 1075}, {454, 6503}, {517, 9928}, {524, 16252}, {525, 8151}, {542, 8549}, {550, 16936}, {568, 7506}, {569, 5891}, {576, 8681}, {578, 5907}, {610, 2323}, {631, 15032}, {648, 1093}, {651, 1068}, {858, 11457}, {916, 8053}, {940, 6862}, {952, 9933}, {1060, 19349}, {1062, 19354}, {1160, 8904}, {1161, 8903}, {1199, 3090}, {1209, 32341}, {1350, 2916}, {1351, 1598}, {1368, 18914}, {1478, 18970}, {1479, 12428}, {1493, 15060}, {1495, 9714}, {1503, 14790}, {1593, 11472}, {1594, 11442}, {1596, 13142}, {1611, 10011}, {1616, 19907}, {1620, 15646}, {1625, 2207}, {1656, 5449}, {1658, 17821}, {1740, 3072}, {1829, 7497}, {1839, 7534}, {1853, 13371}, {1899, 11585}, {1974, 19141}, {1994, 3091}, {1995, 3567}, {2063, 18913}, {2072, 25738}, {2192, 8144}, {2477, 9659}, {2914, 11271}, {2929, 6102}, {2930, 5609}, {2979, 10323}, {2990, 23697}, {3047, 12219}, {3053, 32661}, {3060, 6152}, {3098, 15606}, {3197, 8141}, {3216, 6911}, {3289, 23115}, {3435, 10692}, {3448, 23306}, {3511, 19597}, {3517, 8780}, {3518, 16880}, {3527, 5093}, {3542, 6515}, {3546, 18909}, {3548, 11064}, {3580, 7505}, {3627, 15811}, {3628, 17825}, {3629, 15873}, {3832, 11004}, {3851, 14627}, {4383, 6959}, {4846, 31829}, {5012, 7509}, {5020, 5462}, {5050, 10170}, {5054, 20191}, {5070, 15037}, {5072, 15038}, {5085, 7516}, {5095, 16534}, {5374, 31386}, {5398, 16471}, {5412, 12424}, {5413, 12425}, {5453, 6914}, {5587, 9896}, {5706, 6917}, {5876, 7526}, {5886, 12259}, {5890, 17928}, {5898, 12310}, {5972, 12227}, {6000, 12085}, {6090, 9730}, {6146, 12420}, {6212, 19216}, {6213, 19215}, {6241, 11413}, {6243, 7517}, {6623, 32605}, {6643, 6776}, {6756, 31802}, {6800, 7512}, {6804, 14912}, {6944, 32911}, {7074, 32141}, {7391, 16659}, {7401, 14826}, {7404, 11427}, {7484, 13336}, {7485, 7999}, {7486, 15018}, {7487, 15741}, {7488, 9544}, {7507, 18474}, {7514, 11591}, {7527, 9716}, {7552, 12325}, {7553, 31383}, {7723, 19457}, {8200, 12415}, {8207, 12416}, {8220, 12426}, {8221, 12427}, {8227, 16472}, {8251, 19350}, {8276, 12239}, {8277, 12240}, {8538, 10602}, {8541, 9926}, {8567, 32138}, {8718, 33524}, {8954, 10962}, {8976, 13909}, {9019, 15581}, {9545, 14118}, {9645, 10535}, {9705, 11464}, {9715, 26864}, {9815, 10127}, {9909, 14530}, {9923, 9996}, {9967, 19459}, {10112, 18390}, {10224, 18356}, {10255, 32539}, {10257, 26937}, {10282, 14070}, {10575, 12174}, {10606, 11250}, {10620, 12901}, {10625, 11414}, {10628, 12412}, {10634, 19363}, {10635, 19364}, {10641, 10659}, {10642, 10660}, {10796, 12193}, {10897, 19355}, {10898, 19356}, {10942, 12430}, {10943, 12431}, {11206, 31305}, {11249, 23361}, {11255, 17813}, {11265, 17819}, {11266, 17820}, {11267, 17826}, {11268, 17827}, {11403, 16194}, {11422, 13434}, {11424, 15030}, {11426, 11479}, {11433, 18934}, {11449, 32534}, {11499, 12328}, {11557, 22550}, {11560, 12168}, {11562, 12165}, {11819, 31815}, {11898, 19362}, {12082, 23061}, {12112, 33703}, {12233, 18420}, {12290, 14094}, {12315, 14915}, {12319, 14683}, {12358, 13198}, {12362, 31804}, {12601, 19461}, {12602, 19462}, {12603, 19463}, {12604, 19464}, {12605, 19467}, {12606, 19468}, {12825, 15463}, {12893, 32609}, {12902, 19479}, {13039, 19465}, {13040, 19466}, {13321, 18369}, {13561, 31283}, {13567, 18951}, {13951, 13970}, {14128, 32136}, {14216, 23335}, {14480, 15112}, {15033, 15058}, {15106, 15115}, {15750, 32110}, {15800, 32332}, {15818, 31807}, {16618, 16789}, {17849, 32428}, {17975, 20764}, {18324, 32171}, {18377, 18405}, {18396, 18404}, {18534, 26883}, {19125, 19131}, {19132, 19154}, {19170, 19179}, {19180, 19210}, {19189, 19194}, {19358, 19428}, {19359, 19429}, {19446, 19486}, {19447, 19487}, {19460, 22834}, {22120, 22146}, {22808, 22953}, {23070, 23144}, {23294, 30744}, {26913, 31282}, {26920, 26922}, {26944, 30771}, {32153, 34046}, {32251, 32275}

X(155) = midpoint of X(i) and X(j) for these {i,j}: {3, 12164}, {4, 6193}, {68, 9936}, {1147, 15083}, {1351, 19588}, {6237, 6238}, {12160, 12166}, {12319, 14683}, {12422, 12423}, {16266, 32139}, {17838, 17847}

X(155) = reflection of X(i) in X(j) for these {i,j}: {3, 1147}, {4, 22660}, {6, 19139}, {26, 156}, {64, 12084}, {68, 5}, {1498, 32139}, {2931, 110}, {3448, 23306}, {7387, 6759}, {7689, 12038}, {9927, 5448}, {10620, 12901}, {11411, 12359}, {12085, 13346}, {12163, 3}, {12164, 15083}, {12293, 4}, {12302, 5504}, {12359, 9820}, {12421, 13292}, {12429, 9927}, {12902, 19479}, {14216, 23335}, {14852, 5654}, {15085, 2931}, {15136, 3292}, {16003, 15115}, {17834, 26}, {18356, 10224}, {19458, 12161}, {21651, 12235}, {32140, 13371}

X(155) = isogonal conjugate of X(254)

X(155) = isotomic conjugate of the polar conjugate of X(1609)

X(155) = X(4)-of-tangential-triangle

X(155) = complement of X(11411)

X(155) = anticomplement of X(12359)

X(155) = circumcircle-inverse of X(12095)

X(155) = polar-circle inverse of X(16172)

X(155) = isogonal conjugate of the polar conjugate of X(6515)

X(155) = polar conjugate of the isotomic conjugate of X(6503)

X(155) = excentral-isogonal conjugate of X(2960)

X(155) = tangential-isogonal conjugate of X(26)

X(155) = orthic-isogonal conjugate of X(3)

X(155) = X(155) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 3}, {1993, 6}, {3193, 1}, {3542, 454}, {6193, 9937}, {6515, 1609}, {11441, 1498}, {12134, 2918}, {14516, 2917}, {15242, 18126}

X(155) = X(i)-isoconjugate of X(j) for these (i,j): {1, 254}, {4, 921}, {19, 6504}, {158, 15316}, {2190, 8800}, {13398, 24006}

X(155) = crosspoint of X(4) and X(3542)

X(155) = crosssum of X(i) and X(j) for these (i,j): {3, 15316}, {4, 3147}, {136, 523}, {647, 8754}

X(155) = crossdifference of every pair of points on line {924, 2501}

X(155) = eigencenter of cevian triangle of X(4)

X(155) = eigencenter of anticevian triangle of X(3)

X(155) = X(84)-of-orthic triangle if ABC is acute

X(155) = perspector of orthic triangle and tangential triangle of the MacBeath circumconic, which is also the anticevian triangle of X(3)

X(155) = perspector of orthic triangle and cross-triangle of ABC and 2nd Hyacinth triangle

X(155) = orthic-to-ABC barycentric image of X(3)

X(155) = intouch-to-ABC functional image of X(4)

X(155) = barycentric product X(i)*X(j) for these {i,j}: {3, 6515}, {4, 6503}, {48, 33808}, {63, 920}, {69, 1609}, {343, 8883}, {394, 3542}, {454, 6504}, {3580, 15478}

X(155) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 6504}, {6, 254}, {48, 921}, {216, 8800}, {454, 6515}, {577, 15316}, {920, 92}, {1609, 4}, {3003, 16172}, {3542, 2052}, {6503, 69}, {6515, 264}, {8883, 275}, {15478, 2986}, {18126, 13579}, {32661, 13398}, {33808, 1969}

X(155) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 11411, 12359}, {3, 49, 19357}, {3, 3167, 1147}, {3, 18445, 1181}, {4, 11441, 18451}, {5, 68, 14852}, {5, 12161, 6}, {5, 15068, 17814}, {5, 31831, 1352}, {6, 17814, 5}, {6, 17836, 19458}, {24, 9932, 2931}, {25, 12160, 52}, {25, 12166, 9937}, {26, 156, 154}, {49, 18436, 3}, {51, 21651, 12235}, {52, 10539, 25}, {54, 11459, 7503}, {68, 5654, 5}, {110, 5889, 24}, {143, 13861, 17810}, {154, 17834, 26}, {182, 11793, 7393}, {184, 5562, 3}, {185, 1092, 3}, {185, 3292, 1092}, {381, 12429, 9927}, {389, 9306, 6642}, {394, 1181, 3}, {485, 486, 9722}, {568, 18350, 7506}, {569, 5891, 7395}, {578, 5907, 9818}, {1069, 3157, 1}, {1147, 7689, 12038}, {1147, 12164, 12163}, {1147, 19908, 19357}, {1199, 3090, 5422}, {1351, 1598, 5446}, {1593, 12162, 11472}, {1598, 19588, 12309}, {1614, 11412, 22}, {1993, 6193, 15316}, {1993, 11441, 4}, {3167, 12164, 3}, {3167, 15083, 12163}, {3917, 10984, 3}, {5012, 11444, 7509}, {5020, 11432, 5462}, {5448, 9927, 381}, {5562, 9908, 12163}, {5654, 9936, 68}, {6102, 6644, 9786}, {6193, 22660, 12293}, {6243, 7517, 33586}, {6243, 10540, 7517}, {7395, 11402, 569}, {7488, 9544, 9707}, {7689, 12038, 3}, {9820, 12359, 2}, {10661, 10662, 6}, {10665, 10666, 6}, {11422, 15056, 13434}, {11591, 32046, 7514}, {12161, 15068, 5}, {12162, 13352, 1593}, {12174, 21312, 10575}, {13371, 32140, 1853}, {14516, 22661, 12293}, {15068, 19139, 5654}, {15316, 18451, 12293}, {17814, 17836, 68}, {17814, 19458, 14852}, {18436, 19908, 12163}

u = u(A,B,C) = sin 2A, v = u(B,C,A), w = u(C,A,B);

x = x(A,B,C) = u

Barycentrics a^2 (a^8 - 3 a^6 (b^2 + c^2) + a^4 (3 b^4 + 2 b^2 c^2 + 3 c^4) - a^2 (b^6 + c^6) + b^2 c^2 (b^2 - c^2)^2) : :

Let O_{A} be the reflection of X(3) in BC, and define O_{B} and O_{C} cyclically. Let O'_{A} be the circumcenter of AO_{B}O_{C}, and define O'_{B} and O'_{C} cyclically. X(156) is the circumcenter of O'_{A}O'_{B}O'_{C}. (Randy Hutson, July 11, 2019)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). Then X(156) = X(5)-of-A'B'C'. (Randy Hutson, July 11, 2019)

X(156) lies on these lines: 3,74 4,49 5,184 25,143 26,154 54,381 546,578 550,1092

X(156) is the {X(154),X(155)}-harmonic conjugate of X(26). For a
list of harmonic conjugates, click **Tables** at the top of this
page.

X(156) = midpoint of X(26) and X(155)

X(156) = X(5)-of-tangential-triangle

X(156) = {X(110),X(1614)}-harmonic conjugate of X(3)

X(156) = intouch-to-ABC functional image of X(5)

= g(a,b,c) : g(b,c,a) : g(c,a,b), where

g(a,b,c) = a[a

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(157) lies on these lines: 3,66 6,248 22,183 25,53 161,418 206,216

X(157) = X(264)-Ceva conjugate of X(6)

X(157) = crosssum of X(127) and X(520)

X(157) = X(6)-of-tangential-triangle

X(157) = perspector of circumcircle wrt Schroeter triangle

X(157) = perspector of polar circle wrt tangential triangle

X(157) = intouch-to-ABC functional image of X(6)

= 1/(1 + cos 2A) : 1/(1 + cos 2B) : 1/(1 + cos 2C)

Barycentrics sec A tan A : sec B tan B : sec C tan C

Let A'B'C' be the hexyl triangle. Let A" be the pole, wrt the polar circle, of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(158). (Randy Hutson, October 15, 2018)

Let A'B'C' and A"B"C" be the Euler and anti-Euler triangles, resp. Let A* be the trilinear product A'*A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(158). (Randy Hutson, October 15, 2018)

X(158) lies on these lines: 1,29 3,243 4,65 10,318 37,281 46,412 47,162 75,240 107,759 225,1093 255,775 286,969 823,897 920,921 1068,3542

X(158) = isogonal conjugate of X(255)

X(158) = isotomic conjugate of X(326)

X(158) = X(i)-cross conjugate of X(j) for these (i,j): (19,92), (225,4)

X(158) = crosssum of X(520) and X(1364)

X(158) = crossdifference of every pair of points on line X(680)X(822)

X(158) = X(i)-aleph conjugate of X(j) for these (i,j): (821,158), (1105,255)

X(158) = X(107)-beth conjugate of X(34)

X(158) = trilinear pole of polar of X(63) wrt polar circle (line X(661)X(3064))

X(158) = pole wrt polar circle of trilinear polar of X(63) (line X(521)X(656))

X(158) = polar conjugate of X(63)

X(158) = trilinear product of X(1123) and X(1336)

X(158) = trilinear square of X(4)

f(a,b,c) = a[(a

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = af(a,b,c)

X(159) = X(9)-of-tangential triangle if ABC is acute

X(159) lies on these lines: 3,66 6,25 22,69 23,193 155,511 197,200

X(159) = reflection of X(i) in X(j) for these (i,j): (6,206), (66,141)

X(159) = X(i)-Ceva conjugate of X(j) for these (i,j): (22,3), (69,6)

X(159) = crosssum of X(127) and X(523)

X(159) = tangential-isogonal conjugate of X(25)

X(159) = tangential-isotomic conjugate of X(32445)

X(159) = intouch-to-ABC functional image of X(9)

f(a,b,c) = a[(b

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = af(a,b,c)

X(160) = X(37)-of-tangential triangle if ABC is acute

X(160) lies on these lines: 3,66 6,237 22,325 95,327 154,418 206,57

X(160) = X(95)-Ceva conjugate of X(6)

X(160) = crosssum of X(338) and X(512)

f(a,b,c) = a[(a

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = af(a,b,c)

X(161) = X(63)-of-tangential triangle if ABC is acute

X(161) lies on these lines: 6,25 22,343 26,68 157,418

X(161) = X(68)-Ceva conjugate of X(6)

Trilinears 1/[(b

Trilinears sec A csc(B - C) : :

Trilinears 1/(tan B - tan C) : :

Barycentrics a/(sin 2B - sin 2C) : b/(sin 2C - sin 2A) : c/(sin 2A - sin 2B)

Let La be the A-extraversion of line X(1)X(19), and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(162). (Randy Hutson, January 29, 2018)

Let La be the A-extraversion of line X(8)X(29), and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(162). (Randy Hutson, January 29, 2018)

Let La be the A-extraversion of line X(9)X(21), and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(162). (Randy Hutson, January 29, 2018)

X(162) lies on these lines:

4,270 6,1013 19,897
27,673 28,88 29,58
31,92 47,158 63,204
100,112 107,109 108,110
190,643 238,415 240,896
242,422 255,1099 412,580
799,811

X(162) = isogonal conjugate of X(656)

X(162) = isotomic conjugate of X(14208)

X(162) = X(250)-Ceva conjugate of X(270)

X(162) = cevapoint of X(i) and X(j) for this (i,j): (108,109)

X(162) = X(i)-cross conjugate of X(j) for these (i,j): (108,107), (109,110)

X(162) = crosssum of X(810) and X(822)

X(162) = X(i)-aleph conjugate of X(j) for these (i,j): (28,1052), (107,920), (162,1), (648,63)

X(162) = trilinear pole of line X(1)X(19)

X(162) = trilinear product of X(1113) and X(1114)

X(162) = crossdifference of PU(75)

X(162) = pole wrt polar circle of trilinear polar of X(1577) (line X(1109)X(2632))

X(162) = X(48)-isoconjugate (polar conjugate) of X(1577)

X(162) = X(92)-isoconjugate of X(822)

X(162) = X(6)-isoconjugate of X(525)

X(162) = crosspoint of X(811) and X(823)

X(162) = trilinear product of PU(74)

X(162) = perspector of conic {{A,B,C,PU(74)}}

X(162) = barycentric product of circumcircle intercepts of line X(19)X(27)

Trilinears a

Barycentrics a

Let A'B'C' be the circumcevian triangle of X(512). Let A" be the trilinear product B'*C', and define B" and C" cyclically. A", B", C" are collinear on line X(667)X(788). The lines AA", BB", CC" concur in X(163). (Randy Hutson, August 19, 2019)

X(163) lies on these lines: 1,293 19,563 31,923 32,849 48,1094 99,825 101,110 109,112 284,909 643,1018 692,906 798,1101 813,827]

X(163) = isogonal conjugate of X(1577)

X(163) = isotomic conjugate of X(20948)

X(163) = barycentric product of circumcircle intercepts of line X(1)X(21)

X(163) = crosssum of X(656) and X(661)

X(163) = X(i)-aleph conjugate of X(j) for these (i,j): (648,19), (662,610)

X(163) = trilinear product of PU(2)

X(163) = barycentric product of PU(70)

X(163) = trilinear product of X(58)X(101)

X(163) = trilinear product of the 6 vertices of the 1st and 2nd circumperp triangles

X(163) = trilinear pole of line X(31)X(48)

X(163) = X(92)-isoconjugate of X(656)

X(163) = crossdifference of every pair of points on line X(1109)X(2632)

X(163) = trilinear product X(6)*X(110)

Barycentrics a(sin B/2 + sin C/2 - sin A/2) : b(sin C/2 + sin A/2 - sin B/2) : c(sin A/2 + sin B/2 - sin C/2)

X(164) = X(1)-of-excentral triangle

X(164) lies on these lines: 1,258 3,3659 9,168 40,188 57,177 165,167 173,504 361,503 362,845

X(164) = isogonal conjugate of X(505)

X(164) = X(188)-Ceva conjugate of X(1)

X(164) = X(i)-aleph conjugate of X(j) for these (i,j): (1,361), (2,362), (9,844), (188,164), (366,173)

X(164) = anticomplement of X(21633)

X(164) = X(8)-of-1st-circumperp-triangle

X(164) = X(944)-of-2nd-circumperp-triangle

X(164) = excentral-isogonal conjugate of X(164)

X(164) = excentral-isotomic conjugate of X(844)

X(164) = centroid of curvatures of extraversions of Conway circle

X(164) = ABC-to-excentral trilinear image of X(1)

Trilinears 3a

Trilinears (r/R) - 4 cos A : :

Barycentrics a[tan(B/2) + tan(C/2) - tan(A/2)] : :

X(165) = X(1) - 4 X(3)

If DEF is the pedal triangle of X(165), then |AE| + |AF| = |BF| + |BD| = |CD| + |CE|. (Seiichi Kirikami, October 8, 2010.)

Let A'B'C' be the anticevian triangle, wrt intouch triangle, of X(1). Let A" be the reflection of A' in A, and define B' and C' cyclically. The centroid of A"B"C" is X(165). (Randy Hutson, December 2, 2017)

Let A'B'C' be the excentral triangle. Let A" be the symmedian point of triangle A'BC, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(165). (Randy Hutson, July 31 2018)

If you have The Geometer's Sketchpad, you can view X(165).

X(165) lies on these lines:

1,3 2,516 4,1698 9,910
10,20 32,1571 42,991
43,573 63,100 71,610
105,1054 108,1767 109,212
164,167 166,168 191,1079
210,971 218,1190 220,1615
227,1394 255,1103 269,1253
355,550 371,1703 372,1702
376,515 380,579 386,1695
411,936 479,1323 498,1770
572,1051 574,1572 580,601
612,990 614,902 631,946
750,968 846,1719 950,1788
958,1706 962,1125 1011,1730
1342,1701 1343,1700

X(165) is the {X(3),X(40)}-harmonic conjugate of X(1). For a list of
harmonic conjugates of X(165), click **Tables** at the top of this
page.

X(165) = isogonal conjugate of X(3062)

X(165) = X(9)-Ceva conjugate of X(1)

X(165) = anticomplement of X(3817)

X(165) = X(2)-of-1st-circumperp-triangle

X(165) = homothetic center of ABC and orthic triangle of 1st circumperp triangle

X(165) = homothetic center of excentral triangle and medial triangle of 1st circumperp triangle

X(165) = excentral isogonal conjugate of X(9)

X(165) = excentral isotomic conjugate of X(165)

X(165) = excentral polar conjugate of X(1)

X(165) = Thomson-isogonal conjugate of X(1)

X(165) = reflection of X(1699) in X(2)

X(165) = cyclocevian conjugate of X(1) wrt anticevian triangle of X(1)

X(165) = X(i)-beth conjugate of X(j) for these (i,j): (100,165), (643,200)

X(165) = X(i)-aleph conjugate of X(j) for these (i,j):

(2,169), (9,165), (21,572), (100,101), (188,9), (259,43), (365,978), (366,57), (650,1053)

X(165) = centroid of the triangle with vertices X(1), X(8), X(20)

X(165) = centroid of the triangle with vertices X(4), X(20), X(40)

X(165) = ABC-to-excentral barycentric image of X(2)

f(A,B,C) = (tan A/2)/(cos B/2 + cos C/2 - cos A/2) - (tan B/2)/(cos C/2 + cos A/2 - cos B/2) - (tan C/2)/(cos A/2 + cos B/2 - cos C/2)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = (sin A)f(A,B,C)

X(166) = X(7)-of-excentral triangle

X(166) lies on these lines: 1,1488 165,168 167,188

X(166) = X(266)-cross conjugate of X(57)

X(166) = cevapoint of X(266) and X(289)

f(A,B,C) = s(B,C,A)t(B,C,A) + s(C,A,B)t(C,A,B) - s(A,B,C)t(A,B,C),

where s(A,B,C) = sin(A/2) and t(A,B,C) = (cos B/2 + cos C/2 - cos A/2) sec A/2

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = (sin A)f(A,B,C)

X(167) = X(8)-of-excentral triangle

X(167) lies on these lines: 1,174 164,165 166,188

X(167) = X(9)-aleph conjugate of X(166)

Trilinears b/(1 - sin B/2) + c/(1 - sin C/2) - a/(1 - sin A/2) : :

X(168) lies on these lines: 1,173 9,164 165,166

X(168) = X(188)-aleph conjugate of X(363)

X(168) = X(9)-of-excentral triangleX(168) = homothetic center of the excentral and outer Hutson triangles; see X(363).

X(168) = X(7)-of-1st-circumperp-triangle

X(168) = homothetic center of ABC and orthic triangle of outer Hutson triangle

Trilinears a^3 - a^2 (b + c) + a (b^2 + c^2) - (b - c)^2 (b + c) : :

X(169) lies on these lines: 1,41 3,910 4,9 6,942 46,672 57,277 63,379 65,218 220,517 572,610

X(169) = X(85)-Ceva conjugate of X(1)

X(169) = crosssum of X(6) and X(1473)

X(169) = X(i)-aleph conjugate of X(j) for these (i,j):

(2,165), (85,169), (86,572), (174,43), (188,170), (508,1), (514,1053), (664,101)

f(A,B,C) = - (tan A/2)sec

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = (sin A)f(A,B,C)

X(170) = X(76)-of-excentral triangle

X(170) lies on these lines: 1,7 43,218

X(170) = X(220)-Ceva conjugate of X(1)

X(170) = X(i)-aleph conjugate of X(j) for these (i,j): (9,9), (55,43), (188,169), (220,170), (644,1018)

X(170) = X(664)-beth conjugate of X(170)

Barycentrics a

X(171) lies on these lines: 1,3 2,31 4,601 6,43 7,983 10,58 37,846 42,81 47,498 63,612 72,1046 84,989 98,109 181,511 222,611 292,893 319,757 385,894 388,603 474,978 595,1125 602,631 756,896

X(171) = isogonal conjugate of X(256)

X(171) = isotomic conjugate of X(7018)

X(171) = complement of X(4388)

X(171) = anticomplement of X(3846)

X(171) = perspector of Gemini triangle 33 and cross-triangle of ABC and Gemini triangle 33

X(171) = trilinear pole line X(3287)X(3805) (the perspectrix of ABC and Gemini triangle 34)

X(171) = X(292)-Ceva conjugate of X(238)

X(171) = X(i)-beth conjugate of X(j) for these (i,j): (100,171), (643,42)

X(171) = crosssum of PU(6)

X(171) = crosspoint of PU(8)

X(171) = intersection of tangents at PU(8) to hyperbola {{A,B,C,X(100),PU(8)}}

Barycentrics a

X(172) lies on these lines:

1,32 6,41 12,230 21,37
35,187 36,39 42,199
58,101 65,248 350,384
577,1038 694,904 699,932

X(172) = isogonal conjugate of X(257)

X(172) = crossdifference of every pair of points on line X(522)X(1491)

X(172) = X(101)-beth conjugate of X(172)

X(172) = {X(1),X(32)}-harmonic conjugate of X(1914)

X(172) = intersection of tangents at PU(9) to hyperbola {A,B,C,X(101),PU(9)}

X(172) = crosspoint of PU(9)

X(172) = crosssum of PU(10)

X(172) = homothetic center of anti-tangential midarc triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles

Trilinears tan A/2 + sec A/2 : tan B/2 + sec B/2 : tan C/2 + sec C/2 (M. Iliev, 4/12/07)

Trilinears b(csc B/2) + c(csc C/2) - a(csc A/2) : :

Trilinears cot A'/2 : :, where A'B'C' = excentral triangle

Trilinears b' + c' - a' : :, where a', b', c' are sidelengths of excentral triangle

Trilinears cot A' + csc A' : :, where A'B'C' = excentral triangle

Trilinears [distance from A to far side of A-excircle] : :

Let P_{B} on sideline AC and Q_{C} be equidistant from A, so that AP_{B}Q_{C} is an isosceles triangle. The line P_{B}Q_{C} is called an isoscelizer. The lines P_{B}Q_{C}, P_{C}Q_{A}, P_{A}Q_{B} concur in X(173). (P. Yff, unpublished notes, 1989)

The intouch triangle of the intouch triangle of triangle ABC is perspective to triangle ABC, and X(173) is the perspector. (Eric Danneels, Hyacinthos 7892, 9/13/03)

Also, X(173) = X(1486)-of-the-intouch-triangle. (Darij Grinberg; see notes at X(1485) and X(1486).)

If you have The Geometer's Sketchpad, you can view Congruent Isoscelizers Point.

X(173) lies on these lines: 1,168 9,177 57,174 164,504 180,483 503,844 505,1130

X(173) = isogonal conjugate of X(258)

X(173) = X(174)-Ceva conjugate of X(1)

X(173) = SS(A->A')-of-X(9), where A'B'C' is the excentral triangle

X(173) = X(19)-of-excentral triangle

X(173) = X(19)-of-Yff central triangle

X(173) = homothetic center of excentral triangle and Yff central triangle

X(173) = homothetic center of ABC and orthic triangle of Yff central triangle

X(173) = homothetic center of ABC and extangents triangle of excentral triangle

X(173) = excentral isogonal conjugate of X(845)

X(173) = X(i)-aleph conjugate of X(j) for these (i,j): (1,503), (2,504), (174,173), (188,845), (366,164), (507,1), (508,362), (509,361)

Trilinears [bc/(b + c - a)]

Trilinears csc A' : csc B' : csc C', where A'B'C' = excentral triangle

Trilinears |AA'| : |BB'| : |CC'|, where A'B'C' = excentral triangle

Trilinears (csc A')(cos B' + cos C') : :, where A'B'C' = excentral triangle

Barycentrics sin A/2 : sin B/2 : sin C/2

Let Ea be the ellipse with B and C as foci and passing through X(1), and define Eb and Ec cyclically. Let La be the line tangent to Ea at X(1), and define Lb and Lc cyclically. Let A' = La∩BC, B' = Lb∩CA, C' = Lc∩AB. Then A', B', C' are collinear, and X(174) = trilinear pole of line A'B'C'. The line A'B'C' meets the line at infinity at the isogonal conjugate of X(3659). Alternately, let A" be the trilinear pole of line La, and define B" and C" cyclically. The lines AA", BB" and CC" concur at X(174); see also X(188). The points A", B", C" lie on the circumconic centered at X(9). (Randy Hutson, December 10, 2016)

Let A'B'C' be the excentral triangle. X(174) is the trilinear pole of the Monge line of the incircles of BCA', CAB', ABC'. (Randy Hutson, December 10, 2016) In notes dated 1987, Yff raises this question concerning certain triangles lying within ABC: can isoscelizers (defined at X(173)), P
R_{A} = P_{A}Q_{B}∩P_{B}Q_{C}, R_{B} = P_{B}Q_{C}∩P_{C}Q_{A}, R_{C} = P_{C}Q_{A}∩P_{A}Q_{B},

the following four triangles are congruent:

P_{A}Q_{A}R_{A}, P_{B}Q_{B}R_{B},
P_{C}Q_{C}R_{C}, R_{A}R_{B}R_{C} ?

After proving that the answer is yes, Yff moves the three isoscelizers in such a way that the three outer triangles, stay congruent and the inner triangle (called the Yff central triangle), R_{A}R_{B}R_{C}, shrinks to X(174).

Let D be the point on side BC such that (angle BID) = (angle DIC), and likewise for point E on side CA and point F on side AB. The lines AD, BE, CF concur in X(174). (Seiichi Kirikami, Jan. 29, 2010)

Generalization: if I is replaced by an arbitrary point P = p : q : r (trilinears), then the lines AD, BE, CF concur in the point K(P) = f(p,q,r,A) : f(q,r,p,B) : f(r,p,q,C), where f(p,q,r,A) = (q^{2} + r^{2} + 2qr cos A)^{-1/2}. Moreover, if P* is the inverse of P in the circumcircle, then K(P*) = K(P). (Peter Moses, Feb. 1, 2010, based on Seiichi Kirikami's construction of X(174))

X(174) is the homothetic center of ABC and the extangents triangle of the intouch triangle. (Randy Hutson, 9/23/2011)

If you have The Geometer's Sketchpad, you can view Yff Center of Congruence (1) and Yff Center of Congruence (2) and Yff Center of Congruence (3). For access to a sketch of the Yff central triangle, see X(177).

X(174) lies on these lines: 1,167 2,236 7,234 57,173 175,483 176,1143 188,266 481,1127 558,1489

X(174) = isogonal conjugate of X(259)

X(174) = anticomplement of X(2090)

X(174) = X(508)-Ceva conjugate of X(188)

X(174) = cevapoint of X(i) and X(j) for these (i,j): (1,173), (259,266)

X(174) = X(i)-cross conjugate of X(j) for these (i,j): (1,1488), (177,7), (259,188)

X(174) = crosssum of X(1) and X(503)

X(174) = X(556)-beth conjugate of X(556)

X(174) = SS(A->A')-of-X(2), where A'B'C' is the excentral triangle

X(174) = SS(A->A')-of-X(226), where A'B'C' is the excentral triangle

X(174) = isotomic conjugate of X(556)

X(174) = X(55)-of-intouch triangle

X(174) = X(55)-of-Yff central triangle

X(174) = homothetic center of intouch triangle and Yff central triangle

X(174) = homothetic center of ABC and the intangents triangle of the intouch triangle.

X(174) = {X(8134),X(8136)}-harmonic conjugate of X(8123)

X(174) = {X(8137),X(8139)}-harmonic conjugate of X(8124)

X(174) = X(1824)-of-excentral-triangle

X(174) = excentral-to-ABC trilinear image of X(165)

Barycentrics (sin A)(-1 + sec A/2 cos B/2 cos C/2) : :

Barycentrics -2a + (a + b + c) tan(A/2) : :

Barycentrics a - r

X(175) = 2s*X(1) - (r + 4R)*X(7) = 3X[2]-4X[31534]

The points X(175) and X(176) are discussed in an 1890 article by Emile Lemoine, accessible at Gallica. The article begins on page 111, and the two points are considered beginning on page 128.

A point X is defined as an isoperimetric point of triangle ABC if |XB| + |XC| + |BC| = |XC| + |XA| + |CA| = |XA| + |XB| + |AB|. Veldkamp established that X = X(175), uniquely, for some triangles ABC, but the conditions he gives are not correct. Hajja and Yff proved that the condition tan(A/2) + tan(B/2) + tan(C/2) < 2 is necessary and sufficient. See also X(176) and the 1st and 2nd Eppstein points, X(481), X(482).

In unpublished notes, Yff proved that X(175) is the center of the outer Soddy circle. His proof later appeared in the paper by Hajja and Yff cited below.

Every point on the Soddy line has barycentric coordinates of the form a + k/sa : b + k/sb : c + k/sc, where k is a symmetric function in a,b,c, and sa=(b+c-a)/2, sb=(c+a-b)/2, sc=(a+b-c)/2. Writing S for 4*area(ABC):

X(175) = 2a - S/sa : 2b - S/sb : 2c - S/sc

X(176) = 2a + S/sa : 2b + S/sb : 2c + S/sc

X(481) = a - S/sa : b - S/sb : c - S/sc

X(482) = a + S/sa : b + S/sb : c + S/sc

X(1371) = a + 2S/(3 sa) : b + 2S/(3 sb) : c + 2S/(3 sc)

X(1372) = a - 2S/(3 sa) : b - 2S/(3 sb) : c - 2S/(3 sc)

X(1373) = a + 2S/sa : b + 2S/sb : c + 2S/sc

X(1374) = a - 2S/sa : b - 2S/sb : c - 2S/sc

**Clark Kimberling and R. W. Wagner,** Problem E 3020 and Solution, *American Mathematical Monthly* 93 (1986) 650-652 [proposed 1983].

**G. R. Veldkamp,** "The isoperimetric point and the point(s) of equal detour," *American Mathematical Monthly* 92 (1985) 546-558.

Muwaffaq Hajja and Peter Yff, "The isoperimetric point and the point(s) of equal detour in a triangle," *Journal of Geometry* 87 (2007) 76-82.

There are exactly two points P such that the incircles of the triangles PBC, PCA, PAB are pairwise tangent to one another; the two points are X(175) and X(176). There are exactly two points P such that the radical center of the incircles of PBC, PCA, PAB is P; the two points are X(175) and X(176). (Randy Hutson, 9/23/2011)

X(175) lies on the curves K032, K199, K200, Q074, Q092, Q104 and on these lines: {1,7}, {2,13386}, {4,10905}, {8,1270}, {65,6252}, {69,10908}, {105,30385}, {144,30557}, {174,483}, {226,1131}, {329,3084}, {388,10911}, {490,664}, {651,1335}, {1336,10253}, {1463,7353}, {2082,6203}, {3062,10973}, {3083,9776}, {3297,5228}, {3298,6180}, {4000,7968}, {4644,7969}, {5222,18992}, {5226,5393}, {5261,9907}, {5405,5435}, {5932,31528}, {5933,31530}, {6405,20358}, {7090,10405}, {7585,30325}, {9778,10135}, {9789,21147}, {10580,16663}, {11293,17086}, {13387,20211}, {13389,21454}, {13459,16441}, {15913,31536}, {16214,32081}, {21453,30335}, {30347,31588}, {30413,31547}, {31544,32201}

X(175) = reflection of X(i) and X(j) for these {i,j}: {176, 3160}, {10405, 7090}, {14121, 31534}, {30334, 1}

X(175) = isogonal conjugate of X(30336)

X(175) = anticomplement of X(14121)

X(175) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 13386}, {603, 176}, {1659, 21270}, {1805, 3869}, {2067, 8}, {2362, 4}, {5414, 329}, {6502, 31552}, {13388, 69}, {30557, 3436}

X(175) = X(8)-Ceva conjugate of X(176)

X(175) = X(10135)-cross conjugate of X(7)

X(175) = X(i)-isoconjugate of X(j) for these (i,j): {1, 30336}, {6, 15891}

X(175) = cevapoint of X(10253) and X(32058)

X(175) = crosssum of X(55) and X(19037)

X(175) = reflection of X(175) in the Soddy line

X(175) = X(6406)-of-excentral-triangle

X(175) = X(1152)-of-intouch-triangle

X(175) = X(7353)-of-(inner)-tangential-mid-arc-triangle (TCCT 6.15)

X(175) = X(12224)-of-first-circumperp-triangle (TCCT 6.21)

X(175) = X(6400)-of-second-circumperp-triangle (TCCT 6.22)

X(175) = barycentric product X(i)*X(j) for these {i,j}: {7, 30413}, {8, 16662}, {1659, 31547}

X(175) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 15891}, {6, 30336}, {9778, 30412}, {16662, 7}, {30413, 8}

X(175) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7, 176}, {1, 481, 7}, {1, 482, 17805}, {1, 1372, 481}, {1, 1373, 31538}, {1, 1374, 482}, {1, 17803, 31539}, {1, 30342, 11038}, {1, 31539, 17802}, {1, 31568, 8236}, {7, 176, 21169}, {7, 482, 21170}, {7, 1372, 17801}, {7, 8236, 31566}, {7, 17802, 1}, {7, 17805, 482}, {7, 31601, 1373}, {7, 31602, 481}, {20, 347, 176}, {77, 962, 176}, {176, 21169, 17804}, {176, 21170, 482}, {269, 9785, 176}, {279, 390, 176}, {481, 482, 1374}, {481, 1372, 31602}, {481, 17802, 176}, {481, 17803, 17802}, {481, 31539, 1}, {481, 31602, 17801}, {482, 1374, 7}, {482, 17805, 176}, {482, 21170, 21169}, {1323, 30332, 176}, {1372, 17803, 1}, {1372, 31539, 7}, {1373, 31538, 31601}, {1374, 17805, 21170}, {1442, 4295, 176}, {1443, 30305, 176}, {3600, 3672, 176}, {3638, 3639, 1372}, {3663, 4308, 176}, {3664, 4323, 176}, {3668, 4313, 176}, {3674, 4344, 176}, {4296, 4329, 176}, {4318, 17170, 176}, {4862, 6049, 176}, {5542, 5543, 176}, {5731, 22464, 176}, {7190, 11037, 176}, {8236, 10481, 176}, {10135, 32083, 16662}, {10481, 31568, 31566}, {13388, 13390, 2}, {14121, 31534, 2}, {17801, 17802, 21169}, {17802, 31602, 7}, {30424, 31721, 176}, {31538, 31601, 176}, {31539, 31602, 176}

Barycentrics (sin A)(1 + sec A/2 cos B/2 cos C/2)

Barycentrics 2a + (a + b + c) tan(A/2) : :

Barycentrics a + r

The points X(175) and X(176) are discussed in an 1890 article by Emile Lemoine, accessible at Gallica. The article begins on page 111, and the two points are considered beginning on page 128.

The following construction was found by Elkies: call two circles within ABC *companion circles* if they are the incircles of two triangles formed by dividing ABC into two smaller triangles by passing a line through one of the vertices and some point on the opposite side; chain of circles O(1), O(2), ... such that O(n),O(n+1) are companion incircles for every n consists of at most six distinct circles; there is a unique chain consisting of only three distinct circles; and for this chain, the three subdividing lines concur in X(176).

A point X is defined as a point of equal detour of triangle ABC if |XB| + |XC| - |BC| = |XC| + |XA| - |CA| = |XA| + |XB| - |AB|. Veldkamp established that X = X(176) for some triangles ABC, but the conditions he gives are not correct. Hajja and Yff proved that the condition tan(A/2) + tan(B/2) + tan(C/2) < 2 is necessary and sufficient for the existence of exactly two points of equal detour and that the condition tan(A/2) + tan(B/2) + tan(C/2) = 2 is necessary and sufficient for the existence of exactly one point of equal detour. Yff found that X(176) is also is the center of the inner Soddy circle. See also X(175) and the 1st and 2nd Eppstein points, X(481), X(482).

**G. R. Veldkamp**, "The isoperimetric point and the point(s) of equal detour," *American Mathematical Monthly* 92 (1985) 546-558.

**Noam D. Elkies and Jiro Fukuta**, Problem E 3236 and Solution, *American Mathematical Monthly* 97 (1990) 529-531 [proposed 1987].

Mowaffaq Majja and Peter Yff, "The isoperimetric point and the point(s) of equal detour in a triangle," *Journal of Geometry* 87 (2007) 76-82.

There are exactly two points P such that the incircles of the triangles PBC, PCA, PAB are pairwise tangent to one another; the two points are X(175) and X(176). There are exactly two points P such that the radical center of the incircles of PBC, PCA, PAB is P; the two points are X(175) and X(176). For a point Q, let A' be the incenter of triangle BCQ, and define B' and C' cyclically; then X(176) is the only point Q such that Q is the incenter of A'B'C'. (Randy Hutson, 9/23/2011)

Let Ia, Ib, Ic be the centers of the Elkies companion incircles. Let A' be the trilinear product Ib*Ic, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(482). The lines IaA', IbB', IcC' concur in X(176). (Randy Hutson, December 2, 2017)

X(176) lies on K032, K199, K200, Q074, Q092, Q104 and these lines: {1,7}, {2,1659}, {4,10904}, {8,1271}, {65,6404}, {69,10907}, {105,30386}, {144,30556}, {174,1274}, {226,1132}, {329,3083}, {388,10910}, {489,664}, {651,1124}, {1123,10252}, {1463,7362}, {1587,8953}, {1588,8978}, {2082,6204}, {3062,10972}, {3084,9776}, {3177,31408}, {3297,6180}, {3298,5228}, {3622,8243}, {4000,7969}, {4644,7968}, {5222,18991}, {5226,5405}, {5261,9906}, {5393,5435}, {5932,31529}, {5933,31531}, {6283,20358}, {7586,30324}, {8973,23259}, {9778,10134}, {10405,14121}, {10580,16662}, {11294,17086}, {13386,20211}, {13388,21454}, {13437,16440}, {15913,31537}, {16213,32080}, {21453,30336}, {30346,31589}, {30412,31548}, {31545,32200}

X(176) = reflection of X(i) and X(j) for these {i,j}: {20, 8984}, {175, 3160}, {7090, 31535}, {10405, 14121}, {30333, 1}

X(176) = isogonal conjugate of X(30335)

X(176) = anticomplement of X(7090)

X(176) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 13387}, {603, 175}, {1806, 3869}, {2066, 329}, {2067, 31551}, {6502, 8}, {13389, 69}, {13390, 21270}, {16232, 4}, {30556, 3436}

X(176) = X(8)-Ceva conjugate of X(175)

X(176) = X(10134)-cross conjugate of X(7)

X(176) = X(i)-isoconjugate of X(j) for these (i,j): {1, 30335}, {6, 15892}

X(176) = cevapoint of X(10252) and X(32057)

X(176) = crosssum of X(55) and X(19038)

X(176) = reflection of X(176) in the Soddy line

X(176) = X(6291)-of-excentral-triangle

X(176) = X(1151)-of-intouch-triangle

X(176) = X(7362)-of-(inner)-tangential-mid-arc-triangle (TCCT 6.15)

X(176) = X(12223)-of-first-circumperp-triangle (TCCT 6.21)

X(176) = X(6239)-of-second-circumperp-triangle (TCCT 6.22)

X(176) = barycentric product X(i)*X(j) for these {i,j}: {7, 30412}, {8, 16663}, {13390, 31548}

X(176) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 15892}, {6, 30335}, {174, 5451}, {9778, 30413}, {16663, 7}, {30412, 8}

X(176) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7, 175}, {1, 481, 17802}, {1, 482, 7}, {1, 1371, 482}, {1, 1373, 481}, {1, 1374, 31539}, {1, 17804, 21170}, {1, 17806, 31538}, {1, 21170, 17801}, {1, 21171, 31602}, {1, 30341, 11038}, {1, 31538, 17805}, {1, 31567, 8236}, {1, 31601, 21169}, {7, 482, 21169}, {7, 1371, 17804}, {7, 8236, 31565}, {7, 17802, 481}, {7, 17805, 1}, {7, 21169, 21170}, {7, 31601, 482}, {7, 31602, 1374}, {20, 347, 175}, {77, 962, 175}, {175, 482, 21170}, {175, 17804, 21169}, {175, 21169, 7}, {269, 9785, 175}, {279, 390, 175}, {481, 482, 1373}, {481, 1373, 7}, {481, 17802, 175}, {482, 1371, 31601}, {482, 17805, 175}, {482, 17806, 17805}, {482, 31538, 1}, {482, 31539, 21171}, {482, 31601, 17804}, {1323, 30332, 175}, {1371, 17805, 21169}, {1371, 17806, 1}, {1371, 31538, 7}, {1374, 21171, 7}, {1374, 31539, 31602}, {1442, 4295, 175}, {1443, 30305, 175}, {1659, 13389, 2}, {3600, 3672, 175}, {3638, 3639, 1371}, {3663, 4308, 175}, {3664, 4323, 175}, {3668, 4313, 175}, {3674, 4344, 175}, {4296, 4329, 175}, {4318, 17170, 175}, {4862, 6049, 175}, {5542, 5543, 175}, {5731, 22464, 175}, {7090, 31535, 2}, {7190, 11037, 175}, {8236, 10481, 175}, {10134, 32082, 16663}, {10481, 31567, 31565}, {17804, 21169, 482}, {17805, 31601, 7}, {21171, 31539, 1374}, {30424, 31721, 175}, {31538, 31601, 175}, {31539, 31602, 175}

Trilinears (b' + c')/a' : :, where a', b', c' are sidelengths of excentral triangle

Barycentrics (sin A)(cos B/2 + cos C/2) sec A/2 : :

Let A',B',C' be the first points of intersection of the angle bisectors of ABC with its incircle. The tangents at A', B', C' form a triangle A"B"C", and the lines AA",BB",CC" concur in X(177). Also, X(177) = X(1) of the intouch triangle.

**Clark Kimberling and G. R. Veldkamp,** Problem 1160 and
Solution, *Crux Mathematicorum* 13 (1987) 298-299 [proposed
1986].

X(177) is the perspector of ABC and the Yff central triangle, and X(177) is X(65)-of-the-Yff-central-triangle . (Darij Grinberg, Hyacinthos #7689, 8/25/2003)

If you have The Geometer's Sketchpad, you can view X(177) and Yff Central Triangle.

X(177) lies on these lines: 1,167 7,555 8,556 9,173 57,164

X(177) = isogonal conjugate of X(260)

X(177) = X(7)-Ceva conjugate of X(234)

X(177) = crosspoint of X(7) and X(174)

X(177) = crosssum of X(55) and X(259)

X(177) = SS(A->A')-of-X(10), where A'B'C' is the excentral triangle

X(177) = X(4)-of-mid-arc triangle

X(177) = X(1829)-of-excentral triangle

X(177) = perspector of ABC and mid-triangle of 1st tangential mid-arc triangle and Yff central triangle

Barycentrics cos B/2 + cos C/2 : cos C/2 + cos A/2 : cos A/2 + cos B/2

Let A',B',C' be the first points of intersection of the angle bisectors of ABC with its incircle. Let A",B",C" be the midpoints of segments BC,CA,AB, respectively. The lines A'A",B'B",C'C" concur in X(178).

**Clark Kimberling,** Problem 804, *Nieuw Archikef voor Wiskunde* 6 (1988) 170.

X(178) lies on these lines: 2,188 8,236

X(178) = complement of X(188)

X(178) = crosspoint of X(2) and X(508)

Barycentrics sin A sec

The famous Malfatti Problem is to construct three circles O(A), O(B), O(C) inside ABC such that each is externally tangent to the other two, O(A) is tangent to lines AB and AC, O(B) is tangent to BC and BA, and O(C) is tangent to CA and CB. Let A' = O(B)∩O(C), B' = O(C)∩O(A), C' = O(A)∩O(B). The lines AA',BB',CC' concur in X(179). Trilinears are found in Yff's unpublished notes. See also the Yff-Malfatti Point, X(400), having trilinears csc^{4}(A/4) : csc^{4}(B/4) : csc^{4}(C/4), and the references for historical notes.

**H. Fukagawa and D. Pedoe,** *Japanese Temple Geometry Problems (San Gaku),* The Charles Babbage Research Centre, Winnipeg, Canada, 1989.

**Michael Goldberg,** "On the original Malfatti problem," *Mathematics Magazine,* 40 (1967) 241-247.

**Clark Kimberling and I. G. MacDonald,** Problem E 3251 and Solution, *American Mathematical Monthly* 97 (1990) 612-613.

Let A', B', C' be the centers of the Malfatti circles. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1142). The lines A'A", B'B", C'C" concur in X(179). (Randy Hutson, July 11, 2019)

If you have The Geometer's Sketchpad, you can view X(179).

X(179) lies on these lines: {1,1142}, {485,8242}, {1143,9795}

t(A,B,C) = 1 + 2(sec A/4 cos B/4 cos C/4)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = (sin A)f(A,B,C)

Let A",B",C" be the excenters of ABC, and let A',B',C' be as in the construction of X(179). The lines A'A",B'B",B'B" concur in X(180). Trilinears are found in Yff's unpublished notes. See X(179).

If you have The Geometer's Sketchpad, you can view
**X(180)** and
X(180) External.

X(180) lies on this line: 173,483

= a

Trilinears h(a,b,c) :
h(b,c,a) : h(c,a,b), where h(a,b,c) = [r cos(A/2) + s
sin(A/2)]^{2}, s = semiperimeter, r = inradius

Barycentrics a^{3}cos^{2}(B/2 - C/2) :
b^{3}cos^{2}(C/2 - A/2) :
c^{3}cos^{2}(A/2 - B/2)

Let O(A),O(B),O(C) be the excircles. Apollonius's Problem includes the construction of the circle O tangent to the three excircles and encompassing them. (The circle is called the Apollonius circle.) Let A' = O∩O(A), B'=O∩O(B), C'=O∩O(C). The lines AA',BB',CC' concur in X(181). Yff derived trilinears in 1992.

X(181) is the external center of similitude (or exsimilicenter) of the incircle and Apollonius circle. The internal center is X(1682). (Peter J. C. Moses, 8/22/2003)

X(181) is the isogonal conjugate of the isotomic conjugate of X(12); also, X(181) is the {X(i) ,X(j) }-harmonic conjugate of X(k) for these (i,j,k)): (31,51,3271), (42,1400,1402), (57,1401,1357), (57,1469,1401). (Peter J. C. Moses, 6/20/2014)

A proof of the the concurrence of lines AA',BB',CC' follows.

A =
exsimilicenter(incircle, A-excircle)

A' =
exsimilicenter(A-excircle, Apollonius circle)

Let J =
exsimilicenter(incircle, Apollonius circle).

By Monge's theorem, the points A, A', J are collinear. In particular, J
lies on line AA', and cyclically, J lies on lines BB' and CC'.
Therefore, J = X(181). (Darij Grinberg, Hyacinthos, 7461, 8/10/03)

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B', C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the intouch triangle at X(181). (Randy Hutson, March 21, 2019)

See also **Clark Kimberling, Shiko Iwata, and Hidetosi Fukagawa,** Problem
1091 and Solution, *Crux Mathematicorum* 13 (1987) 128-129;
217-218. [proposed 1985].

X(181) lies on these lines:

{1,970}, {6,197}, {8,959}, {10,12}, {11,2051}, {25,2175}, {31,51}, {33,3022}, {42,228}, {43,57}, {44,375}, {55,573}, {56,386}, {58,1324}, {81,5061}, {171,511}, {182,5329}, {200,3779}, {213,2333}, {373,748}, {389,3072}, {518,3687}, {553,1463}, {575,5363}, {612,3688}, {750,3917}, {756,2171}, {942,5530}, {994,1361}, {1124,1685}, {1254,1425}, {1317,3032}, {1335,1686}, {1356,5213}, {1358,3034}, {1364,5348}, {1376,4259}, {1395,1843}, {1672,1683}, {1673,1684}, {1674,1693}, {1675,1694}, {1695,1697}, {2007,2019}, {2008,2020}, {2330,5285}, {2534,2538}, {2535,2539}, {3027,3029}, {3028,3031}, {3056,5269}, {3340,4517}, {3781,5268}, {3792,3819}, {4276,5172}

X(181) = isogonal conjugate of X(261)

X(181) = X(i)-Ceva conjugate of X(j) for these (i,j): (12,2197), (59,4559), (65,2171), (2171,1500)

X(181) = X(i)-cross conjugate of X(j) for these (i,j): (872,1500), (2643,512)

X(181) = crosspoint of X(i) and X(j) for these (i,j): (42,1824), (59,4559), (65,1400), (1354,2171)

X(181) = crosssum of X(i) and X(j) for these (i,j): (2,2975), (11,4560), (21,333), (81,4225), (86,1444), (1098,2185)

X(181) = crossdifference of X(3904) and X(3910)

X(181) = X(i)-beth conjugate of X(j) for these (i,j): (42,181), (660,181), (756,756)

X(181) = {X(1),X(970)}-harmonic conjugate of X(1682)

X(181) = X(60)-isoconjugate of X(75)

X(181) = trilinear product of vertices of extangents triangle

X(181) = trilinear product of X(i) and X(j) for these {I,J}:

{1,181}, {6,2171}, {7,872}, {10,1402}, {12,31}, {19,2197}, {25,201}, {33,1425}, {34,3690}, {37,1400}, {42,65}, {55,1254}, {56,756}, {57,1500}, {59,2643}, {71,1880}, {73,1824}, {109,4705}, {115,2149}, {210,1042}, {213,226}, {225,228}, {227,2357}, {349,2205}, {512,4551}, {594,604}, {608,3949}, {651,4079}, {661,4559}, {762,1412}, {798,4552}, {1020,3709}, {1089,1397}, {1110,1365}, {1214,2333}, {1334,1427}, {1395,3695}, {1409,1826}, {1415,4024}, {1426,2318}, {1441,1918}, {3063,4605}, {3124,4564}, {4017,4557}

X(181) = barycentric product of X(i) and X(j) for these {I,J}:

{{1,2171}, {2,181}, {4,2197}, {6,12}, {7,1500}, {9,1254}, {10,1400}, {19,201}, {34,3949}, {37,65}, {42,226}, {56,594}, {57,756}, {59,115}, {71,225}, {72,1880}, {73,1826}, {85,872}, {109,4024}, {210,1427}, {213,1441}, {227,1903}, {278,3690}, {281,1425}, {307,2333}, {321,1402}, {349,1918}, {512,4552}, {523,4559}, {604,1089}, {608,3695}, {651,4705}, {661,4551}, {663,4605}, {664,4079}, {762,1014}, {1018,4017}, {1020,4041}, {1042,2321}, {1091,2150}, {1109,2149}, {1214,1824}, {1252,1365}, {1262,4092}, {1334,3668}, {1404,4013}, {1411,4053}, {1415,4036}, {1426,3694}, {2222,2610}, {2643,4564}, {3124,4998}, {3709,4566}

X(181) = X(i)-isoconjugate of X(j) for these (i,j):

(1,261), (2,2185), (7,1098), (8,757), (9,1509), (21,86), (27,1812), (28,332), (29,1444), (55,873), (58,314), (60,75), (69,270), (76,2150), (81,333), (99,3737), (200,552), (249,4858), (274,284), (283,286), (304,2189), (310,2194), (312,593), (348,2326), (514,4612), (645,1019), (649,4631), (650,4610), (662,4560), (663,4623), (693,4636), (763,2321), (849,3596), (1014,1043), (1021,4573), (1434,2287), (2170,4590), (4391,4556)

Trilinears cos A + sin A tan ω : cos B + sin B tan ω : cos C + sin C tan ω

Trilinears sin A - sin(A - 2ω) : sin B - sin(B - 2ω) : sin C - sin(C - 2ω)

Trilinears cos A + cos(A - 2ω) : cos B + cos(B - 2ω) : cos C + cos(C - 2ω) (cf., X(39))

Trilinears a + 2R cot ω cos A : b + 2R cot ω cos B: c + 2R cot ω cos C (cf., X(1350), X(1351))

Trilinears sin A + cos A cot ω : sin B + cos B cot ω : sin C + cos C cot ω (cf., X(575), X(576),,X(1350), X(1351))

Trilinears cos A + (2 - 2 cot ω) sin A : cos B + (2 - 2 cot ω) sin B : cos C + (2 - 2 cot ω) sin C

Barycentrics sin A cos(A - ω) : sin B cos(B - ω) : sin C cos(C -ω)

Barycentrics a^2(a^4 - a^2b^2 - a^2c^2 - 2b^2c^2) : :

X(182) = radical center of Lucas(2 tan ω) circles, where 2 tan ω is the value of t for which the Brocard circle is the radical circle of the Lucas(t) circles. (Randy Hutson, January 29, 2015)

X(182) lies on these lines:

1,983 2,98 3,6 4,83
5,206 10,1678 22,51
24,1843 25,3066 30,597 36,1469
40,1700 54,69 55,613
56,611 111,353 140,141
171,1397 373,1495 474,1437
517,1386 518,1385 524,549 691,2698
692,1001 727,1293 729,1296
952,996

X(182) is the {X(371),X(372)}-harmonic conjugate of X(39). For a list of other harmonic conjugates of X(182), click **Tables** at the top of this page.

X(182) = midpoint of X(3) and X(6)

X(182) = reflection of X(i) in X(j) for these (i,j): (6,575), (141,140), (576,6)

X(182) = isogonal conjugate of X(262)

X(182) = isotomic conjugate of X(327)

X(182) = complement of X(1352)

X(182) = X(3)-of-1st-Brocard triangle

X(182) = X(3)- of 2nd Brocard triangle

X(182) = X(182)-of-circumsymmedial triangle

X(182) = {X(3),X(6)}-harmonic conjugate of X(511)

X(182) = {X(6),X(1350)}-harmonic conjugate of X(1351)

X(182) = {X(1340),X(1341)}-harmonic conjugate of X(3)

X(182) = {X(1687),X(1688)}-harmonic conjugate of X(6)

X(182) = inverse-in-circumcircle of X(2080)

X(182) = inverse-in-2nd-Brocard-circle of X(3095)

X(182) = inverse-in-circle-{{X(3102),X(3103),PU(1)}} of X(32452)

X(182) = exsimilicenter of circle centered at X(371) through X(1151) and circle centered at X(1152) through X(372)

X(182) = exsimilicenter of circle centered at X(372) through X(1152) and circle centered at X(1151) through X(371)

X(182) = radical trace of circles with diameters X(371)X(372) and X(1151)X(1152)

X(182) = harmonic center of 1st and 2nd Kenmotu circles

X(182) = {X(15),X(16)}-harmonic conjugate of X(574)

X(182) = harmonic center of Lucas radical circle and Lucas(-1) radical circle

X(182) = harmonic center of Lucas inner circle and Lucas(-1) inner circle

X(182) = harmonic center of 2nd Lemoine circle and circle {{X(1687),X(1688),PU(1),PU(2)}}

X(182) = radical trace of circles O(15,16) and O(61,62)

X(182) = exsimilicenter of circles {X(15),X(62),PU(1)} and {X(16),X(61),PU(1)}; the insimilicenter is X(32)

X(182) = X(2)-of-1st-Ehrmann-triangle

X(182) = {X(9738),X(9739)}-harmonic conjugate of X(9737)

X(182) = Artzt-to-McCay similarity image of X(381)

X(182) = X(3)-of-6th-anti-Brocard-triangle

X(182) = X(5476)-of-4th-anti-Brocard-triangle

X(182) = homothetic center of 5th anti-Brocard triangle and cevian triangle of X(3)

X(182) = homothetic center of 6th anti-Brocard triangle and 1st Brocard triangle

X(182) = endo-homothetic center of 6th Brocard triangle and 1st anti-Brocard triangle

X(182) = perspector of 1st Neuberg triangle and cross-triangle of 1st and 2nd Neuberg triangles

X(182) = Cundy-Parry Phi transform of X(39)

X(182) = Cundy-Parry Psi transform of X(83)

X(182) = endo-similarity image of reflection triangles of PU(1); the similitude center of these triangles is X(6)

Barycentrics csc A cos(A - ω) : csc B cos(B - ω) : csc C cos(C - ω)

Barycentrics cot A + tan ω : :

Let A'B'C' be the circummedial triangle. Let La be the line through A parallel to B'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines A'A", B'B", C'C" concur in X(183). (Randy Hutson, December 26, 2015)

X(183) lies on these lines:

2,6 3,76 5,315 22,157
25,264 55,350 95,305
187,1003 274,474 316,381
317,427 383,621 538,574
622,1080 668,956

X(183) is the {X(2),X(69)}-harmonic conjugate of X(325). For a list
of other harmonic conjugates of X(183), click **Tables** at the top
of this page.

X(183) = isogonal conjugate of X(263)

X(183) = isotomic conjugate of X(262)

X(183) = X(645)-beth conjugate of X(183)

X(183) = {X(2),X(69)}-harmonic conjugate of X(325)

X(183) = X(6)-of-circummedial-triangle

X(183) = pole wrt circumcircle of trilinear polar of X(3114) (line X(669)X(804))

X(183) = insimilicenter of Artzt and anti-Artzt circles; the exsimilicenter is X(2)

X(183) = crossdifference of every pair of points on the line through [U(2) of pedal triangle of P(1)] and [P(2) of pedal triangle of U(1)]

X(183) = X(5034)-of-6th-Brocard-triangle

Trilinears sin A sin 2A : :

Barycentrics a

X(184) is the homothetic center of triangles ABC and A'B'C', the
latter defined as follows: let B_{1} and C_{1} be the
points where the perpendicular bisector BC meets sidelines CA and AB,
and cyclically define C_{2}, A_{2}; A_{3},
B_{3}. Then A'B'C' is formed by the perpendicular bisectors of
segments B_{1}C_{1}, C_{2}A_{2},
A_{3}B_{3}. (Fred Lang, Hyacinthos #1190)

X(184) is the subject of Hyacinthos messages 5423-5441 (May, 2002).
In #5423, Alexei Myakishev notes that X(184) serves as a common vertex
of three triangles inside ABC, mutually congruent and similar to ABC.
(The triangles can be labeled XB_{C}C_{B},
XC_{A}A_{C}, XA_{B}B_{A}, with
B_{C} and C_{B} on side BC, C_{A} and
A_{C} on side CA, and A_{B} and B_{A} on side
AB.) See

Alexei Myakishev, On the Procircumcenter and Related Points ,
*Forum Geometricorum* 3 (2003) 29-34.

In #5435, Paul Yiu cites Fred Lang's construction of X(184) and notes that the three triangles are then easily constructed from X(184). The triangles determine three other triangles with common vertex X(184); in #5437, Nikos Dergiades notes that the vertex angles of these are 4A - π, 4B - π, 4C - π, and that

if ABC is acute, then X(184) = X(63)-of-the-orthic-triangle = X(226)-of-the-tangential-triangle

X(184) = homothetic center of the orthic triangle and the medial triangle of the tangential triangle.

Randy Hutson notes that X(184) is the exsimilicenter of the circumcircle and sine-triple-angle circle. (December 14, 2014)

Let A'B'C' be the intersections, other than X(3), of the X(3)-cevians and the Brocard circle. Let A"B"C" be the intersections, other than X(6), of the X(6)-cevians and the Brocard circle. Then A'B'C' and A"B"C" are perspective at X(184). Also, X(184) = U∩V, where U = isotomic conjugate of polar conjugate of Brocard axis (i.e., line X(3)X(49)), and V = polar conjugate of isotomic conjugate of Brocard axis (i.e., line X(6)X(25)). Let DEF be the orthic triangle. Let D' be the isotomic conjugate of X(4) wrt AEF, and define E' and F' cyclically; then the lines AD', BE', CF' concur in X(184). (Randy Hutson, June 1, 2015)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to ABC at X(184).

X(184) lies on these lines:

2,98 3,49 4,54 5,156
6,25 23,576 24,389
26,52 22,511 31,604
32,211 48,212 55,215
157,570 160,571 199,573
205,213 251,263 351,686
381,567 397,463 398,462
418,577 572,1011 647,878

X(184) is the {X(6),X(25)}-harmonic conjugate of X(51). For a list of other harmonic conjugates of X(184), click **Tables** at the top of this page.

X(184) = isogonal conjugate of X(264)

X(184) = isotomic conjugate of X(18022)

X(184) = complement of X(11442)

X(184) = inverse-in-Brocard-circle of X(125)

X(184) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,32), (54,6), (74,50)

X(184) = X(217)-cross conjugate of X(6)

X(184) = crosspoint of X(3) and X(6)

X(184) = crosssum of X(i) and X(j) for these (i,j): (2,4), (5,324), (6, 157), (92,318), (273,342), (338,523), (339,850), (427,1235), (491,492)

X(184) = crossdifference of every pair of points on line X(297)X(525)

X(184) = X(32)-Hirst inverse of X(237)

X(184) = X(i)-beth conjugate of X(j) for these (i,j): (212,212), (692,184)

X(184) = X(22) of 1st Brocard triangle

X(184) = trilinear product of PU(19)

X(184) = {X(3),X(49)}-harmonic conjugate of X(1147)

X(184) = vertex conjugate of PU(157) (the polar conjugates of PU(38)

X(184) = X(75)-isoconjugate of X(4)

X(184) = {X(8880),X(8881)}-harmonic conjugate of X(25)

X(184) = homothetIc center of orthic triangle and X(3)-Ehrmann triangle; see X(25)

X(184) = perspector of ABC and unary cofactor triangle of tangential-of-tangential triangle

X(184) = perspector of ABC and unary cofactor triangle of MacBeath triangle

Trilinears (cos A)(cos^2 B + cos^2 C) : :

Trilinears a(b^2 + c^2 - a^2)[2a^2(b^2 - c^2)^2 - a^4(b^2 + c^2) - (b^2 - c^2)^2(b^2 + c^2)] : :

Barycentrics (sin 2A)[1 - cos A cos(B - C)] : (sin 2B)[1 - cos B cos(C - A)] : (sin 2C)[1 - cos C cos(A - B)]

Alexei Myakishev has noted that X(185) is the Nagel point of the orthic triangle only is ABC is an acute triangle.

Let Ha be the foot of the A-altitude. Let Ba and Ca be the feet of perpendiculars from Ha to CA and AB, respectively. Let Ga be the centroid of HaBaCa. Define Gb and Gc cyclically. The lines HaGa, HbGb, HcGc concur in X(185). (Randy Hutson, December 26, 2015)

Let Ha, Hb, Hc be the orthocenters of the A-, B-, and C-altimedial triangles. X(185) is the orthocenter of HaHbHc. (Randy Hutson, March 25, 2016)

Let P be a point on the circumcircle. Let Pa be the orthogonal projection of P on the A-altitude, and define Pb, Pc cyclically. The locus of the orthocenter of PaPbPc as P varies is an ellipse centered at X(185). See also X(9730). (Randy Hutson, March 25, 2016)

X(185) lies on these lines:

1,296 3,49 4,51 5,113
6,64 20,193 25,1498
30,52 39,217 54,74
72,916 287,384 378,578
382,568 411,970 648,1105

X(185) = reflection of X(i) in X(j) for these (i,j): (4,389), (125,974)

X(185) = isogonal conjugate of X(1105)

X(185) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,417), (4,235)

X(185) = crosspoint of X(3) and X(4)

X(185) = crosssum of X(i) and X(j) for these (i,j): (3,4), (25,1249)

X(185) = anticomplement of X(5907)

X(185) = bicentric sum of PU(17)

X(185) = PU(17)-harmonic conjugate of X(647)

X(185) = orthology center of orthic and half-altitude triangles

X(185) = half-altitude isogonal conjugate of X(4)

X(185) = orthic-isogonal conjugate of X(235)

X(185) = orthic-isotomic conjugate of X(1843)

X(185) = X(20)-of-X(4)-Brocard-triangle

X(185) = anticomplement of X(4) wrt orthic triangle

X(185) = X(4)-of-tangential-triangle-of-Jerabek-hyperbola

X(185) = eigencenter of cevian triangle of X(648)

X(185) = eigencenter of anticevian triangle of X(647)

X(185) = trilinear product of vertices of 2nd Hyacinth triangle

X(185) = X(10)-of-circumorthic-triangle if ABC is acute

X(185) = excentral-to-ABC functional image of X(8)

X(185) = polar-circle-inverse of X(34170)

Trilinears sin 3A csc 2A : sin 3B csc 2B : sin 3C csc 2C

Barycentrics (sin A)(4 cos A - sec A) : (sin B)(4 cos B - sec B) : (sin C)(4 cos C - sec C)

Barycentrics a^2 ((a^2 - b^2 - c^2)^2 - b^2 c^2) / (a^2 - b^2 - c^2) : :

Tripolars |cos A| : :

X(186) = 3*X(2) + (J^2 - 3)*X(3) = 2 X(3) + X(23) = (a^2 - b^2 - c^2)(a^2 + b^2 - c^2)(a^2 - b^2 + c^2)*X(3) + (a^2b^2c^2)*X(4)

As a point on the Euler line, X(186) has Shinagawa coefficients (4F, -E - 4F).

Let AA_{1}A_{2}, BB_{1}B_{2}, CC_{1}C_{2} be the circumcircle-inscribed equilateral triangles used in the construction of the Trinh triangle. Let A' be the cevapoint of A_{1} and A_{2}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(186). (Randy Hutson, October 8, 2019

X(186) lies on these lines: 2,3 54,389 93,252 98,935 107,477 112,187 249,250 2931,3580

X(186) is the {X(3),X(24)}-harmonic conjugate of X(4). For a list of other harmonic conjugates of X(100), click **Tables** at the top of this page.

X(186) = reflection of X(i) in X(j) for these (i,j): (4,403), (403,468)

X(186) = isogonal conjugate of X(265)

X(186) = isotomic conjugate of X(328)

X(186) = complement of X(3153)

X(186) = anticomplement of X(2072)

X(186) = inverse-in-circumcircle of X(4)

X(186) = X(340)-Ceva conjugate of X(323)

X(186) = X(50)-cross conjugate of X(323)

X(186) = crosspoint of X(54) and X(74)

X(186) = crosssum of X(i) and X(j) for these (i,j): (5,30), (621,622)

X(186) = crossdifference of every pair of points on line X(216)X(647)

X(186) = inverse-in-polar-circle of X(5)

X(186) = pole wrt polar circle of trilinear polar of X(94) (line X(5)X(523))

X(186) = X(48)-isoconjugate (polar conjugate) of X(94)

X(186) = perspector of ABC and the reflection of the circumorthic triangle in the Euler line

X(186) = perspector of ABC and the reflection of the Kosnita triangle in the Euler line

X(186) = perspector of ABC and the reflection of the orthic triangle in the orthic axis

X(186) = reflection of X(403) in the orthic axis

X(186) = crosspoint of X(3) and X(2931) wrt both the excentral and tangential triangles

X(186) = homothetic center of circumorthic and Kosnita triangles

X(186) = inverse-in-Kosnita-circle of X(3)

X(186) = perspector of circumconic through polar conjugates of PU(5)

X(186) = Hofstadter 3 point

X(186) = antigonal image of X(5962

)
X(186) = X(484)-of-orthic-triangle if ABC is acute

X(186) = Thomson-isogonal conjugate of X(15131)

X(186) = Ehrmann-vertex-to-orthic similarity image of X(3153)

X(186) = {X(3),X(4)}-harmonic conjugate of X(3520)

X(186) = intersection of the tangent to hyperbola {{A,B,C,X(3),X(15)}} at X(15) and the tangent to hyperbola {{A,B,C,X(3),X(16)}} at X(16)

Trilinears sin A - 3 cos A tan ω : :

Trilinears 2 sin(A - 2ω) - sin(A + 2ω) + sin A : :

Trilinears sin A + sin A cos 2ω - 3 cos A sin 2ω : :

Trilinears cos(A + ω) sin 2ω - e^2 sin(A - ω) : :

Barycentrics a

Tripolars b c Sqrt[2(b^2 + c^2) - a^2] : :

Let L denote the line having trilinears of X(187) as coefficients. Then L is the line passing through X(2) perpendicular to the Euler line.

Let A'B'C' be the 1st Brocard triangle. Let A"B"C" be the 2nd Brocard triangle. Let A* = Λ((A',A"), and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(187). (Randy Hutson, December 26, 2015)

Let (O_{A}) be the circumcircle of BCX(2). Let P_{A} be the perspector of (O_{A}). Let L_{A} be the polar of P_{A} wrt (O_{A}). Define L_{B} and L_{C} cyclically. Let A' = L_{B}∩L_{C}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(187). (Randy Hutson, June 7, 2019)

If you have The Geometer's Sketchpad, you can view X(1316), which includes X(187).

X(187) lies on the Darboux quintic and these lines:

2,316 3,6 23,111 30,115
35,172 36,1015 74,248
99,385 110,352 112,186
183,1003 237,351 249,323
325,620 353,3117 395,531 396,530 729,805

X(187) is the {X(3),X(6)}-harmonic conjugate of X(574). For a list of other harmonic conjugates of X(187), click **Tables** at the top of this page.

X(187) = midpoint of X(i) and X(j) for these (i,j): (15,16), (99,385)

X(187) = reflection of X(i) in X(j) for these (i,j): (115,230), (316,625), (325,620)

X(187) = isogonal conjugate of X(671)

X(187) = isotomic conjugate of X(18023)

X(187) = inverse-in-circumcircle of X(6)

X(187) = inverse-in-Brocard-circle of X(574)

X(187) = inverse-in-van-Lamoen-circle-of-X(2)

X(187) = radical trace of the circumcircle and Brocard circle

X(187) = complement of X(316)

X(187) = anticomplement of X(625)

X(187) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,6593), (111,6)

X(187) = crosspoint of X(i) and X(j) for these (i,j): (2,67), (6,111), (468,524)

X(187) = crosssum of X(i) and X(j) for these (i,j): (2,524), (6,23), (111,895), (115,690)

X(187) = crossdifference of every pair of points on line X(2)X(523)

X(187) = X(55)-beth conjugate of X(187)

X(187) = inverse-in-Moses-radical-circle of X(1495)

X(187) = radical trace of Moses radical circle and Parry circle

X(187) = radical trace of Lucas radical circle and Lucas(-1) radical circle

X(187) = radical trace of Lucas inner and Lucas(-1) inner circle

X(187) = radical trace of circles {{P(1),U(2),U(39)}} and {{U(1),P(2),P(39)}}

X(187) = intersection of Brocard axis and Lemoine axis

X(187) = intersection of Brocard axis (or Lemoine axis) and non-transverse axis of hyperbola {{A,B,C,PU(2)}}

X(187) = intersection of Brocard axis (or Lemoine axis) and tangent at X(691) to hyperbola {{A,B,C,PU(2)}}

X(187) = midpoint of PU(2)

X(187) = bicentric sum of PU(2)

X(187) = perspector of ABC and the reflection of the circumsymmedial triangle in the Brocard axis

X(187) = perspector of ABC and the reflection of the circumsymmedial triangle in the Lemoine axis

X(187) = intersection of tangents to hyperbola {A,B,C,X(2),X(6)} at X(6) and X(111)

X(187) = inverse-in-Parry-circle of X(2502)

X(187) = X(187)-of-2nd-Brocard-triangle

X(187) = X(187)-of-circumsymmedial-triangle

X(187) = reflection of X(5107) in X(6)

X(187) = X(92)-isoconjugate of X(895)

X(187) = X(1577)-isoconjugate of X(691)

X(187) = {X(1687),X(1688)}-harmonic conjugate of X(2080)

X(187) = trilinear pole of PU(107)

X(187) = inverse-in-Parry-isodynamic-circle of X(351); see X(2)

X(187) = radical trace of 3rd and 4th Lozada circles

X(187) = radical trace of 6th and 7th Lozada circles

X(187) = radical trace of 8th and 9th Lozada circles

X(187) = radical trace of 10th and 11th Lozada circles

X(187) = radical trace of circumcircles of outer and inner Grebe triangles

X(187) = X(115)-of-4th-anti-Brocard-triangle

X(187) = X(187)-of-X(3)PU(1)

X(187) = Thomson-isogonal conjugate of X(6054)

X(187) = Cundy-Parry Phi transform of X(576)

X(187) = Cundy-Parry Psi transform of X(7607)

X(187) = homothetic center of Trinh triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles

X(187) = QA-P4 (Isogonal Center of quadrangle ABCX(2); see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/25-qa-p4.html)

Trilinears [bc(b + c - a)]

Trilinears csc A cos A/2 : :

Trilinears |AX(1)| : |BX(1)| : |CX(1)|

Trilinears sec A' : sec B', sec C', where A'B'C' = excentral triangle

Barycentrics 1/(sin(B/2) sin(C/2) + sin(A/2)) : : c.f., X(5451)

Barycentrics 1/(csc(B/2) csc(C/2) + csc(A/2)) : : c.f., X(5451)

Barycentrics cos A/2 : cos B/2 : cos C/2

Let A'B'C' be the excentral triangle of ABC, so that A' = -1 : 1 : 1 (trilinears). Let A'' be the point where the bisector of angle BA'C meets the line BC. Define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(188). (Seiichi Kirikami, February 14, 2010)

Let Ea be the ellipse with B and C as foci and passing through the A-excenter, and define Eb and Ec cyclically. Let La be the line tangent to Ea at the A-excenter, and define Lb and Lc cyclically. Let A' = La∩BC, B' = Lb∩CA, C' = Lc∩AB. Then A', B', C' are collinear, and the trilinear pole of line A'B'C' = X(188). Note: The triangle formed by La, Lb, Lc is also the excentral triangle of the excentral triangle. Alternately, let A" be the trilinear pole of line La, and define B", C" cyclically. The lines AA", BB" and CC" concur at X(188); see also X(174). (Randy Hutson, December 2, 2017)

X(188) lies on these lines: 1,361 2,178 9,173 40,164 166,167 174,266

X(188) = isogonal conjugate of X(266)

X(188) = isotomic conjugate of X(4146)

X(188) = anticomplement of X(178)

X(188) = X(2)-Ceva conjugate of X(236)

X(188) = cevapoint of X(1) and X(164)

X(188) = X(259)-cross conjugate of X(174)

X(188) = crosssum of X(1) and X(361)

X(188) = X(188)-beth conjugate of X(266)

X(188) = SS(A->A') of X(4), where A'B'C' is the excentral triangle

X(188) = isotomic conjugate of X(4146)

X(188) = X(65)-of-excentral-triangle

X(188) = perspector of circumconic centered at X(236)

X(188) = center of circumconic that is locus of trilinear poles of lines passing through X(236)

X(188) = excentral-to-ABC trilinear image of X(1)

X(188) = excentral-to-ABC barycentric image of X(164)

Barycentrics 1/(cos B + cos C - cos A - 1) : 1/(cos C + cos A - cos B - 1) : 1/(cos A + cos B - cos C - 1)

X(189) is the perspector of triangle ABC and the pedal triangle of X(84).

X(189) lies on the Lucas cubic and these lines: 2,77 7,92 8,20 29,81 69,309 222,281

X(189) = isogonal conjugate of X(198)

X(189) = isotomic conjugate of X(329)

X(189) = cyclocevian conjugate of X(8)

X(189) = anticomplement of X(223)

X(189) = X(309)-Ceva conjugate of X(280)

X(189) = cevapoint of X(84) and X(282)

X(189) = X(i)-cross conjugate of X(j) for these (i,j): (4,7), (57,2), (282,280)

X(189) = trilinear pole of line X(522)X(905)

X(189) = perspector of ABC and the reflection in X(282) of the pedal triangle of X(282)

Barycentrics 1/(b - c) : 1/(c - a) : 1/(a - b)

In unpublished notes, Yff has studied the parabola tangent to sidelines BC, CA, AB and having focus X(101). If A',B',C' are the respective points of tangency, then the lines AA', BB', CC' concur in X(190).

The line X(100)X(190) is tangent to the Steiner circumellipse at X(190) and to the circumcircle at X(100). (Peter Moses, July 7, 2009)

Let Ha be the hyperbola passing through A, and with foci at B and C. This is called the A-Soddy hyperbola in (Paul Yiu, *Introduction to the Geometry of the Triangle,* 2002; 12.4 The Soddy hyperbolas, p. 143). Let La be the polar of X(8) with respect to Ha. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The triangle A'B'C' is perspective to ABC, and the perspector is X(190). (Randy Hutson, December 26, 2015)

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(190) = X(238) of IaIbIc. (Randy Hutson, December 26, 2015)

Let A_{5}B_{5}C_{5} and A_{6}B_{6}C_{6} be Gemini triangles 5 and 6, resp. Let L_{A} be the tangent at A to conic {{A,B_{5},C_{5},B_{6},C_{6}}}, and define L_{B} and L_{C} cyclically. Let A' = L_{B}∩L_{C}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(190). (Randy Hutson, January 15, 2019)

If you have The Geometer's Sketchpad, you can view X(190).

X(190) lies on the Steiner circumellipse and these lines:

1,537 2,45 6,192 7,344
8,528 9,75 10,671 37,86
40,341 44,239 63,312
69,144 71,290 72,1043
99,101 100,659 110,835
162,643 191,1089 238,726
320,527 321,333 329,345
350,672 513,660 514,1016
522,666 644,651 646,668
649,889 658,1020 670,799
789,813 872,1045 1222, 3057

X(190) = reflection of X(i) in X(j) for these (i,j): (239,44), (335,37), (673,9), (903,2)

X(190) = isogonal conjugate of X(649)

X(190) = isotomic conjugate of X(514)

X(190) = complement of X(4440)

X(190) = anticomplement of X(1086)

X(190) = X(i)-Ceva conjugate of X(j) for these (i,j): (99,100), (666,3570)

X(190) = cevapoint of X(i) and X(j) for these (i,j): (2,514), (9,522), (37,513), (440,525)

X(190) = X(i)-cross conjugate of X(j) for these (i,j): (513,86), (514,2), (522,75)

X(190) = crosssum of X(512) and X(798)

X(190) = crossdifference of every pair of points on line X(1015)X(1960)

X(190) = X(i)-aleph conjugate of X(j) for these (i,j): (2,1052), (190,1), (645,411), (668,63), (1016,100)

X(190) = X(i)-beth conjugate of X(j) for these (i,j): (9,292), (190,651), (333,88), (645,190), (646,646), (1016,190)

X(190) = trilinear pole of the line X(1)X(2)

X(190) = Steiner-circumellipse-antipode of X(903)

X(190) = barycentric product of PU(24)

X(190) = crossdifference of PU(25)

X(190) = trilinear product of PU(58)

X(190) = perspector of ABC and tangential triangle (wrt excentral triangle) of hyperbola passing through X(1), X(9) and the excenters (the Jerabek hyperbola of the excentral triangle)

X(190) = X(6)-isoconjugate of X(513)

X(190) = perspector of ABC and vertex-triangle of Gemini triangles 5 and 6

X(190) = ABC-to-Gemini-triangle-19 parallelogic center

are Ceva conjugates. The P-Ceva conjugate of Q is the perspector

of the cevian triangle of P and the anticevian triangle of Q.

Trilinears S

**
X(191) = X(1) - 2 X(21)
**

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(191) = X(21) of IaIbIc. (Randy Hutson, December 2, 2017)

Let IaIbIc be the excentral triangle. Let Na be the nine-point center of BCIa, and define Nb and Nc cyclically. The lines IaNa, IbNb, IcNc concur in X(191); c.f. X(5506). (Randy Hutson, December 2, 2017)

X(191) lies on these lines:

1,21 9,46 10,267 30,40
35,72 36,960 109,201
165,1079 190,1089 329,498

X(191) = reflection of X(i) in X(j) for these (i,j): (1,21), (79,442)

X(191) = isogonal conjugate of X(267)

X(191) = X(10)-Ceva conjugate of X(1)

X(191) = crosspoint of X(i) and X(j) for these (i,j): (10,502)

X(191) = crosssum of X(58) and X(501)

X(191) = excentral-isogonal conjugate of X(3)

X(191) = X(i)-aleph conjugate of X(j) for these (i,j): (2,2), (8,20), (10,191), (37,1045), (188,3), (366,6)

X(191) = X(643)-beth conjugate of X(191)

X(191) = crossdifference of every pair of points on line X(661)X(2605)

X(191) = X(54)-of-excentral-triangle

X(191) = perspector of excentral triangle and Fuhrmann triangle

X(191) = intersection of Euler lines of outer and inner Garcia triangles

X(191) = {X(1),X(21)}-harmonic conjugate of X(5426)

X(191) = complement of X(14450)

X(191) = anticomplement of X(11263)

(CONGRUENT PARALLELIANS POINT)

Barycentrics ca + ab - bc : ab + bc - ca : bc + ca - ab

The segments through X(192) parallel to the sidelines with endpoints on the sidelines have equal length. For references as early as 1881, see Hyacinthos message 2929 (Paul Yiu, May 29, 2001). See also

**Sabrina Bier,** "Equilateral Triangles Intercepted by Oriented
Parallelians," *Forum Geometricorum* 1 (2001) 25-32.

X(192) lies on these lines:

1,87 2,37 6,190 7,335
8,256 9,239 55,385
69,742 144,145 315,746 869,1045

X(192) = reflection of X(i) in X(j) for these (i,j): (8,984), (75,37), (1278,75)

X(192) = isogonal conjugate of X(2162)

X(192) = isotomic conjugate of X(330)

X(192) = complement of X(1278)

X(192) = anticomplement of X(75)

X(192) = X(1)-Ceva conjugate of X(2)

X(192) = crosspoint of X(1) and X(43)

X(192) = crosssum of X(1) and X(87)

X(192) = X(9)-Hirst inverse of X(239)

X(192) = X(646)-beth conjugate of X(192)

X(192) = perspector of anticomplementary triangle and Gemini triangle 4

X(192) = perspector of Gemini triangles 4 and 6

X(192) = trilinear pole of line X(3835)X(4083) (the perspectrix of ABC and Gemini triangle 16)

Trilinears (S

Barycentrics cot B + cot C - cot A : cot C + cot A - cot B : cot A + cot B - cot C

= 3a

Let A' be the reflection of the midpoint of segment BC in X(6), and define B' and C' cyclically. The lines AA', BB', CC' concur in X(193). (Randy Hutson, 9/23/2011)

Let A' be the trilinear pole of the perpendicular bisector of BC, and define B', C' cyclically. A'B'C' is also the anticomplement of the anticomplement of the midheight triangle. A'B'C' is perspective to the orthic and anticomplementary triangles at X(193). (Randy Hutson, January 29, 2018)

X(193) lies on these lines:

2,6 4,1351 7,239 8,894 20,185 23,159 44,344 66,895 144,145 146,148 253,287 317,393 330,959 371,488 372,487 608,651 1839,3187

X(193) = reflection of X(i) in X(j) for these (i,j): (3,1353), (4,1351), (69,6), (1352,576)

X(193) = isogonal conjugate of X(8770)

X(193) = isotomic conjugate of X(2996)

X(193) = complement of X(20080)

X(193) = anticomplement of X(69)

X(193) = anticomplementary conjugate of X(1370)

X(193) = X(4)-Ceva conjugate of X(2)

X(193) = X(2)-Hirst inverse of X(230)

X(193) = polar conjugate of X(34208)

X(193) = X(i)-beth conjugate of X(j) for these (i,j): (645,193), (662,608)

X(193) = perspector of pedal and antipedal triangles of X(4) (orthic and anticomplementary triangles)

X(193) = perspector, wrt orthic triangle, of polar circle

X(193) = anticomplementary isotomic conjugate of X(20)

X(193) = orthic-isogonal conjugate of X(2)

X(193) = trilinear pole of polar, wrt complement of polar circle, of X(69)

X(193) = pole of orthic axis wrt Steiner circumellipse

X(193) = {X(385),X(7774)}-harmonic conjugate of X(2)

X(193) = endo-homothetic center of 3rd and 4th tri-squares central triangles

X(193) = perspector, wrt anticomplementary triangle, of polar circle

Barycentrics a

Barycentrics cot

Let Oa be the circle through A and tangent to BC at its midpoint. Define Ob and Oc cyclically. The radical center of Oa, Ob, Oc is X(194). (Randy Hutson, December 26, 2015)

Let AaBaCa, AbBbCb, AcBcCc be the A-, B- and C-anti-altimedial triangles. Let A' be the trilinear product Aa*Ab*Ac, and define B', C' cyclically. Triangle A'B'C' is the anticomplementary triangle of the 1st Brocard triangle, and is perspective to ABC at X(4), and to the anticomplementary triangle at X(194). (Randy Hutson, November 2, 2017)

X(194) lies on these lines:

{1,87}, {2,39}, {3,385}, {4,147}, {6,384}, {8,730}, {10,3097}, {20,185}, {32,99}, {63,239}, {69,695}, {75,1107}, {83,3734}, {183,5013}, {184,3492}, {190,2176}, {257,986}, {262,2996}, {263,3498}, {304,3797}, {315,736}, {325,5025}, {350,2275}, {401,1993}, {487,1587}, {488,1588}, {548,3793}, {574,1078}, {616,3104}, {617,3105}, {627,3106}, {628,3107}, {648,1968}, {712,4393}, {1007,2023}, {1593,1941}, {1654,4201}, {1670,2547}, {1671,2546}, {1909,2276}, {2128,2285}, {3096,4045}, {3212,3503}, {3314,3933}, {3413,3557}, {3414,3558}, {3522,5188}, {3770,4261}, {3906,5652}, {3972,5007}

X(194) is the {X(39),X(76)}-harmonic conjugate of X(2). For a list
of other harmonic conjugates of X(194), click **Tables** at the top
of this page.

X(194) = reflection of X(76) in X(39)

X(194) = isogonal conjugate of X(3224)

X(194) = isotomic conjugate of X(2998)

X(194) = isotomic conjugate of the complement of X(32747)

X(194) = complement of X(20081)

X(194) = anticomplement of X(76)

X(194) = anticomplementary conjugate of X(315)

X(194) = eigencenter of cevian triangle of X(6)

X(194) = eigencenter of anticevian triangle of X(2)

X(194) = radical center of the Neuberg circles.

X(194) = X(6)-Ceva conjugate of X(2)

X(194) = X(3)-Hirst inverse of X(385)

X(194) = anticomplementary-isotomic conjugate of X(69)

X(194) = X(6374)-cross conjugate of X(2)

X(194) = vertex conjugate of PU(140)

X(194) = 1st-Brocard-to-6th-Brocard similarity image of X(6)

X(194) = X(99)-of-6th-Brocard-triangle

X(194) = circumcircle-inverse of X(32517)

X(194) = polar circle inverse of X(32527)

X(194) = anticomplementary-circle-inverse of X(32528)

X(194) = orthoptic-circle-of-Steiner-inellipse-inverse of X(32530)

X(194) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(32526)

X(194) = de Longchamps-circle-inverse of X(32529)

X(194) = 2nd-Brocard-circle-inverse of X(32531)

X(194) = perspector of 3rd Brocard triangle and 1st Brocard-reflected triangle

Trilinears a[a^8 + b^8 + c^8 - 4a^6(b^2 + c^2) + a^4(6b^4 + 6c^4 + 5b^2c^2) - a^2(4b^6 + 4c^6 - b^4c^2 - b^2c^4) - 2b^2c^2(b^4 + c^4 - b^2c^2)] : :

Barycentrics 4 cos 2A + cot

Let A' be the isogonal conjugate of the A-vertex of the outer Napoleon triangle, and define B' and C' cyclically. Let A" be the isogonal conjugate of the A-vertex of the inner Napoleon triangle, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(195). (Randy Hutson, November 18, 2015)

The Napoleon axis and Napoleon-Feuerbach cubic K005 meet in three points: X(17), X(18), and X(195). (Randy Hutson, November 18, 2015)

A construction of X(195) is given by Antreas Hatipolakis and Angel Montesdeoca at 24180.

X(195) lies on the Napoleon cubic and these lines:

1,3467 3,54 4,399
5,3459 6,17 49,52
110,143 140,323 155,381
382,1498 2121,3462 3461,3468

X(195) = reflection of X(i) in X(j) for these (i,j): (3,54), (54,1493), (3519,1209)

X(195) = isogonal conjugate of X(3459)

X(195) = complement of X(12325)

X(195) = anticomplement of X(21230)

X(195) = X(5)-Ceva conjugate of X(3)

X(195) = crosssum of X(137) and X(523)

X(195) = X(3)-of-reflection-triangle

X(195) = X(79)-of-tangential-triangle if ABC is acute

X(195) = tangential isogonal conjugate of X(2937)

X(195) = 2nd isogonal perspector of X(5); see X(36)

X(195) = Yiu-isogonal conjugate of X(1157)

X(195) = perspector of [cross-triangle of ABC and outer Napoleon triangle] and [cross-triangle of ABC and inner-Napoleon triangle]

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cos B + cos C - cos A - 1) tan A tan A/2

X(196) lies on these lines:

1,207 2,653 4,65 7,92
19,57 34,937 40,208
55,108 226,281 329,342

X(196) = isogonal conjugate of X(268)

X(196) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,4), (92,278)

X(196) = cevapoint of X(19) and X(207)

X(196) = X(221)-cross conjugate of X(347)

X(196) = X(i)-beth conjugate of X(j) for these (i,j): (648,2), (653,196)

X(196) = Danneels point of X(653)

X(196) = pole wrt polar circle of trilinear polar of X(280) (line X(521)X(3239))

X(196) = polar conjugate of X(280)

Let A'B'C' be the extouch triangle. Let A" be the crosspoint of the circumcircle intercepts of line B'C', and define B", C" cyclically. The lines A'A", B'B", C'C" concur in X(197). (Randy Hutson, July 31 2018)

For a construction of X(197), see Dasari Naga Vijay Krishna, "On A Simple Construction of Triangle Centers X(8), X(197), X(K) & X(594)", Scientific Inquiry and Review, Vol. 2, Issue 3, July 2018.

X(197) lies on these lines:

3,10 6,181 19,25 22,100 42,48 56,227 159,200

X(197) = X(8)-Ceva conjugate of X(6)

X(197) = crosssum of X(124) and X(514)

X(197) = isogonal conjugate of X(8048)

X(197) = crossdifference of every pair of points on line X(905)X(3910)

X(197) = crosspoint of circumcircle intercepts of excircles radical circle

Trilinears a (a^3 + a^2 (b + c) - a (b + c)^2 - (b - c)^2 (b + c)) : :

Trilinears s cos A - r cot(A/2) : :

X(198) lies on these lines:

3,9 6,41 19,25 5,1030
64,71 100,346 101,102
154,212 208,227 218,579
284,859 478,577 958,966

X(198) = isogonal conjugate of X(189)

X(198) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,55), (9,6), (223,221)

X(198) = crosspoint of X(40) and X(223)

X(198) = crosssum of X(i) and X(j) for these (i,j): (57,1422), (84,282), (513,1146), (650,1364), (1433,1436)

X(198) = crossdifference of every pair of points on line X(522)X(905)

X(198) = X(i)-beth conjugate of X(j) for these (i,j): (9,19), (101,198)

X(198) = perspector of Apus and tangential triangles

Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

As a point on the Euler line, X(199) has Shinagawa coefficients (E + 2F + $bc$, -2E - 2F + $bc$).

Let I_{A}, I_{B}, I_{C} be the excenters. Let (O_{A}) be the circle tangent to the circumcircle at A and passing through I_{A}. Let A' be the antipode of A in (O_{A}). Let L_{A} be the tangent to (O_{A}) at A'. Define L_{B} and L_{C} cyclically. Let T_{A} = L_{B}∩L_{C}, and define T_{B} and T_{C} cyclically. Triangle T_{A}T_{B}T_{C} is homothetic to the tangential triangle at X(199). (Randy Hutson, June 7, 2019)

X(199) lies on these lines: 2,3 42,172 51,572 55,1030 184,573

X(199) = isogonal conjugate of X(8044)

X(199) = anticomplement of X(34119)

X(199) = X(10)-Ceva conjugate of X(6)

X(199) = crosspoint of X(101) and X(250)

X(199) = crosssum of X(125) and X(514)

X(199) = tangential isogonal conjugate of X(8053)

X(199) = orthic-to-tangential similarity image of X(430)

Trilinears (b + c - a)

Trilinears (1 + cos A)/(1 - cos A) : : (Randy Hutson, 9/23/2011)

Trilinears 1 - csc^2(A/2) : :

Trilinears squared distance of A to Gergonne line : :

Barycentrics a(b + c - a)

Let A'B'C' be the extouch triangle. Let A" be the trilinear product of the circumcircle intercepts of line B'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(200). (Randy Hutson, July 31 2018)

Let A' be the trilinear product of the circumcircle intercepts of the A-excircle. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(200). (Randy Hutson, July 31 2018)

X(200) lies on these lines:

1,2 3,963 9,55 33,281
40,64 46,1004 57,518
63,100 69,269 159,197
219,282 220,728 255,271
318,1089 319,326 329,516
341,1043 756,968

X(200) is the {X(8),X(78)}-harmonic conjugate of X(1). For a list of
harmonic conjugates of X(200), click **Tables** at the top of this
page.

X(200) = reflection of X(i) in X(j) for these (i,j): (1,997), (57,1376)

X(200) = isogonal conjugate of X(269)

X(200) = isotomic conjugate of X(1088)

X(200) = X(8)-Ceva conjugate of X(9)

X(200) = cevapoint of X(220) and X(480)

X(200) = X(220)-cross conjugate of X(9)

X(200) = crosspoint of X(8) and X(346)

X(200) = crosssum of X(i) and X(j) for these (i,j): (56,1407), (57,1420), (1042,1427)

X(200) = X(i)-beth conjugate of X(j) for these (i,j): (100,223), (200,55), (643,165)

X(200) = {X(1),X(8)}-harmonic conjugate of X(4853)

X(200) = {X(2),X(8)}-harmonic conjugate of X(4847)

X(200) = homothetic center of anticomplementary triangle and 3rd antipedal triangle of X(1)

X(200) = homothetic center of ABC and medial triangle of 3rd antipedal triangle of X(1)

X(200) = Danneels point of X(8)

X(200) = polar conjugate of X(1847)

X(200) = trilinear square of X(9)

X(200) = trilinear product of the circumcircle intercepts with the excircles

X(200) = X(1899)-of-excentral-triangle

X(200) = complement of polar conjugate of isogonal conjugate of X(22153)

X(200) = excentral-to-ABC barycentric image of X(57)

X(200) = mixtilinear-incentral-to-ABC barycentric image of X(1)

X(200) = mixtilinear-excentral-to-ABC barycentric image of X(1)

Barycentrics (sin 2A)[1 + cos(B - C)] : (sin 2B)[1 + cos(C - A)] : (sin 2C)[1 + cos(A - B)]

X(201) lies on these lines:

1,212 9,34 10,225
12,756 33,40 37,65
38,56 55,774 57,975
63,603 72,73 109,191
210,227 220,221 255,1060
337,348 388,984 601,920

X(201) = isogonal conjugate of X(270)

X(201) = X(10)-Ceva conjugate of X(12)

X(201) = crosspoint of X(10) and X(72)

X(201) = crosssum of X(i) and X(j) for these (i,j): (1,580),
(28,58)

X(201) = X(i)-beth conjugate of X(j) for these (i,j): (72,201),
(1018,201)

u = u(A,B,C) = sin(A + π/3), v = u(B,C,A), w = u(C,A,B)

Trilinears 1 - cos(A + π/3) : 1 - cos(B + π/3) : 1 - cos(C + π/3) (Joe Goggins, Oct. 19, 2005)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(202) lies on these lines:

1,62 6,101 11,13 12,18
15,36 16,55 17,499
56,61 395,495 397,496

X(202) = X(1)-Ceva conjugate of X(15)

u = u(A,B,C) = sin(A - π/3), v = u(B,C,A), w = u(C,A,B)

Trilinears 1 + cos(A + 2π/3) : 1 + cos(B + 2π/3) : 1 + cos(C + 2π/3) (Joe Goggins, Oct. 19, 2005)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(203) lies on these lines:

1,61 6,101 11,14 12,17
15,55 16,36 18,499
56,62 396,495 398,496

X(203) = X(1)-Ceva conjugate of X(16)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(204) lies on these lines: 6,33 19,31 25,34 55,1033 63,162 108,223 207,221

X(204) = X(1)-Ceva conjugate of X(19)

X(204) = X(i)-beth conjugate of X(j) for these (i,j): (108,204),
(162,223)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(205) lies on these lines: 25,41 37,48 78,101 154,220 184,213

X(205) = X(9)-Ceva conjugate of X(31)

Trilinears (sin A)(sin 2A - tan ω) : :

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung); then X(206) = X(6)-of-A'B'C'. (Randy Hutson, July 31 2018)

X(206) lies on these lines:

2,66 5,182 6,25 26,511
69,110 157,216 160,577
219,692 237,571

X(206) = midpoint of X(i) and X(j) for these (i,j): (6,159), (110,1177)

X(206) = complement of X(66)

X(206) = complementary conjugate of X(427)

X(206) = X(2)-Ceva conjugate of X(32)

X(206) = crosspoint of X(2) and X(315)

X(206) = crosssum of X(339) and X(523)

X(206) = X(66)-of-medial triangle

X(206) = perspector of circumconic centered at X(32)

X(206) = isogonal conjugate of the isotomic conjugate of X(22)

X(206) = center of conic that is the locus of centers of conics passing through X(6) and the vertices of the tangential triangle

X(206) = centroid of X(6) plus the vertices of the tangential triangle

X(206) = crosssum of circumcircle intercepts of de Longchamps line

X(206) = polar conjugate of isogonal conjugate of X(22075)

X(206) = center of circumconic that is locus of trilinear poles of lines passing through X(32); this conic is the isogonal conjugate of the de Longchamps line

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(207) lies on these lines: 1,196 19,56 33,64 34,1042 40,108 78,653 204,221

X(207) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,34), (196,19)

X(207) = X(1)-beth conjugate of X(64)

X(207) = trilinear product X(34)*X(1490)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(208) lies on these lines:

1,102 4,57 19,225 25,34
33,64 40,196 198,227
226,406 318,653

X(208) = isogonal conjugate of X(271)

X(208) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,34), (57,19),
(342,223)

X(208) = crosssum of X(3) and X(1433)

X(208) = X(i)-beth conjugate of X(j) for these (i,j): (108,208),
(162,1)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(209) lies on these lines: 6,31 10,12 44,51 306,518

X(209) = isogonal conjugate of X(272)

X(209) = X(4)-Ceva conjugate of X(37)

Trilinears (cos B - cos C)^2 + (cos A + 1)(cos B + cos C - 2) : :

Barycentrics a(b + c)(b + c - a) : b(c + a)(c + a - b) : c(a + b)(a + b - c) : :

X(210) lies on these lines:

1,2334 2,354 6,612 8,312 9,55
10,12 31,44 33,220
37,42 38,899 43,984
45,968 51,374 56,936
63,1004 78,958 165,971
201,227 213,762 381,517
392,519 430,594 869,1107
956,997 976,1104

X(210) = X(2)-of-extouch triangle, so that X(210)X(1158) = Euler line of the extouch triangle

X(210) = reflection of X(i) in X(j) for these (i,j): (51,375), (354,2)

X(210) = isogonal conjugate of X(1014)

X(210) = complement of X(3873)

X(210) = anticomplement of X(3742)

X(210) = X(10)-Ceva conjugate of X(37)

X(210) = crosspoint of X(8) and X(9)

X(210) = crosssum of X(i) and X(j) for these (i,j): (56,57), (58,1412)

X(210) = crossdifference of every pair of points on line X(1019)X(1429)

X(210) = X(i)-beth conjugate of X(j) for these (i,j): (200,210), (210,42)

X(210) = centroid of Bevan circle intercepts with sidelines of ABC

X(210) = centroid of AbAcBcBaCaCb as defined at X(3588)

X(210) = centroid of AbAcBcBaCaCb as used in the construction of the inner-Conway triangle; see preamble before X(11677)

X(210) = trilinear pole of line X(3709)X(4041)

X(210) = excentral-to-ABC barycentric image of X(2)

Barycentrics a^4*(b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 + b^2*c^2 - c^4) : :

X(211) lies on the cubic K1088 and these lines: {5, 141}, {6, 11360}, {32, 184}, {51, 1506}, {52, 114}, {232, 15897}, {263, 2548}, {571, 2909}, {632, 15510}, {688, 2518}, {754, 3491}, {1078, 11673}, {2794, 13419}, {2979, 3096}, {3060, 7752}, {3202, 20960}, {3456, 17970}, {3917, 6292}, {4173, 5007}, {5017, 20987}, {5167, 7747}, {7807, 14962}, {11674, 12110}, {14575, 18796}

X(211) = X(4)-Ceva conjugate of X(39)

X(211) = crossdifference of every pair of points on line {850, 3050}

X(211) = barycentric product X(i)*X(j) for these {i,j}: {39, 3060}, {1964, 18041}, {3051, 7752}

X(211) = barycentric quotient X(i)/X(j) for these {i,j}: {3060, 308}, {18041, 18833}

X(211) = {X(3051),X(23208)}-harmonic conjugate of X(3203)

= (cos A)cos

= a

Barycentrics (sin 2A)(1 + cos A) : (sin 2B)(1 + cos B) : (sin 2C)(1 + cos C)

The trilinear polar of X(212) passes through X(1946). (Randy Hutson, June 7, 2019)

X(212) lies on these lines:

1,201 3,73 6,31 9,33
11,748 34,40 35,47
48,184 56,939 63,1040
78,283 109,165 154,198
238,497 312,643 582,942

X(212) = isogonal conjugate of X(273)

X(212) = crossdifference of every pair of points on line X(514)X(3064)

X(212) = X(4)-isoconjugate of X(7)

X(212) = X(57)-isoconjugate of X(92)

X(212) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,48), (9,41), (283,219)

X(212) = X(228)-cross conjugate of X(55)

X(212) = crosspoint of X(i) and X(j) for these (i,j): (3,219), (9,78)

X(212) = crosssum of X(i) and X(j) for these (i,j): (4,278), (34,57)

X(212) = X(212)-beth conjugate of X(184)

Trilinears a

Barycentrics (b + c)a

X(213) lies on these lines: 1,6 8,981 31,32 39,672 58,101 63,980 83,239 100,729 184,205 274,894 607,1096 667,875 692,923

X(213) = isogonal conjugate of X(274)

X(213) = isotomic conjugate of X(6385)

X(213) = complement of X(17137)

X(213) = anticomplement of X(21240)

X(213) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,42), (37,228)

X(213) = crosspoint of X(6) and X(31)

X(213) = crosssum of X(i) and X(j) for these (i,j): (2,75), (81,1444), (85,348)

X(213) = crossdifference of every pair of points on line X(320)X(350)

X(213) = X(i)-beth conjugate of X(j) for these (i,j): (41,213), (101,65), (644,213)

X(213) = bicentric sum of PU(9)

X(213) = PU(9)-harmonic conjugate of X(667)

X(213) = barycentric product of PU(85)

X(213) = trilinear pole of line X(669)X(798)

X(213) = X(92)-isoconjugate of X(1444)

X(213) = {X(1),X(9)}-harmonic conjugate of X(5283)

X(214) lies on the bicevian conic of X(1) and X(2), which is also QA-Co1 (Nine-point Conic) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/other-quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/76-qa-co1.html) (Randy Hutson, July 20, 2016)

X(214) lies on these lines: 1,88 2,80 9,48 10,140 11,442 36,758 44,1017 119,515 142,528 535,908 662,759 1015,1100

X(214) = midpoint of X(1) and X(100)

X(214) = reflection of X(11) in X(1125)

X(214) = isogonal conjugate of X(1168)

X(214) = complement of X(80)

X(214) = X(2)-Ceva conjugate of X(44)

X(214) = crosspoint of X(2) and X(320)

X(214) = X(21)-beth conjugate of X(244)

X(214) = perspector of circumconic centered at X(44)

X(214) = center of circumconic that is locus of trilinear poles of lines passing through X(44)

X(214) = X(36) of X(1)-Brocard triangle

X(214) = inner-Garcia-to-ABC similarity image of X(10)

X(214) = X(7687)-of-excentral-triangle

X(214) = QA-P3 (Gergonne-Steiner Point) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/quadrangle-objects/10-mathematics/quadrangle-objects/18-qa-p3.html)

Trilinears cos

Trilinears 1 + cos 3A : 1 + cos 3B : 1+ cos 3C (M. Iliev, 4/12/07)

Trilinears a^3 (a^2 - b^2 - c^2 + b c)^2 (a - b - c) : :

X(215) is the insimilicenter of the incircle and the sine-triple-angle circle. (Randy Hutson, December 14, 2014)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(215) lies on these lines: 1,49 11,110 12,54 55,184

X(215) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,50)

Trilinears cos A (tan A + tan B + tan C) + sin A : :

Trilinears cos A + sin A (cot A cot B cot C) : :

Trilinears (sin 2B + sin 2C) cos A : :

Trilinears sin A (cos 2B + cos 2C) : :

Trilinears (sin A)(1 - sin^2 B - sin^2 C) : :

Trilinears sec B sec C + csc B csc C : :

Trilinears cos(A + T) : :, T as at X(389)

Barycentrics csc 2B + csc 2C : :

Barycentrics a^2(b^2 + c^2 - a^2)[a^2(b^2 + c^2) - (b^2 - c^2)^2] : :

Barycentrics (sin A)(sin 2A)cos(B - C) : :

X(216) is the perspector of triangle ABCand the tangential triangle of the Johnson circumconic. (Randy Hutson, 9/23/2011)

Let Ea be the ellipse with B and C as foci and passing through X(5), and define Eb, Ec cyclically. Let La be the line tangent to Ea at X(5), and define Lb, Lc cyclically. Let A' be the trilinear pole of line La, and define B', C' cyclically. A', B', C' lie on the circumconic centered at X(216). (Randy Hutson, July 20, 2016)

X(216) = intersection of isogonal conjugate of polar conjugate of Euler line (i.e., line X(3)X(6)) and the polar conjugate of isogonal conjugate of Euler line (i.e., line X(2)X(216)) (Randy Hutson, July 20, 2016)

X(216) lies on hyperbola {{X(2),X(6),X(216),X(233),X(1249),X(1560),X(3162)}}, which is a circumconic of the medial triangle, as well as the locus of the perspector of circumconics centered at a point on the Euler line. Also, this hyperbola is tangent to Euler line at X(2). (Randy Hutson, July 20, 2016)

X(216) lies on these lines:

2,232 3,6 5,53 51,418
95,648 97,288 115,131
157,206 373,852 395,465
396,466 631,1075 1015,1060 2493,3054

X(216) = isogonal conjugate of X(275)

X(216) = isotomic conjugate of X(276)

X(216) = inverse-in-Brocard-circle of X(577)

X(216) = complement of X(264)

X(216) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,5), (3,418), (5,51), (324,52)

X(216) = cevapoint of X(217) and X(418)

X(216) = X(217)-cross conjugate of X(51)

X(216) = crosspoint of X(i) and X(j) for these (i,j): (2,3), (5,343)

X(216) = crosssum of X(4) and X(6)

X(216) = crossdifference of every pair of points on line X(186)X(523)

X(216) = inverse-in-Brocard-circle of X(577)

X(216) = center of circumconic that is locus of trilinear poles of lines passing through X(5)

X(216) = intersection of trilinear polars of any 2 points on the Johnson circumconic

X(216) = perspector of cevian triangle of X(3) and tangential triangle, wrt cevian triangle of X(3), of circumconic of cevian triangle of X(3) centered at X(3)

X(216) = pole wrt polar circle of trilinear polar of X(8795)

X(216) = X(48)-isoconjugate (polar conjugate) of X(8795)

X(216) = X(92)-isoconjugate of X(54)

X(216) = X(1577)-isoconjugate of X(933)

X(216) = X(573)-of-orthic-triangle if ABC is acute

X(216) = perspector of ABC and unary cofactor triangle of circumorthic triangle

Barycentrics a^4*(a^2 - b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(217) lies on the cubic K1088 and these lines: {3, 3289}, {4, 6}, {5, 1625}, {32, 184}, {39, 185}, {51, 3199}, {54, 112}, {83, 287}, {125, 1506}, {187, 13367}, {213, 20230}, {216, 5562}, {232, 389}, {263, 10790}, {394, 3785}, {574, 1204}, {577, 10984}, {578, 1968}, {648, 9291}, {1015, 1425}, {1404, 1409}, {1500, 3270}, {1562, 7765}, {1614, 1971}, {1691, 11674}, {1692, 5167}, {1899, 2548}, {1993, 20065}, {1994, 20088}, {2909, 20968}, {2965, 14533}, {3053, 19357}, {3172, 11402}, {3398, 17974}, {4173, 6752}, {4846, 15075}, {5007, 8779}, {5013, 10605}, {5038, 5622}, {5052, 6467}, {5058, 21640}, {5062, 21641}, {5889, 22240}, {6423, 19356}, {6424, 19355}, {6759, 10311}, {7736, 18909}, {7737, 19467}, {7747, 21659}, {7757, 9289}, {8571, 10254}, {10547, 14600}, {10986, 26882}, {12111, 26216}, {13330, 15073}, {13366, 14581}, {13754, 22416}, {15043, 15355}, {16502, 19349}, {18445, 23128}

X(217) = isogonal conjugate of X(276)

X(217) = X(i)-Ceva conjugate of X(j) for these (i,j): {6, 51}, {216, 418}, {1625, 15451}, {1987, 237}

X(217) = X(i)-isoconjugate of X(j) for these (i,j): {1, 276}, {54, 1969}, {63, 8795}, {75, 275}, {76, 2190}, {92, 95}, {264, 2167}, {304, 8884}, {326, 8794}, {561, 8882}, {811, 15412}, {933, 20948}, {1577, 18831}, {2148, 18022}, {2169, 18027}, {2616, 6331}, {14208, 16813}

X(217) = crosspoint of X(i) and X(j) for these (i,j): {6, 184}, {51, 216}

X(217) = crossdifference of every pair of points on line {340, 520}

X(217) = crosssum of X(i) and X(j) for these (i,j): {2, 264}, {76, 7763}, {95, 275}

X(217) = X(92)-isoconjugate of X(95)

X(217) = barycentric product X(i)*X(j) for these {i,j}: {3, 51}, {4, 418}, {5, 184}, {6, 216}, {25, 5562}, {32, 343}, {48, 1953}, {52, 2351}, {53, 577}, {63, 2179}, {110, 15451}, {112, 17434}, {154, 8798}, {212, 1393}, {213, 16697}, {228, 18180}, {255, 2181}, {311, 14575}, {324, 14585}, {394, 3199}, {512, 23181}, {560, 18695}, {603, 7069}, {647, 1625}, {810, 2617}, {1092, 14569}, {1437, 21807}, {1576, 6368}, {1820, 2180}, {2200, 17167}, {3049, 14570}, {3078, 20574}, {3527, 26907}, {9247, 14213}, {13450, 23606}, {14587, 24862}, {17500, 20775}

X(217) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 18022}, {6, 276}, {25, 8795}, {32, 275}, {51, 264}, {53, 18027}, {184, 95}, {216, 76}, {343, 1502}, {418, 69}, {560, 2190}, {1501, 8882}, {1576, 18831}, {1625, 6331}, {1953, 1969}, {1974, 8884}, {2179, 92}, {2207, 8794}, {3049, 15412}, {3199, 2052}, {5562, 305}, {9247, 2167}, {9418, 19189}, {14574, 933}, {14575, 54}, {14585, 97}, {15451, 850}, {16697, 6385}, {17434, 3267}, {18695, 1928}, {23181, 670}

X(217) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 184, 14585), (39, 185, 3269), (54, 112, 1970), (1614, 10312, 1971), (7592, 8743, 6)

f(A,B,C) = cos

Barycentrics a^2 (a^2 + b^2 + c^2 - 2 a b - 2 a c) : :

Let A' be the center of the conic through the contact points of the B- and C- excircles with the sidelines of ABC. Define B' and C' cyclically. The triangle A'B'C' is perspective to the intouch triangle at X(218). See also X(6), X(25), X(222), X(940), X(1743). (Randy Hutson, July 23, 2015)

A'B'C' is also the unary cofactor triangle of the intangents triangle. (Randy Hutson, June 7, 2019)

X(218) lies on these lines:

1,6 3,41 4,294 7,277
32,906 43,170 46,910
56,101 65,169 145,644
198,579 222,241 279,651

X(218) = isogonal conjugate of X(277)

X(218) = eigencenter of cevian triangle of X(7)

X(218) = eigencenter of anticevian triangle of X(55)

X(218) = X(7)-Ceva conjugate of X(55)

X(218) = crosssum of X(650) and X(1086)

X(218) = X(644)-beth conjugate of X(218)

X(218) = crossdifference of every pair of points on the de Longchamps line of the intouch triangle

X(218) = perspector of 2nd mixtilinear triangle and unary cofactor triangle of 5th mixtilinear triangle

X(218) = perspector of excentral triangle and unary cofactor triangle of inverse-in-incircle triangle

Trilinears (sin A)/(1 - sec A) : :

Trilinears 1/(csc A - 2 csc 2A) : :

Trilinears a(b + c - a)(b

Trilinears (b + c - a) cos A : :

Barycentrics sin 2A cot A/2 : :

Let A'B'C' be the extouch triangle. Let A" be the barycentric product of the circumcircle intercepts of line B'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(219). (Randy Hutson, July 31 2018)

X(219) lies on these lines:

1,6 3,48 8,29 10,965
19,517 40,610 41,1036
55,284 56,579 63,77 69,1332
101,102 144,347 200,282
206,692 255,268 278,329
332,345 346,644 572,947
577,906 604,672 1993,3219

X(219) = isogonal conjugate of X(278)

X(219) = isotomic conjugate of X(331)

X(219) = X(i)-Ceva conjugate of X(j) for these (i,j): (8,55), (63,3), (283,212)

X(219) = X(i)-cross conjugate of X(j) for these (i,j): (48,268), (71,9), (212,3)

X(219) = crosspoint of X(i) and X(j) for these (i,j): (8,345), (64,78)

X(219) = crosssum of X(i) and X(j) for these (i,j): (19,34), (56,608)

X(219) = X(i)-beth conjugate of X(j) for these (i,j): (101,478), (219,48), (644,219)

X(219) = trilinear pole of line X(652)X(1946)

X(219) = crossdifference of every pair of points on line X(513)X(1835)

X(219) = X(92)-isoconjugate of X(56)

X(219) = perspector of extouch triangle and unary cofactor triangle of intouch triangle

X(219) = perspector of ABC and unary cofactor triangle of Gemini triangle 37

Trilinears (1 + cos A)

Barycentrics a

The trilinear polar of X(220) passes through X(657) (Randy Hutson, July 20, 2016)

Let A' be the barycentric product of the circumcircle intercepts of the A-excircle. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(220). (Randy Hutson, July 11, 2019)

X(220) lies on these lines:

1,6 3,101 8,294 33,210
40,910 41,55 48,963
63,241 64,71 78,949
144,279 154,205 169,517
200,728 201,221 268,577
277,1086 281,594 329,948
346,1043

X(220) = isogonal conjugate of X(279)

X(220) = X(i)-Ceva conjugate of X(j) for these (i,j): (9,55), (200,480)

X(220) = cevapoint of X(1) and X(170)

X(220) = crosspoint of X(9) and X(200)

X(220) = crosssum of X(57) and X(269)

X(220) = crossdifference of every pair of points on line X(513)X(676)

X(220) = X(i)-beth conjugate of X(j) for these (i,j): (101,221), (220,41), (644,220), (728,728)

X(220) = {X(1),X(9)}-harmonic conjugate of X(1212)

X(220) = perspector of ABC and unary cofactor triangle of inverse-in-incircle triangle

X(220) = barycentric square of X(9)

Trilinears (1 - cos A)(1 + cos A - cos B - cos C) : :

X(221) lies on these lines:

1,84 3,102 6,19 8,651
31,56 40,223 55,64
201,220 204,207 960,1038

X(221) = isogonal conjugate of X(280)

X(221) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,56), (222,6), (223,198)

X(221) = crosspoint of X(i) and X(j) for these (i,j): (1,40), (196,347)

X(221) = crosssum of X(1) and X(84)

X(221) = X(i)-beth conjugate of X(j) for these (i,j): (1,34), (40,40), (101,220), (109,221), (110,3)

X(221) = X(92)-isoconjugate of X(268

) X(221) = perspector of unary cofactor triangles of 1st and 3rd extouch triangles

= 1/(csc A + 2 csc 2A) : 1/(csc B + 2 csc 2B) : 1/(csc A + 2 csc 2C)

= a(b

Barycentrics a^{2}/(1 + sec A) : b^{2}/(1
+ sec B) : c^{2}/(1 + sec C)

Let A' be the center of the conic through the contact points of the incircle and the A-excircle with the sidelines of ABC. Define B' and C' cyclically. The triangle A'B'C' is perspective to the intouch triangle at X(222). See also X(6), X(25), X(218), X(940), X(1743). (Randy Hutson, July 23, 2015)

Let A'B'C' be the intouch triangle. Let A" be the barycentric product of the circumcircle intercepts of line B'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(222). (Randy Hutson, July 31 2018)

X(222) lies on these lines:

1,84 2,651 3,73 6,57
7,27 33,971 34,942
46,227 55,103 56,58
63,77 72,1038 171,611
189,281 218,241 226,478
268,1073 581,1035 601,1066
613,982 912,1060 1355,1363 1993,3218

X(222) = isogonal conjugate of X(281)

X(222) = isotomic conjugate of X(7017)

X(222) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,56), (77,3), (81,57)

X(222) = cevapoint of X(6) and X(221)

X(222) = X(i)-cross conjugate of X(j) for these (i,j): (48,3), (73,77)

X(222) = crosspoint of X(7) and X(348)

X(222) = crosssum of X(i) and X(j) for these (i,j): (55,607), (650,1146)

X(222) = crossdifference of every pair of points on line X(3064)X(3700)

X(222) = trilinear pole of line X(1459)X(1946)

X(222) = Danneels point of X(651) (see notes at X(3078))

X(222) = X(4)-isoconjugate of X(9)

X(222) = intouch-isogonal conjugate of X(12723)

X(222) = perspector of intouch triangle and unary cofactor triangle of extouch triangle

X(222) = perspector of ABC and unary cofactor triangle of Gemini triangle 38

X(222) = vertex conjugate of foci of inconic that is the isotomic conjugate of the polar conjugate of the incircle (centered at X(17073))

X(222) = X(i)-beth conjugate of X(j) for these (i,j): (21,1012), (63,63), (110,222), (287,222), (648,222), (651,222), (662,2), (895,222)

Trilinears sec B + sec C - sec A + 1 : :

Trilinears (a^3 + a^2 b - a b^2 - b^3 + a^2 c - 2 a b c + b^2 c - a c^2 + b c^2 - c^3)/(b + c - a) : :

Let A' be the homothetic center of ABC and the orthic triangle of the A-extouch triangle and define B' and C' cyclically. Triangle A'B'C' is perspective to the 3rd extouch triangle at X(223). (Randy Hutson, September 14, 2016)

Let V_{a}, V_{b}, V_{c} be the antipodes of V=X(40) in the circles (VBC), (VCA), (VAB), respectively. The triangle V_{a}V_{b}V_{c} is here named the **Bevan antipodal triangle** (Hyacinthos #29638; however, see next note). The point X(223) is the unique finite fixed point of the affine transformation that maps the reference triangle ABC onto V_{a}V_{b}V_{c}. (Angel Montesdeoca, October 15, 2019)

The construction of the triangle VaVbVc can be generalized for an arbitrary point P (instead of V=X(40)), and is, in fact, the antipedal triangle of P (Randy Hutson, message in Anopolis Group). Thus, "Bevan antipodal triangle" = antipedal triangle of X(40). (Angel Montesdeoca, October 16, 2019)

X(223) lies on the Thomson cubic and these lines: 4 2,77 3,1035 6,57 9,1073 40,221 55,1456 56,937 63,651 108,204 109,165 312,664 329,347 380,608 580,603 936,1038 1249,3352 3341,3349 3351,3356

X(223) = isogonal conjugate of X(282)

X(223) = complement of X(189)

X(223) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,57), (77,1), (342,208), (347,40)

X(223) = cevapoint of X(198) and X(221)

X(223) = X(i)-cross conjugate of X(j) for these (i,j): (198,40), (227,347)

X(223) = crosspoint of X(2) and X(329)

X(223) = crosssum of X(6) and X(1436)

X(223) = perspector of ABC and antipedal triangle of X(3345)

X(223) = perspector of pedal and anticevian triangles of X(3182)

X(223) = perspector of ABC and medial triangle of pedal triangle of X(1490)

X(223) = perspector of circumconic centered at X(57)

X(223) = center of circumconic that is locus of trilinear poles of lines passing through X(57)

X(223) = X(92)-isoconjugate of X(2188)

X(223) = {X(1),X(1745)}-harmonic conjugate of X(1490)

X(223) = pole wrt polar circle of trilinear polar of X(7020)

X(223) = X(48)-isoconjugate (polar conjugate) of X(7020)

X(223) = trilinear product X(40)*X(57)

X(223) also lies on line 55,1456.
X(223) = X(i)-aleph conjugate of X(j) for these (i,j):

(63,1079), (77,223), (81,580), (174,46), (651,109)

X(223) = X(i)-beth conjugate of X(j) for these (i,j):

(2,278), (100,200), (162,204), (329,329), (651,223), (662,63)

Trilinears (b^2 + c^2 - a^2) (a^4 - b^4 - c^4 - 2 a^3 (b + c) - 2 a^2 b c + 2 a (b^3 + c^3) + 2 b^2 c^2) : :

X(224) lies on these lines: 1,377 3,63 8,914 21,90 46,100 65,1004 908,1079

X(224) = X(7)-Ceva conjugate of X(63)

X(224) = barycentric product of vertices of Gemini triangle 7

X(224) = barycentric product of vertices of Gemini triangle 8

Barycentrics (tan A)(cos B + cos C) : (tan B)(cos C + cos A) : (tan C)(cos A + cos B)

Barycentrics (b + c)/((b^2 + c^2 - a^2) (b + c - a)) : :

X(225) lies on these lines:

1,4 3,1074 7,969 10,201
12,37 19,208 28,108
46,254 65,407 75,264
91,847 158,1093 377,1038
412,775 653,897

X(225) = isogonal conjugate of X(283)

X(225) = isotomic conjugate of X(332)

X(225) = X(4)-Ceva conjugate of X(65)

X(225) = X(407)-cross conjugate of X(4)

X(225) = crosspoint of X(i) and X(j) for these (i,j): (4,158), (273,278)

X(225) = crosssum of X(i) and X(j) for these (i,j): (3,255), (212,219)

X(225) = X(i)-beth conjugate of X(j) for these (i,j): (4,225), (10,227), (108,1042), (318,10)

X(225) = pole wrt polar circle of trilinear polar of X(333) (line X(522)X(663))

X(225) = polar conjugate of X(333)

Trilinears bc(b + c)/(b + c - a) : ca(c + a)/(c + a - b) : ab(a + b)/(a + b - c)

Trilinears cos(angle BIC) : cos(angle CIA) : cos(angle AIB)

Barycentrics (b + c)/(b + c - a) : (c + a)/(c + a - b) : (a + b)/(a + b - c)

Barycentrics area(A'BC) : : , where A'B'C' is the 2nd extouch triangle

Barycentrics r

X(226) is the homothetic center of the intouch triangle and the triangle formed by the lines of the external pairs of extouch points of the excircles. (Randy Hutson, 9/23/2011)

Let A' be the radical center of the incircle and the B- and C-excircles; define B' and C' cyclically. A'B'C' is also the complement of the excentral triangle, and the triangle formed by the radical axes of the incircle and each excircle. Then X(226) is the homothetic center of A'B'C' and the intouch triangle. (Randy Hutson, December 26, 2015)

Let A' be the homothetic center of the orthic triangles of the intouch and A-extouch triangles, and define B' and C' cyclically. Let AaBaCa be the orthic triangle of the A-extouch triangle, and define AbBbCb, and AcBcCc cyclically. Let A" be the centroid of AaAbAc, and define B" and C" cyclically. Then A'B'C' and A"B"C" are homothetic to each other and to the medial triangle and the orthic triangle of the intouch triangle at X(226). (Randy Hutson, December 26, 2015)

Let (A') be the pedal circle of the A-vertex of the hexyl triangle, and define (B') and (C') cyclically. Then X(226) is the radical center of circles (A'), (B'), (C'). (Randy Hutson, December 26, 2015)

Let IaIbIc be the reflection triangle of X(1). Let A' be the trilinear pole of line IbIc, and define B', C' cyclically. The lines AA', BB', CC' concur in X(226). (Randy Hutson, July 20, 2016)

Let A_{13}B_{13}C_{13} be Gemini triangle 13. Let A' be the perspector of conic {{A,B,C,B_{13},C_{13}}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(226). (Randy Hutson, January 15, 2019)

Let A_{14}B_{14}C_{14} be Gemini triangle 14. Let A' be the center of conic {{A,B,C,B_{14},C_{14}}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(226). (Randy Hutson, January 15, 2019)

X(226) lies on these lines:

1,4 2,7 5,912 10,12
11,118 13,1082 14,554
27,284 29,951 35,79
36,1006 37,440 41,379
46,498 55,516 56,405
76,85 78,377 81,651
83,1429 86,1412 92,342
98,109 102,1065 175,1131
176,1132 196,281 208,406
222,478 228,851 262,982
273,469 306,321 429,1426
443,936 452,1420 474,1466
481,485 482,486 495,517
535,551 664,671 673,1174
748,1471 857,1446 975,1038
990,1040 1029,1442 1260,1376
1284,1402 1401,1463

X(226) = reflection of X(993) in X(1125)

X(226) = isogonal conjugate of X(284)

X(226) = isotomic conjugate of X(333)

X(226) = complement of X(63)

X(226) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,65), (349,307)

X(226) = cevapoint of X(37) and X(65)

X(226) = X(i)-cross conjugate of X(j) for these (i,j): (37,10), (73,307)

X(226) = crosspoint of X(2) and X(92)

X(226) = crosssum of X(i) and X(j) for these (i,j): (6,48), (41,55)

X(226) = crossdifference of every pair of points on line X(652)X(663)

X(226) = X(63)-of-medial-triangle

X(226) = bicentric sum of PU(20)

X(226) = midpoint of PU(20)

X(226) = trilinear pole of line X(523)X(656) (the polar of X(29) wrt polar circle)

X(226) = pole wrt polar circle of trilinear polar of X(29) (the line X(243)X(522))

X(226) = X(48)-isoconjugate (polar conjugate) of X(29)

X(226) = X(6)-isoconjugate of X(21)

X(226) = X(184)-of-2nd-extouch-triangle

X(226) = {X(2),X(57)}-harmonic conjugate of X(3911)

X(226) = {X(9),X(57)}-harmonic conjugate of X(1708)

X(226) = homothetic center of intouch triangle and the complement of excentral triangle)

X(226) = homothetic center of 3rd Euler tringle and inverse-in-incircle triangle

X(226) = perspector of intouch triangle and Gergonne line extraversion triangle

X(226) = perspector of 2nd extouch triangle and Gergonne line extraversion triangle

X(226) = X(i)-beth conjugate of X(j) for these (i,j): (2,226), (21,1064), (100,42), (190,226), (312,306), (321,321), (335,226), (835,226)

X(226) = barycentric product X(109)*X(850)

X(226) = perspector of Gemini triangle 9 and cross-triangle of ABC and Gemini triangle 9

X(226) = trilinear pole of perspectrix of ABC and Gemini triangle 10

X(226) = perspector of Gemini triangle 40 and cross-triangle of ABC and Gemini triangle 40

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(227) lies on these lines:

12,37 34,55 40,221
42,65 46,222 56,197
198,208 201,210 322,347 607,910

X(227) = isogonal conjugate of X(285)

X(227) = X(10)-Ceva conjugate of X(65)

X(227) = crosspoint of X(223) and X(347)

X(227) = crosssum of X(84) and X(1433)

X(227) = X(i)-beth conjugate of X(j) for these (i,j): (10,225),
(40,227), (100,72)

Barycentrics (sin A sin 2A)(sin B + sin C) : (sin B sin 2B)(sin C + sin A) : (sin C sin 2C)(sin A + sin B)

X(228) lies on these lines:

3,63 9,1011 12,407
19,25 28,943 31,32
35,846 42,181 48,184
73,408 98,100 226,851

X(228) = isogonal conjugate of X(286)

X(228) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,71), (37,213), (55,42)

X(228) = crosspoint of X(i) and X(j) for these (i,j): (3,48), (37,72), (55,212), (71,73)

X(228) = crosssum of X(i) and X(j) for these (i,j): (4,92), (7,273), (27,29), (28,81)

X(228) = crossdifference of every pair of points on line X(693)X(905)

X(228) = X(212)-beth conjugate of X(228)

X(228) = X(28)-isoconjugate of X(75)

X(228) = X(81)-isoconjugate of X(92)

X(228) = trilinear pole of line X(810)X(3049)

X(228) = perspector of unary cofactor triangles of Gemini triangles 1 and 2

u = u(a,b,c) = a(b + c - a)/(b + c), v = u(b,c,a), w = u(c,a,b)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(229) lies on these lines: 1,267 21,36 28,60 58,244 65,110 593,1104

X(229) = midpoint of X(1) and X(267)

X(229) = X(7)-Ceva conjugate of X(81)

X(230) = X(13) + X(14) + X(15) + X(16)

X(230) is the midpoint of the centers of the (equilateral) pedal triangles of X(15) and X(16).

X(230) = QA-P6 (Parabola Axes Crosspoint) of quadrangle ABCX(2); see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/27-qa-p6.html

X(230) lies on these lines:

2,6 5,32 12,172 25,53
30,115 39,140 50,858
111,476 112,403 231,232
393,459 427,571 538,620
549,574 625,754 2482,5215

X(230) = midpoint of X(i) and X(j) for these (i,j): (115,187), (325,385), (395,396)

X(230) = isogonal conjugate of X(2987)

X(230) = complement of X(325)

X(230) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,114), (297,1503)

X(230) = crosspoint of X(2) and X(98)

X(230) = crosssum of X(6) and X(511)

X(230) = crossdifference of every pair of points on line X(3)X(512)

X(230) = X(2)-Hirst inverse of X(193)

X(230) = X(i)-beth conjugate of X(j) for these (i,j): (281,230), (645,230)

X(230) = centroid of quadrangle X(13)X(14)X(15)X(16)

X(230) = radical center of cirumcircle, nine-point circle and Lester circle

X(230) = radical center of cirumcircle, nine-point circle and Hutson-Parry circle

X(230) = perspector of circumconic centered at X(114)

X(230) = center of circumconic that is locus of trilinear poles of lines passing through X(114)

X(230) = inverse-in-Steiner-inellipse of X(6)

X(230) = isotomic conjugate of X(8781)

X(230) = X(910)-of-orthic-triangle if ABC is acute

X(230) = PU(4)-harmonic conjugate of X(2501)

Trilinears (cos B) (4 sin^2 C - 1) (cos A sin B - sin A cos B) - (cos C) (4 sin^2 B - 1) (cos C sin A - sin C cos A) : :

X(231) lies on these lines: 4,96 6,17 50,115 230,232

X(231) = complement of X(1273)

X(231) = X(2)-Ceva conjugate of X(128)

X(231) = crosssum of X(6) and X(1154)

X(231) = crossdifference of every pair of points on line X(3)X(1510)

X(231) = X(281)-beth conjugate of X(230)

X(231) = perspector of circumconic centered at X(128)

X(231) = center of circumconic that is locus of trilinear poles of lines passing through X(128)

X(231) = {X(17),X(18)}-harmonic conjugate of X(1209)

Barycentrics sin A tan A cos(A + ω) : sin B tan B cos(B + ω) : sin C tan C cos(C + ω)

X(232) lies on these lines:

{2,216}, {3,1968}, {4,39}, {6,25}, {19,444}, {22,577}, {23,250},
{24,32}, {33,2276}, {34,2275}, {50,3447}, {53,427}, {111,1304},
{112,186}, {115,403}, {132,1513}, {217,389}, {230,231}, {235,5254},
{297,325}, {378,574}, {385,648}, {406,5283}, {420,3229}, {428,5421},
{511,2211}, {566,5094}, {571,1485}, {800,1196}, {1015,1870},
{1033,1611}, {1172,2092}, {1180,3087}, {1235,3934}, {1506,1594},
{1560,3258}, {1575,1861}, {1593,5013}, {1597,5024}, {1609,3162},
{1692,2065}, {1783,5291}, {1995,5158}, {2971,5140}, {3053,3172},
{3089,5286}, {3269,3331}, {3518,5007}, {3542,3767}, {4220,5317},
{4232,5304}

X(232) = midpoint of X(3269) and X(3331)

X(232) = isogonal conjugate of X(287)

X(232) = complement of X(30737)

X(232) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,132), (297,511)

X(232) = X(237)-cross conjugate of X(511)

X(232) = crosssum of X(2) and X(401)

X(232) = crossdifference of every pair of points on line X(3)X(525)

X(232) = orthojoin of X(132)

X(232) = X(6)-Hirst inverse of X(25)

X(232) = X(281)-beth conjugate of X(232)

X(232) = perspector of hyperbola {{A,B,C,X(4),X(112),PU(39)}} (centered at X(132))

X(232) = center of circumconic that is locus of trilinear poles of lines passing through X(132)

X(232) = intersection of trilinear polars of X(112), P(39), and U(39)

X(232) = crossdifference of PU(37)

X(232) = PU(4)-harmonic conjugate of X(647)

X(232) = pole wrt polar circle of trilinear polar of X(290) (line X(2)X(647))

X(232) = X(48)-isoconjugate (polar conjugate) of X(290)

X(232) = inverse-in-Moses-radical-circle of X(468)

X(232) = pole of Euler line wrt Moses radical circle

X(232) = polar-circle-inverse of X(34235)

Trilinears b sec(A - B) + c sec(A - C) : :

Barycentrics (a^2 (b^2 + c^2) - (b^2 - c^2)^2) (2 a^4 - 3 a^2 (b^2 + c^2) + (b^2 - c^2)^2) : :

X(233) lies on hyperbola {{X(2),X(6),X(216),X(233),X(1249),X(1560),X(3162)}}. This hyperbola is a circumconic of the medial triangle, and the locus of perspectors of circumconics centered at a point on the Euler line. It is tangent to Euler line at X(2). (Randy Hutson, March 21, 2019)

X(233) lies on these lines: 2,95 5,53 6,17 115,128 122,138

X(233) = isogonal conjugate of X(288)

X(233) = isotomic conjugate of X(31617)

X(233) = perspector of circumconic centered at X(140)

X(233) = complement of X(95)

X(233) = X(2)-Ceva conjugate of X(140)

X(233) = crosspoint of X(2) and X(5)

X(233) = crosssum of X(6) and X(54)

X(233) = crossdifference of every pair of points on line X(1157)X(1510)

X(233) = polar conjugate of the isogonal conjugate of X(32078)

X(233) = center of circumconic that is locus of trilinear poles of lines passing through X(140)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(234) lies on these lines: 2,178 7,174 57,362 75,556 555,1088

X(234) = X(7)-Ceva conjugate of X(177)

X(234) = X(31)-of-intouch-triangle

Trilinears (sec A)(cos

Barycentrics (tan A)(cos

Barycentrics (a^4 (b^2 + c^2) - 2 a^2 (b^2 - c^2)^2 + (b^2 - c^2)^2 (b^2 + c^2))/(a^2 - b^2 - c^2) : :

As a point on the Euler line, X(235) has Shinagawa coefficients (F, F - E).

X(235) lies on these lines: 2,3 11,34 12,33 52,113 133,136

X(235) = midpoint of X(4) and X(24)

X(235) = complement of X(11413)

X(235) = anticomplement of X(16196)

X(235) = excentral-to-ABC functional image of X(55)

X(235) = X(4)-Ceva conjugate of X(185)

X(235) = crosssum of X(3) and X(1092)

X(235) = orthic-isogonal conjugate of X(185)

X(235) = X(56) of orthic triangle if ABC is acute

X(235) = insimilicenter of nine-point circle and incircle of orthic triangle if ABC is acute; the exsimilicenter is X(427)

X(235) = pole wrt polar circle of trilinear polar of X(801) (line X(523)X(2071))

X(235) = X(48)-isoconjugate (polar conjugate) of X(801)

X(235) = perspector of ABC and cross-triangle of ABC and 2nd Hyacinth triangle

X(235) = radical center of the polar-circle-inverses of the power circles

X(235) = intersection of tangents to Hatzipolakis-Lozada hyperbola at X(4) and X(155)

X(235) = crosspoint, wrt orthic triangle, of X(4) and X(155)

Trilinears [1 + sin(A/2)]/sin A : [1 + sin(B/2)]/sin B : [1 + sin(C/2)]//sin C (M. Iliev, 4/12/07)

X(236) lies on these lines: 2,174 8,178 9,173

X(236) = isogonal conjugate of X(289)

X(236) = complement of X(7048)

X(236) = X(2)-Ceva conjugate of X(188)

X(236) = perspector of circumconic centered at X(188)

X(236) = center of circumconic that is locus of trilinear poles of lines passing through X(188)

are line conjugates. The P-line conjugate of Q is the point

where line PQ meets the trilinear polar of the isogonal conjugate of Q.

= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a

Barycentrics a^{3}cos(A + ω) : b^{3}cos(B + ω) : c^{3}cos(C + ω)

As a point on the Euler line, X(237) has Shinagawa coefficients (EF + F^{2} + S^{2}, -(E + F)^{2} - S^{2}).

X(237) is the point of intersection of the Euler line and the Lemoine axis (defined as the radical axis of the circumcircle and the Brocard circle).

If you have The Geometer's Sketchpad, you can view X(1316), which includes X(237).

X(237) lies on these lines: 2,3 6,160 31,904 32,184 39,51 154,682 187,351 206,571

X(237) is the {X(1113),X(1114)}-harmonic conjugate of X(1316). For a
list of other harmonic conjugates of X(237), click **Tables** at the
top of this page.]

X(237) = isogonal conjugate of X(290)

X(237) = isotomic conjugate of X(18024)

X(237) = complement X(14957)

X(237) = anticomplement of X(21531)

X(237) = X(98)-Ceva conjugate of X(6)

X(237) = crosspoint of X(i) and X(j) for these (i,j): (6,98), (232,511)

X(237) = crosssum of X(i) and X(j) for these (i,j): (2,511), (98,287)

X(237) = crossdifference of every pair of points on line X(2)X(647)

X(237) = X(32)-Hirst inverse of X(184)

X(237) = X(3)-line conjugate of X(2)

X(237) = X(55)-beth conjugate of X(237)

X(237) = crosspoint of X(3) and X(3511) wrt excentral triangle

X(237) = crosspoint of X(3) and X(3511) wrt tangential triangle

X(237) = X(92)-isoconjugate of X(287)

X(237) = trilinear pole of PU(89)

Barycentrics a

X(238) is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(8) and U(8) of bicentric points (see the notes just before X(1908). (Randy Hutson, 9/23/2011)

X(238) lies on these lines:

1,6 2,31 3,978 4,602
7,1471 8,983 10,82
21,256 36,513 40,1722
42,1621 43,55 47,499
56,87 57,1707 58,86
63,614 71,1244 100,899
105,291 106,898 162,415
190,726 212,497 239,740
241,1456 242,419 244,896
390,1253 459,1395 484,1739
516,673 517,1052 519,765
580,946 601,631 651,1458
662,1326 942,1046 987,1472
992,1009 993,995 1006,1064
1040,1711 1054,1155 1284,1428
1465,1758 1479,1714 1699,1754

X(238) = midpoint of X(1) and X(1279)

X(238) = reflection of X(1) in X(1297)

X(238) = isogonal conjugate of X(291)

X(238) = isotomic conjugate of X(334)

X(238) = anticomplement of X(3836)

X(238) = X(i)-Ceva conjugate of X(j) for these (i,j): (105,1), (292,171)

X(238) = X(659)-cross conjugate of X(3573)

X(238) = crosssum of X(i) and X(j) for these (i,j): (10,726), (42,672), (239,894)

X(238) = crossdifference of every pair of points on line X(37)X(513)

X(238) = X(i)-Hirst inverse of X(j) for these (i,j): (1,6), (43,55)

X(238) = X(1)-line conjugate of X(37)

X(238) = X(105)-aleph conjugate of X(238)

X(238) = X(i)-beth conjugate of X(j) for these (i,j): (21,238), (643,902), (644,238), (932,238)

X(238) = {X(1),X(9)}-harmonic conjugate of X(984)

X(238) = intersection of trilinear polars of PU(8)

X(238) = inverse-in-circumconic-centered-at-X(9) of X(6)

X(238) = crossdifference of PU(i) for these i: 6, 52, 53

X(238) = trilinear product of PU(134)

X(238) = X(6530)-of-excentral-triangle

X(238) = trilinear pole of line X(659)X(4435) (the perspectrix of ABC and Gemini triangle 33)

X(238) = perspector of Gemini triangle 34 and cross-triangle of ABC and Gemini triangle 34

Barycentrics a

X(239) is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(6) and U(6) of bicentric points (see the notes just before X(1908). (Randy Hutson, 9/23/2011)

X(239) is the point of intersection of the following lines:

X(1)X(2) = trilinear polar of X(190)

trilinear polar of cevapoint{X(1), X(2)}, which is X(239)X(514)

UV, where U = X(1)-Ceva-conjugate-of-(2) = X(192), and V = X(2)-Ceva-conjugate-of-X(1) = X(9)

(Randy Hutson, December 26, 2015)

X(239) lies on these lines:

1,2 6,75 7,193 9,192
44,190 57,330 63,194
81,274 83,213 86,1100
92,607 141,319 238,740
241,664 257,333 294,666
318,458 320,524 335,518
514,649 1043,1104

X(239) = reflection of X(i) in X(j) for these (i,j): (190,44), (320,1086)

X(239) = isogonal conjugate of X(292)

X(239) = isotomic conjugate of X(335)

X(239) = complement of X(6542)

X(239) = anticomplement of X(3912)

X(239) = crosspoint of X(256) and X(291)

X(239) = crosssum of X(i) and X(j) for these (i,j): (3,255), (212,219)

X(239) = crossdifference of every pair of points on line X(42)X(649)

X(239) = X(i)-Hirst inverse of X(j) for these (i,j): (171,238), (665,1015)

X(239) = X(1)-line conjugate of X(42)

X(239) = X(i)-beth conjugate of X(j) for these (i,j): (333,239), (645,44)

X(239) = perspector of conic {A,B,C,X(86),X(190)}

X(239) = inverse-in-Steiner-circumellipse of X(1)

X(239) = trilinear pole of line X(659)X(812)

X(239) = crossdifference of PU(8)

X(239) = intersection of trilinear polars of PU(6) (the 1st and 2nd bicentrics of the Lemoine axis)

X(239) = X(2)-Ceva conjugate of X(6651)

X(239) = trilinear pole of PU(134)

X(239) = barycentric square root of X(4366)

Barycentrics tan A cos(A + ω) : tan B cos(B + ω) : tan C cos(C + ω)

X(240) lies on these lines: 1,19 4,256 38,92 63,1096 75,158 162,896 278,982 281,984 522,656 607,611 608,613

X(240) = isogonal conjugate of X(293)

X(240) = isotomic conjugate of X(336)

X(240) = crossdifference of every pair of points on line X(48)X(656)

X(240) = X(1)-Hirst inverse of X(19)

X(240) = X(1)-line conjugate of X(48)

X(240) = X(318)-beth conjugate of X(240)

X(240) = crossdifference of PU(22)

X(240) = perspector of hyperbola {A,B,C,PU(23)}

X(240) = intersection of trilinear polars of P(23) and U(23)

X(240) = pole wrt polar circle of trilinear polar of X(1821)

X(240) = X(48)-isoconjugate (polar conjugate) of X(1821)

Trilinears (b^2 + c^2 - a b - a c)/(a - b - c) : :

Barycentrics (b^2 + c^2)(1 - cos A) - a^2(cos B + cos C) : :

X(241) lies on these lines: 1,3 2,85 6,77 7,37 9,269 44,651 63,220 141,307 218,222 239,664 277,278 294,910 347,1108 514,650 960,1042

X(241) = isogonal conjugate of X(294)

X(241) = crosssum of X(i) and X(j) for these (i,j): (6,910), (518,1376

X(241) = crossdifference of every pair of points on line X(55)X(650)

X(241) = X(1)-Hirst inverse of X(57)

X(241) = X(1)-line conjugate of X(55)

X(241) = X(i)-beth conjugate of X(j) for these (i,j): (2,241), (100,241), (1025,241), (1026,241)

X(241) = trilinear pole of line X(926)X(1362)

X(241) = X(237)-of-intouch-triangle

X(241) = perspector of hyperbola {A,B,C,PU(46)}

X(241) = crossdifference of PU(112)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(242) lies on these lines: 4,9 25,92 28,261 29,257 34,87 162,422 238,419 278,459 915,929

X(242) = isogonal conjugate of X(295)

X(242) = isotomic conjugate of X(337)

X(242) = crossdifference of every pair of points on line X(71)X(1459)

X(242) = X(4)-Hirst inverse of X(19)

X(242) = X(4)-line conjugate of X(71)

X(242) = inverse-in-polar-circle of X(10)

X(242) = pole wrt polar circle of the line X(10)X(514)

X(242) = X(48)-isoconjugate (polar conjugate) of X(335)

X(243) is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(15) and U(15) of bicentric points (see the notes just before X(1908). (Randy Hutson, 9/23/2011)

X(243) is the point of intersection of the following lines:

trilinear polars of P(15) and U(15)

X(1)X(4)

trilinear polar of cevapoint{X(1),X(4)}

UV, where U = X(1)-Ceva-conjugate-of-(4) = X(1148), and V = X(4)-Ceva-conjugate-of-X(1) = X(46)

(Randy Hutson, December 26, 2015)

X(243) lies on these lines: 1,4 3,158 55,92 65,412 318,958 411,821 425,662 522,652 920,1075 1040,1096

X(243) = isogonal conjugate of X(296)

X(243) = crossdifference of every pair of points on line X(73)X(652)

X(243) = X(i)-Hirst inverse of X(j) for these (i,j): (1,4), (46,1148)

X(243) = X(1)-line conjugate of X(73)

X(243) = perspector of conic {A,B,C,X(29),X(653),PU(15)}

X(243) = crossdifference of PU(16)

X(243) = pole wrt polar circle of the line X(226)X(522)

X(243) = X(48)-isoconjugate (polar conjugate) of X(1952)

Trilinears [1 - cos(B - C)]sin

Trilinears d(a,b,c) : : , where d(a,b,c) = (distance from A to Nagel line)

Barycentrics a(b - c)

Let O* be a circle with center X(3) and variable radius R*. Let La be the radical axis of O* and the A-excircle, and define Lb and Lc cyclically. Let A'=Lb∩Lc, B'=Lc∩La, C'=La∩Lb. Then A'B'C' is perspective to ABC, and the locus of the perspector as R* varies is the hyperbola {{A,B,C,X(1),X(10)}}, which has center X(244). Also, X(244) lies in the inellipse centered at X(10), as well as the Hofstadter ellpse E(1/2), which is the incentral inellipse. (Randy Hutson, Decembe 26, 2016)

X(244) lies on aforementioned ellipses and these lines: 1,88 2,38 11,867 31,57 34,1106 42,354 58,229 63,748 238,896 474,976 518,899 596,1089 665,866

X(244) = isogonal conjugate of X(765)

X(244) = isotomic conjugate of X(7035)

X(244) = anticomplement of X(24003)

X(244) = perspector of anti-Aquila and 2nd Sharygin triangles

X(244) = barycentric product of vertices of Garcia reflection triangle

X(244) = trilinear pole of line X(764)X(2087)

X(244) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,513), (75,514)

X(244) = crosspoint of X(1) and X(513)

X(244) = crosssum of X(i) and X(j) for these (i,j): (1,100), (31,101), (78,1331), (109,1420), (200,644), (651,1445), (678,1023), (756,1018)

X(244) = crossdifference of every pair of points on line X(100)X(101)

X(244) = X(1)-Hirst inverse of X(1054)

X(244) = X(1)-line conjugate of X(100)

X(244) = complement of X(3952)

X(244) = antipode of X(4738) in inellipse centered at X(10)

X(244) = reflection of X(4738) in X(10)

X(244) = bicentric difference of PU(34)

X(244) = PU(34)-harmonic conjugate of X(1635)

X(244) = tripolar centroid of X(1022)

X(244) = perspector of circumconic centered at X(661)

X(244) = center of circumconic that is locus of trilinear poles of lines passing through X(661)

X(244) = X(2)-Ceva conjugate of X(661)

X(244) = trilinear pole wrt incentral triangle of line X(1)X(6)

X(244) = intersection of tangents to Steiner inellipse at X(1015) and X(1086)

X(244) = crosspoint wrt medial triangle of X(1015) and X(1086)

X(244) = trilinear square of X(513)

f(A,B,C) = csc

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(245) lies on these lines: 1,60 115,125

X(245) = X(1)-line conjugate of X(110)

f(A,B,C) = csc(B-A)[cos A csc(B - A) + cos C csc(B - C)] + csc(C - A) u(A,B,C),

u(A,B,C) = [cos A csc(C - A) + cos B csc(C - B)]

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(246) lies on these lines: 3,74 115,125

X(246) = X(3)-line conjugate of X(110)

f(A,B,C) = csc(B-A)[sec A csc(B - A) + sec C csc(B - C)] + csc(C - A) u(A,B,C),

u(A,B,C) = [sec A csc(C - A) + sec B csc(C - B)]

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(247) lies on these lines: 4,110 115,125

X(247) = crossdifference of every pair of points on line
X(110)X(686)

X(247) = X(4)-line conjugate of X(110)

Barycentrics sin A sin 2A sec(A + ω) : sin B sin 2B sec(B + ω) : sin C sin 2C sec(C + ω)

Barycentrics a^2 (a^2 - b^2 - c^2)/(b^4 + c^4 - a^2 b^2 - a^2 c^2) : :

X(248) lies on these lines:

4,32 6,157 39,54 50,67
65,172 66,571 69,287
72,293 74,187 290,385 682,695

X(248) = isogonal conjugate of X(297)

X(248) = crosspoint of X(98) and X(287)

X(248) = crosssum of X(232) and X(511)

X(248) = crossdifference of every pair of points on line X(114)X(132)

X(248) = X(4)-line conjugate of X(132)

X(248) = trilinear pole of line X(184)X(647)

X(248) = crossdifference of every pair of points on line X(114)X(132)

X(248) = X(92)-isoconjugate of X(511)

X(248) = barycentric product X(3)*X(98)

X(248) = barycentric product of circumcircle intercepts of line X(3)X(525)

are isogonal conjugates of previously listed centers.

= a/(b

Barycentrics csc^{2}(B - C) : csc^{2}(C -
A) : csc^{2}(A - B)

X(249) is the vertex conjugate of the foci of the inellipse that is the barycentric square of line X(2)X(6). The center of this inellipse is X(620), and its perspector is X(4590). (Randy Hutson, October 15, 2018)

X(249) is the trilinear pole of line X(110)X(351), which is the tangent to the circumcircle at X(110), and the locus of trilinear poles of tangents at P to hyperbola {{A,B,C,X(6),P}}, as P moves on the Brocard axis. (Randy Hutson, October 15, 2018)

X(249) lies on these lines: 99,525 110,512 186,250 187,323 297,316 648,687 805,827 849,1110

X(249) = isogonal conjugate of X(115)

X(249) = isotomic conjugate of X(338)

X(249) = cevapoint of X(i) and X(j) for these (i,j): (6,110), (24,112), (1379,1380)

X(249) = X(i)-cross conjugate of X(j) for these (i,j): (3,99), (6,110)

X(249) = perspector of ABC and reflection of symmedial triangle in the Brocard axis

X(249) = polar conjugate of X(2970)

X(249) = X(92)-isoconjugate of X(125)

X(249) = crossdifference of every pair of points on line X(1648)X(8029)

= (a

Barycentrics (tan A)csc^{2}(B - C) : (tan
B)csc^{2}(C - A) : (tan C)csc^{2}(A - B)

X(250) is the trilinear pole of line X(110)X(112), which is the tangent to the MacBeath circumconic at X(110), and the locus of trilinear poles of tangents at P to conic {{A,B,C,X(3),P}}, as P moves on the Brocard axis. (Randy Hutson, March 21, 2019)

X(250) lies on these lines: 23,232 107,687 110,520 112,691 186,249 325,340 476,933 523,648 827,935

X(250) = isogonal conjugate of X(125)

X(250) = isotomic conjugate of X(339)

X(250) = cevapoint of X(i) and X(j) for these (i,j): (3,110), (25,112), (162,270), (1113,1114)

X(250) = X(i)-cross conjugate of X(j) for these (i,j): (3,110), (22,99), (24,107), (25,112), (199,101)

X(250) = polar conjugate of X(338)

X(250) = circumcenter of reflection triangle of X(125)

X(250) = barycentric product X(99)*X(112)

X(250) = cevapoint of MacBeath circumconic intercepts of Brocard axis

= a/(b

Barycentrics a^{3}csc(A + ω) :
b^{3}csc(B + ω) : c^{3}csc(C + ω)

Let K be the symmedian point of ABC and let A' be the symmedian point of the triangle BCK; define B' and C' cyclically. The lines AA', BB', CC' concur in X(251). (Randy Hutson, 9/23/2011)

Let A5'B5'C5' be the 5th anti-Brocard triangle. The radical center of the circumcircles of BCA5', CAB5', ABC5' is X(251). (Randy Hutson, July 20, 2016)

X(251) lies on these lines: 2,32 6,22 37,82 110,694 112,427 184,263 308,385 609,614 689,699

X(251) = isogonal conjugate of X(141)

X(251) = complement of X(1369)

X(251) = cevapoint of X(6) and X(32)

X(251) = X(i)-cross conjugate of X(j) for these (i,j): (6,83), (23,111), (523,112)

X(251) = isotomic conjugate of X(8024)

X(251) = similitude center of ABC and 1st orthosymmedial triangle

X(251) = pole wrt polar circle of trilinear polar of X(1235)

X(251) = X(48)-isoconjugate (polar conjugate) of X(1235)

X(251) = barycentric product of vertices of circummedial triangle

X(251) = perspector of ABC and cross-triangle of ABC and circummedial triangle

X(251) = Kosnita(X(6),X(6)) point

X(251) = homothetic center of 1st orthosymmedial and 1st anti-orthosymmedial triangles

f(A,B,C) = sec(B - C)/[1 - 2 cos(2A)]

Trilinears h(A,B,C) : h(B,C,A) : h(C,A,B), where

h(A,B,C) = cos A sec(3A) sec(B - C) (Manol Iliev, 4/01/07)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C) f(C,A,B) Let A'B'C' be the orthic triangle. Let O

Let H_{A} be the reflection of X(4) in the Euler line of BCX(4), and define H_{B} and H_{C} cyclically. The lines AH_{A}, BH_{B}, CH_{C} concur in X(252). (Randy Hutson, June 7, 2019)

X(252) lies on these lines: 3,930 54,140 93,186

X(252) = isogonal conjugate of X(143)

X(252) = anticomplement of X(31376)

X(252) = X(5)-isoconjugate of X(2964)