ENCYCLOPEDIA OF TRIANGLE CENTERS

Long ago, someone drew a triangle and three segments across it. Each segment started at a vertex and stopped at the midpoint of the opposite side. The segments met in a point. The person was impressed and repeated the experiment on a different shape of triangle. Again the segments met in a point. The person drew yet a third triangle, very carefully, with the same result. He told his friends. To their surprise and delight, the coincidence worked for them, too. Word spread, and the magic of the three segments was regarded as the work of a higher power.

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).

HOW TO USE ETC

You won't have to scroll down very far to find well known centers. Other named centers can be found using your computer's searcher - for example, search for "Apollonius" to find "Apollonius point" as X(181).

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).

NOTATION AND COORDINATES

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.

Introduced on December 28, 2016: Index of Triangles Referenced in ETC

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.

Introduced on March 21, 2015: Shinagawa coefficients for triangle centers on the Euler line

Suppose that X is a triangle center given by barycentric coordinates f(a,b,c) : f(b,c,a) : f(c,a,b). The Shinagawa coefficients of X are the functions G(a,b,c) and H(a,b,c) such that

f(a,b,c) = G(a,b,c)*S² + H(a,b,c)*S_BS_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)/S², so that E = (abc/S)² = 4R²
F = S_AS_BS_C/S², so that F = (a² + b² + c²)/2 - 4R² = S_ω - 4R²

Examples:
X(2) has Shinagawa coefficients (1, 0); i.e., X(2) = 1*S² + 0*S_BS_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, $aS_BS_C$ abbreviates aS_BS_C + bS_CS_A + cS_AS_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 = (4R² - 36|GO|²)/8 and F = R²( 1 - J²)/2, where J = |OH|/R.

Introduced on November 1, 2011: Combos

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 = (b² + c² - a²)/2 = bc cos A
S_B = (c² + a² - b²)/2 = ca cos B
S_C = (a² + b² - c²)/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) = (a² + b² + c²)/(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).

X(1) = INCENTER

Trilinears    1 : 1 : 1
Barycentrics   a : b : c
Barycentrics  sin A : sin B : sin C
X(1) = 3R*X(2) + r*X(3) + s*cot(ω)*X(6)
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 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 607,949 631,1000 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)
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) = inverse-in-circumcircle of X(36)
X(1) = inverse-in-Fuhrmann-circle of X(80)
X(1) = inverse-in-Bevan-circle 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(2) = CENTROID

Trilinears    1/a : 1/b : 1/c
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)
Barycentrics  1 : 1 : 1
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.

The Parry isodynamic circle is here introduced as the circle centered at X(2502) that passes through the isodynamic points, X(15) and X(16). This circle is orthogonal to both the circumcircle and Parry circle. (Randy Hutson, February 10, 2016)

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 homothetic to the midheight triangle at X(2). (Randy Hutson, January 29, 2018)

Let A'B'C' be the excentral triangle. Let Oa be the A'-McCay circle of triangle A'BC, and define Ob, Oc cyclically. X(2) is the radical center of Oa, Ob, Oc. (Randy Hutson, June 27, 2018)

X(2) lies on the Parry circle, Thomson cubic, and these lines:
1,8 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) = inverse-in-circumcircle of X(23)
X(2) = inverse-in-nine-point-circle of X(858)
X(2) = inverse-in-Brocard-circle 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) = 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), (32,206), (39,141), (44,214), (57,223), (114,230), (115,523), (128,231), (132,232), (140,233), (188,236)

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(3) = CIRCUMCENTER

Trilinears    cos A : cos B : cos C
Trilinears    a(b² + c² - a²) : b(c² + a² - b²) : c(a² + b² - c²)
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 : :

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) = {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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X(4) = ORTHOCENTER

Trilinears    sec A : sec B : sec C
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
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

Barycentrics tan A : tan B : tan C
Barycentrics 1/(b² + c² - a²) : 1/(c² + a² - b²) : 1/(a² + b² - c²)
X(4) = (1 + J) X(1113) + (1 - J) X(1114)

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:
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{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) = inverse-in-circumcircle of X(186)
X(4) = inverse-in-nine-point-circle 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(5) = NINE-POINT CENTER

Trilinears    cos(B - C) : cos(C - A) : cos(A - B)
Trilinears    cos A + 2 cos B cos C : :
Trilinears    cos A - 2 sin B sin C
Trilinears    bc[a²(b² + c²) - (b² - c²)²] : :
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   a²(b² + c²) - (b² - c²)²
Barycentrics   S^2 + SB SC : :
Barycentrics   1 + cot B cot C : :
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 Nine-point center and Euler line.

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) = inverse-in-circumcircle of X(2070)
X(5) = inverse-in-orthocentroidal-circle 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) = inverse-in-polar-circle 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) = 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)

X(6) = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT)

Trilinears    a : b : c
Trilinears    sin A : sin B : sin C
Barycentrics    a² : b² : c²
X(6) = (r² + s² + 2rR)*X(1) - 6rR*X(2) -2r²*X(3)
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 x² + y² + z².

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)

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) = inverse-in-circumcircle of X(187)
X(6) = inverse-in-orthocentroidal-circle of X(115)
X(6) = inverse-in-1st-Lemoine-circle 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)

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) = 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

X(7) = GERGONNE POINT

Trilinears    bc/(b + c - a) : ca/(c + a - b) : ab/(a + b - c)
Trilinears    sec²(A/2) : sec²(B/2) : sec²(C/2)
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_A)/a : (ca - S_B)/b : (ab - S_C)/c
Trilinears    (1 - cos A)a^-2 : :
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_A : ca - S_B : ab - S_C
X(7) = (2r + 4R)*X(1) + 3r*X(2) - 4r*X(3)
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 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 1354,1367 1365,1366 1386,1456 1419,1449 1435,1848 1486,1602 1617,1621

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) = inverse-in-incircle 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 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) = endo-homothetic center of Ehrmann vertex-triangle and Ehrmann mid-triangle; the homothetic center is X(3818)

X(8) = NAGEL POINT

Trilinears    (b + c - a)/a : (c + a - b)/b : (a + b - c)/c
Trilinears    csc²(A/2) : csc²(B/2) : csc²(C/2)
Trilinears    (bc + S_A)/a : (ca + S_B)/b : (ab + S_C)/c
Trilinears    (1 + cos A)a^-2 : :
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    bc + S_A : ca + S_B : ab + S_C
X(8) = 2*X(1) - 3*X(2) = 3*X(2) - 4*X(10) = 4X(10) - X(145)
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)

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

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) = 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) = 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) = 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}

X(9) = MITTENPUNKT

Trilinears    b + c - a : c + a - b : a + b - c
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
Barycentrics   a(b + c - a) : b(c + a - b) : c(a + b - c)
Barycentrics   1 + cos A : 1 + cos B : 1 + cos C
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', 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)

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)

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

X(9) lies on the Thomson cubic and these lines:
1,6 2,7 3,84 4,10 5,1729 11,3254 8,346 21,41 31,612 32,987 33,212 34,201 35,90 38,614 39,978 42,941 43,256 46,79 48,101 55,200 56,1696 58,975 100,1005 164,168 165,910 171,1707 173,177 192,239 223,1073 228,1011 241,269 261,645 294,1253 312,314 318,1896 321,1751 342,653 348,738 364,366 374,517 393,1785 440,1211 478,1038 498,920 522,657 604,1420 607,1039 608,1041 609,1333 644,1320 654,1639 750,896 943,1802 986,1722 991,1818 1088,1223 1125,1732 1174,1621 1249,1712 1377,1703 1378,1702 1479,1752 1571,1574 1572,1573 1678,1705 1679,1704 1680,1701 1681,1700 3341,3344 3343,3352 3349,3351

X(9) is the {X(44),X(45)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(9), click Tables at the top of this page. X(9) is the internal center of similitude of the Bevan circle and Spieker circle; the external center is X(1706).

X(9) = midpoint of X(i) and X(j) for these (i,j): (7,144), (8,390)
X(9) = reflection of X(i) in X(j) for these (i,j): (1,1001), (7,142)
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) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1), (8,200), (21,55), (63,40), (190,522), (312,78), (318,33), (329, 1490), (333,8)
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): (6,282), (37,281), (41,33), (55,1), (71,219), (210,8), (212,78), (220,200)
X(9) = crosspoint of X(i) and X(j) for these (i,j): (2,8), (21,333), (63,271), (312,318)
X(9) = crosssum of X(i) and X(j) for these (i,j): (6,56), (19,208), (65,1400), (244,649), (603,604), (1418,1475)
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), (192,239)
X(9) = X(6)-of-excentral-triangle
X(9) = X(i)-aleph conjugate of X(j) for these (i,j): (1,43), (2,9), (9,170), (188,165), (190,1018), (366,1), (507,361), (508,57), (509,978)
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) = perspector of ABC and extraversion triangle of X(57)
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) = barycentric product of PU(59)
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) = X(159) of intouch triangle
X(9) = X(6) of 2nd extouch triangle
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) = X(i)-isoconjugate of X(j) for these {i,j}: {1,57}, {6,7}, {31,85}, {48,273}, {75,604}, {92,603}
X(9) = antigonal image of X(3256)
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) = excentral-isogonal conjugate of X(165)
X(9) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1743,6), (2,63,57), (10,40,1706)
X(9) = Thomson-isogonal conjugate of X(3576)
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(10) = SPIEKER CENTER

Trilinears    bc(b + c) : ca(c + a) : ab(a + b)
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
X(10) = X(1) - 3X(2) = 3X(2) + X(8) = X(7) + X(8) + 2 X(9)

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)

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) = inverse-in-circumcircle of X(1324)
X(10) = inverse-in-nine-point-circle 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) = 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(11) = FEUERBACH POINT

Trilinears    1 - cos(B - C) : 1 - cos(C - A) : 1 - cos(A - B)
Trilinears    sin²(B/2 - C/2) : :
Trilinears    bc(b + c - a)(b - c)² : :
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)² : :

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) = 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(12) = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11)

Trilinears ;  1 + cos(B - C) : 1 + cos(C - A) : 1 + cos(A -B) : :
Trilinears ;   cos²(B/2 - C/2) : :
Trilinears ;   1 + cos A + 2 cos B cos C : :
Trilinears ;   1 - cos A + 2 sin B sin C : :
Trilinears ;   bc(b + c)²/(b + c - a) : :
Barycentrics   a(1 + cos(B - C)) : :
Barycentrics   (b + c)²/(b + c - a) : :
X(12) = R*X(1) + 3r*X(2) - r*X(3)

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) = 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(13) = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)

Trilinears    csc(A + π/3) : csc(B + π/3) : csc(C + π/3)
Trilinears    sec(A - π/6) : sec(B - π/6) : sec(C - π/6)
Barycentrics   a⁴ - 2(b² - c²)² + a²(b² + c² + 4*sqrt(3)*Area(ABC)) : :
X(13) = 3^1/2(r² + 2rR + s²)*X(1) - 6r(3^1/2R - 2s)*X(2) - 2r(3^1/2r + 3s)*X(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 c²xy² - b²xz² + (a² - b² + c²)y²z - (a² + b² - c²)yz² = 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)S²)^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_BCL'_CA, L_CAL'_AB, L_ABL'_BC concur. (Dao Thanh Oai, 2014)

Let F be X(13) or X(14). Let A₀, B₀, C₀ be points on BC, CA, AB, respectively, such that the directed angles FA₀-to-FC₀ = π/3 and FC₀-to-FB₀ = π/3; if you have GeoGebra, see Figure 13B. The points A₀, B₀, C₀ 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) = inverse-in-orthocentroidal-circle 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(14) = 2nd ISOGONIC CENTER

Trilinears csc(A - π/3) : csc(B - π/3) : csc(C - π/3)
= 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) = a⁴ - 2(b² - c²)² + a²(b² + c² - 4*sqrt(3)*Area(ABC))

X(14) = 3^1/2(r² + 2rR + s²)*X(1) - 6r(3^1/2R + 2s)*X(2) + 2r(3^1/2r - 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', 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)S²)^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) = inverse-in-orthocentroidal-circle of X(13)
X(14) = complement of X(617)
X(14) = anticomplement of X(619)
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(15) = 1st ISODYNAMIC POINT

Trilinears    sin(A + π/3) : sin(B + π/3) : sin(C + π/3)
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)
X(15) = (r² + 2rR + s²)*X(1) - 6rR*X(2) - 2r(r - 3^1/2s)*X(3) = sqrt(3)*X(3) + (cot ω)*X(6)    (Peter Moses, April 2, 2013)

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)S²)^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) = inverse-in-circumcircle of X(16)
X(15) = inverse-in-Brocard-circle of X(16)
X(15) = complement of X(621)
X(15) = anticomplement of X(623)
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)

X(16) = 2nd ISODYNAMIC POINT

Trilinears    sin(A- π/3) : sin(B - π/3) : sin(C - π/3)
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)
X(16) = -(r² + 2rR + s²)*X(1) + 6rR*X(2) + 2r(r + 3^1/2s)*X(3) = sqrt(3)*X(3) - (cot ω)*X(6)    (Peter Moses, April 2, 2013)

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)S²)^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) = inverse-in-circumcircle of X(15)
X(16) = inverse-in-Brocard-circle of X(15)
X(16) = complement of X(622)
X(16) = anticomplement of X(624)
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)

X(17) = 1st NAPOLEON POINT

Trilinears csc(A + π/6) : csc(B + π/6) : csc(C + π/6)
= 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)

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).

Dao Thanh Oai, Equilateral Triangles and Kiepert Perspectors in Complex Numbers, Forum Geometricorum 15 (2015) 105-114.

Dao Thanh Oai, A family of Napoleon triangles associated with the Kiepert configuration, The Mathematical Gazette 99 (March 2015) 151-153.

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) = 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) = Cundy-Parry Phi transform of X(13)
X(17) = Cundy-Parry Psi transform of X(15)

X(18) = 2nd NAPOLEON POINT

Trilinears csc(A - π/6) : csc(B - π/6) : csc(C - π/6)
= 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)

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) = 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)

X(19) = CLAWSON POINT

Trilinears    tan A : tan B : tan C
Trilinears    sin 2B + sin 2C - sin 2A : :
Trilinears    1/(b² + c² - a²) : :
Trilinears    a² - S² + S_A : :
Trilinears    SB*SC : :
Barycentrics  a tan A : b tan B : c tan C
X(19) = (r + 2R - s)(r + 2R + s)*X(1) - 6R(r + 2R)*X(2) - 2(r² + 2rR - s²)*X(3)    (Peter Moses, April 2, 2013)

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 ŕ l'étude des antiparallčles 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) = 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}, {92,255}
X(19) = X(571)-of-excentral-triangle
X(19) = perspector, wrt excentral triangle, of polar circle

Centers 20- 30,
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.
underbar

X(20) = DE LONGCHAMPS POINT

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    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    [-3a⁴ + 2a²(b² + c²) + (b² - c²)²] : :
X(20) = 2(r + 2R)*X(1) - (r +4R)*X(7) = 3X(2) - 4X(3)
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) = inverse-in-circumcircle of X(2071)
X(20) = inverse-in-orthocentroidal-circle of X(3091)
X(20) = inverse-in-Steiner-circle of X(858)
X(20) = inverse-in-polar-circle of X(10151)
X(20) = inverse-in-orthoptic-circle-of-Steiner-inelipse of X(5159)
X(20) = inverse-in-orthoptic-circle-of-Steiner-circumelipse of X(858)
X(20) = inverse-in-anticomplement-of-circumcircle 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) = 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)

X(21) = SCHIFFLER POINT

Trilinears       1/(cos B + cos C) : 1/(cos C + cos A) : 1/(cos A + cos B)
                        = 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) = 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(22) = EXETER POINT

Trilinears   a(b⁴ + c⁴ - a⁴) : b(c⁴ + a⁴ - b⁴) : c(a⁴ + b⁴ - c⁴)
Barycentrics  a²(b⁴ + c⁴ - a⁴) : b²(c⁴ + a⁴ - b⁴) : c²(a⁴ + b⁴ - c⁴)
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 Exeter point.

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) = 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)

X(23) = FAR-OUT POINT

Trilinears a[b⁴ + c⁴ - a⁴ - b²c²] : :
Barycentrics 2 sin 2A - 3 tan ω : 2 sin 2B - 3 tan ω : 2 sin 2C - 3 tan ω (M. Iliev, 5/13/07)
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 Far-out point.

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) = 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) = 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(24) = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE

Trilinears    sec A cos 2A : sec B cos 2B : sec C cos 2C
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) : :
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) = 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)

X(25) = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES

Trilinears    sin A tan A : :
Trilinears    a/(b² + c² - a²) : :
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) = 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) = {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)

X(26) = CIRCUMCENTER OF THE TANGENTIAL TRIANGLE

Trilinears    a[b²cos 2B + c²cos 2C - a²cos 2A] : :
Trilinears    (J² - 3) cos A + 4 cos B cos C : : , where J is as at X(1113)
Barycentrics    a²(b²cos 2B + c²cos 2C - a²cos 2A) : :
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) = inverse-in-circumcircle of X(2072)
X(26) = crosssum of X(125) and X(924)

X(27) = CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER

Trilinears (sec A)/(b + c) : (sec B)/(c + a) : (sec C)/(a + b)
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).

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(28) = CEVAPOINT OF X(19) AND X(25)

Trilinears (tan A)/(b + c) : (tan B)/(c + a) : (tan C)/(a + b)
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).

X(28) lies on these lines:
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) = 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)

X(29) = CEVAPOINT OF INCENTER AND ORTHOCENTER

Trilinears (sec A)/(cos B + cos C) : (sec B)/(cos C + cos A) : (sec C)/(cos A + cos B)
Barycentrics (tan A)/(cos B + cos C) : (tan B)/(cos C + cos A) : (tan C)/(cos A + cos B)

As a point on the Euler line, X(29) has Shinagawa coefficients (F*S², $bcS_BS_C$ - F*S²).

If you have The Geometer's Sketchpad, you can view X(29).

If you have GeoGebra, you can view X(29) and X(29) cevapoint..

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) = 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(30) = EULER INFINITY POINT

Trilinears    cos A - 2 cos B cos C : cos B - 2 cos C cos A : cos C - 2 cos A cosB
Trilinears    bc[2a⁴ - (b² - c²)² - a²(b² + c²)] : :
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) = 2a⁴ - (b² - c²)² - a²(b² + c²)
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(31) = 2nd POWER POINT

Trilinears    a² : b² : c²
Trilinears    1 - cos 2A : 1 - cos 2B : 1 - cos 2C
Trilinears    cot B + cot C : :
Trilinears    S_B + S_C : :
Trilinears    d(a,b,c) : : , where d(a,b,c) = distance between A and de Longchamps line
Barycentrics    a³ : b³ : c³

X(31) = (r² + s²)*X(1) - 6rR*X(2) - 4r²*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 A₁B₁C₁ and A₂B₂C₂ be the 1st and 2nd Kenmotu diagonals triangles. Let A' be the trilinear product A₁*A₂, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(31). (Randy Hutson, February 10, 2016)

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).

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) = 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(32) = 3rd POWER POINT

Trilinears    a³ : b³ : c³
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    a⁴ : b⁴ : c⁴
X(32) = -(r² + 4rR - s²)(r² + 2rR + s²)*X(1) + 6rR(r² + 4rR - s²)*X(2) + 2r²(r² + 4rR - 3s²)*X(3)   (Peter Moses, April 2, 2013)

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', 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) 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(33) = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES

Trilinears       1 + sec A : 1 + sec B : 1 + sec C = tan A cot(A/2) : tan B cot(B/2) : tan C cot(C/2)
                        = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b² + c² - a²)
                        = g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec A cos²(A/2)

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 cos²(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).

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) = 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(34) = X(4)-BETH CONJUGATE OF X(4)

Trilinears       1 - sec A : 1 - sec B : 1 - sec C
                        = tan A tan(A/2) : tan B tan(B/2) : tan C tan(C/2)
                        = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(b + c - a)(b² + c² - a²)]
                        = g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec A sin²(A/2)

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 sin²(A/2)

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).

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) = 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}

X(35) = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36)

Trilinears    1 + 2 cos A : 1 + 2 cos B : 1 + 2 cos C
Trilinears    a(b² + c² - a² + bc) : :
Trilinears    sin(3A/2) csc(A/2) : :
Barycentrics    sin A + sin 2A : :
Barycentrics    a²(b² + c² - a² + bc) : :

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).

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) = 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)

X(36) = INVERSE-IN-CIRCUMCIRCLE OF INCENTER

Trilinears    1 - 2 cos A : 1 - 2 cos B : 1 - 2 cos C
Trilinears    a(b² + c² - a² - bc)
Trilinears    sec(A/2) cos(3A/2) : :
Barycentrics   sin A - sin 2A : :
Barycentrics   a²(b² + c² - a² - bc) : :

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) = 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

X(37) = CROSSPOINT OF INCENTER AND CENTROID

Trilinears       b + c : c + a : a + b
Trilinears       ar - S : br - S : cr - S
Barycentrics  a(b + c) : b(c + a) : c(a + b)
X(37) = (r² + 2rR - s²)*X(1) - 6rR*X(2) - 2r²*X(3)   (Peter Moses, April 2, 2013)

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). (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

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) = 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(38) = CROSSPOINT OF X(1) AND X(75)

Trilinears       b² + c² : c² + a² : a² + b²
                         = csc A sin(A + ω) : csc B sin(B + ω) : csc C sin(C + ω)
                         = S_A + S_ω : S_B + S_ω : S_C + S_ω

Barycentrics  a(b² + c²) : b(c² + a²) : c(a² + b²)
                        = sin(A + ω) : sin(B + ω) : sin(C + ω)
X(38) = 3r² + 8rR - s²)*X(1) - 6rR*X(2) - 4r²*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)

X(39) = BROCARD MIDPOINT

Trilinears    a(b² + c²) : b(c² + a²) : c(a² + b²)
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    a²(b² + c²) : b²(c² + a²) : c²(a² + b²)
X(39) = (r² + 4rR - s²)*(r² + 2rR + s²)X(1) - 6rR(r² + 4rR - s²)*X(2) - 2r²(r² + 4rR + s²)*X(3)   (Peter Moses, April 2, 2013)
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) = 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)

X(40) = BEVAN POINT

Trilinears cos B + cos C - cos A - 1 : :
Trilinears    b/(c + a - b) + c/(a + b - c) - a/(b + c -a) : :
Trilinears    sin²(B/2) + sin²(C/2) - sin²(A/2) : :
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 The Geometer's Sketchpad, you can view X(40).
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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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}, {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) = 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)

X(41) = X(6)-CEVA CONJUGATE OF X(31)

Trilinears    a²(b + c - a) : b²(c + a - b) : c²(a + b - c)
Trilinears    a²cot(A/2) : b²cot(B/2) : c²cot(C/2)
Trilinears    a²(a - s) : b²(b - s) : c²(c - s)
Trilinears    a tan A' : : , where A'B'C' is the excentral triangle
Barycentrics    a³(b + c - a) : b³(c + a - b) : c³(a + b - c)
X(41) = (r + 2R)(r² + 4rR + s²)*X(1) - 6rR(r + 4R)*X(2) -2r(² + 4rR - s²)*X(3)   (Peter Moses, April 2, 2013)

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) = 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(42) CROSSPOINT OF INCENTER AND SYMMEDIAN POINT

Trilinears    a(b + c) : b(c + a) : c(a + b)
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    a²(b + c) : b²(c + a) : c²(a + b)

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) = 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(43) = X(6)-CEVA CONJUGATE OF X(1)

Trilinears ab + ac - bc : bc + ba - ca : ca + cb - ab
= 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)

X(43) lies on 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) = 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) = 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(44) = X(6)-LINE CONJUGATE OF X(1)

Trilinears       b + c - 2a : c + a - 2b : a + b - 2c
Barycentrics  a(b + c - 2a) : b(c + a - 2b) : c(a + b - 2c)

X(44) = (3r² + 6rR + s²)*X(1) - 18rR*X(2) - 6r²*X(3)    (Peter Moses, April 2, 2013)

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) = 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(45) = X(9)-BETH CONJUGATE OF X(1)

Trilinears       2b + 2c - a : 2c + 2a - b : 2a + 2b - c
Barycentrics  a(2b + 2c - a) : b(2c + 2a - b) : c(2a + 2b - c)

X(45) = (3r² + 6rR - s²)*X(1) - 18rR*X(2) - 6r²*X(3)    (Peter Moses, April 2, 2013)

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) = crosssum of X(6) and X(999)
X(45) = X(i)-beth conjugate of X(j) for these (i,j): (9,1), (644,45)

X(46) X(4)-CEVA CONJUGATE OF X(1)

Trilinears    cos B + cos C - cos A : :
Trilinears    a^3 + a^2(b + c) - a(b^2 + c^2) - (b - c)^2(b + c) : :
Barycentrics    a(cos B + cos C - cos A) : :

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', 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) = inverse-in-Bevan-circle 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(47) = X(110)-BETH CONJUGATE OF X(34)

Trilinears       cos 2A : cos 2B : cos 2C = f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = a²(a⁴ + b⁴ + c⁴ - 2a²b² - 2a²c²)
Trilinears        a² - 2R² : b² - 2R² : c² - 2R²
Barycentrics  a cos 2A : b cos 2B : c cos 2C

X(47) = (r² - R² + s²)*X(1) - 6rR*X(2) - 4r²*X(3) (Peter Moses, April 2, 2013)

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) = 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)

X(48) = CROSSPOINT OF X(1) AND X(63)

Trilinears    tan B + tan C : :
Trilinears    a²(b² + c² - a²) ::
Trilinears    S_BS_C - S² : S_CS_A - S² : S_AS_B - S²
Trilinears    1 - cot B cot C : :

X(48) = (r² + 4rR² + 4R² + s²)*X(1) - 6R(2R + r)*X(2) - 2(r² + 2rR - s²)*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) = 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)

X(49) = CENTER OF SINE-TRIPLE-ANGLE CIRCLE

Trilinears       cos 3A : cos 3B : cos 3C
Barycentrics  sin A cos 3A : sin B cos 3B : sin C cos 3C
X(49) = (r² + 2rR² + s²)*X(1) - 6R(R + r)*X(2) - 3(r² + 4rR + R² - s²)*X(3)    (Peter Moses, April 2, 2013)

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_AQ_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_AM_BM_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_AM_BM_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) = 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)

X(50) = X(74)-CEVA CONJUGATE OF X(184)

Trilinears    sin 3A : :
Trilinears    cos A sin 2A + sin A cos 2A : :
Trilinears     sin A + cos A cot D/2 : : , where where cot D/2 = (4*area)/(6R² - a² - b² - c²), where R = abc/(4*area)       (Peter Moses, 12/19/2011; cf. X(568))
Trilinears         a(1 - 4 cos²A) : b(1 - 4 cos²B) : c(1 - 4 cos²C)
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) : :

X(50) = -(r² + 2rR + s²)(r² + 4rR + 3R² - s²)*X(1) + 6rR(r² + 4rR + 3R² - s²)*X(2) + 2r²(r² + 4rR + 3R² - 3s²)*X(3) (Peter Moses, April 2, 2013)

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 AA₁A₂, BB₁B₂, CC₁C₂ be the circumcircle-inscribed equilateral triangles used in the construction of the Trinh triangle. Let A' be the barycentric product A₁*A₂, 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', 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) = inverse-in-Brocard-circle 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)

X(51) = CENTROID OF ORTHIC TRIANGLE

Trilinears    a²cos(B - C) : :
Trilinears    a[a²(b² + c²) - (b² - c²)²] : :
Trilinears    sin A (sin 2B + sin 2C) : :
Trilinears    sec A (csc 2B + csc 2C) : :
Barycentrics    a³cos(B - C) : b³cos(C - A) : c³cos(A - B)

X(51) = (r² + 2rR + s²)*X(1) + 6R(R - r)*X(2) - (r² + 4rR - s²)*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(52) = ORTHOCENTER OF ORTHIC TRIANGLE

Trilinears    cos 2A cos(B - C) : cos 2B cos(C - A) : cos 2C cos(A - B)
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] : :
X(52) = (r² + 2rR + s²)*X(1) - 6rR*X(2) - (r² - 4rR - 2R² + s²)*X(3)    (Peter Moses, April 2, 2013)

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) = inverse-in-Brocard-circle 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(53) = SYMMEDIAN POINT OF ORTHIC TRIANGLE

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

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(54) = KOSNITA POINT

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

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))

Let (Na) be the reflection of the nine-point circle in BC, and define (Nb) and (Nc) cyclically. X(54) is the radical center of (Na), (Nb), (Nc). The tangents at A, B, C to the Napoleon-Feuerbach cubic K005 concur in X(54). Let A'B'C' be the reflection triangle. Let La be the trilinear polar of A', and define Lb and Lc cyclically. Let A" be Lb∩Lc, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(54). (Randy Hutson, July 23, 2015)

If you have GeoGebra, you can view X(54).

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) = 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

X(55) = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)

Trilinears    a(b + c - a) : b(c + a - b) : c(a + b - c)
Trilinears    1 + cos A : 1 + cos B : 1 + cos C
Trilinears    cos²(A/2) : cos²(B/2) : cos²(B/2)
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    a²/(1 - cos A) : :
Trilinears    a(2ar - S) : :
Barycentrics   a²(b + c - a) : b²(c + a - b) : c²(a + b - c) : :
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) = 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

X(56) = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)

Trilinears    a/(b + c - a) : b/(c + a - b) : c/(a + b - c)
Trilinears    1 - cos A : 1 - cos B : 1 - cos C
Trilinears    sin²(A/2) : sin²(B/2) : sin²(C/2)
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    a²/(b + c - a) : b²/(c + a - b) : c²/(a + b - c)

Barycentrics area(A'BC) : : , where A'B'C' = 2nd circumperp triangle
X(56) = R*X(1) - r*X(3)

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

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) = 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(57) ISOGONAL CONJUGATE OF X(9)

Trilinears    1/(b + c - a) : 1/(c + a - b) : 1/(a + b - c)
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    cos²(B/2) + cos²(C/2) - cos²(A/2) ::
Trilinears    S_A - bc : S_B - ca : S_C - ab : :
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
X(57) = (Ra + Rb + Rc)*X(1) - r*Ja - r*Jb - r*Jc, where Ja, Jb, Jc are excenters, and Ra, Rb, Rc are the exradii

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)

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) = 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)

X(58) ISOGONAL CONJUGATE OF X(10)

Trilinears    a/(b + c) : b/(c + a) : c/(a + b)
Trilinears    (1 - cos A)/(cos B + cos C) : :
Trilinears    sa² + SR : sb² + SR : sc² + SR
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 : :
Barycentrics a²/(b + c) : b²/(c + a) : c²/(a + b)

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) = {X(1),X(21)}-harmonic conjugate of X(4653)

X(59) ISOGONAL CONJUGATE OF X(11)

Trilinears 1/[1 - cos(B - C)] : 1/[1 - cos(C - A)] : 1/[1 - cos(A - B)]
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)
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(60) = ISOGONAL CONJUGATE OF X(12)

Trilinears 1/[1 + cos(B - C)] : 1/[1 + cos(C - A)] : 1/[1 + cos(A - B)]
Barycentrics a/[1 + cos(B - C)] : b/[1 + cos(C - A)] : c/[1 + cos(A - B)]

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(61) = ISOGONAL CONJUGATE OF X(17)

Trilinears    sin(A + π/6) : sin(B + π/6) : sin(C + π/6)
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(62) = ISOGONAL CONJUGATE OF X(18)

Trilinears    sin(A - π/6) : sin(B - π/6) : sin(C - π/6)
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(63) = ISOGONAL CONJUGATE OF X(19)

Trilinears    cot A : cot B : cot C
Trilinears    b² + c² - a² : c² + a² - b² : a² + b² - c²
Trilinears    S_A : S_B : S_C
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) = (r + 2R)*X(1) - 3R*X(2) - 2r*X(3)    (Peter Moses, April 2, 2013)
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) = 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)

X(64) = ISOGONAL CONJUGATE OF X(20)

Trilinears    1/(cos A - cos B cos C) : 1/(cos B - cos C cos A) : 1/(cos C - cos A cos B)
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) = anticomplement of X(2883)
X(64) = X(25)-cross conjugate of X(6)
X(64) = X(1)-beth conjugate of X(207)

X(64) = crosssum of X(3) and X(1498)
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(65) = ORTHOCENTER OF THE INTOUCH TRIANGLE

Trilinears    cos B + cos C : cos C + cos A : cos A + cos B
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', 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', 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", C" cyclically. Equivalently, A" is the reflection of A in B'C', and cyclically for B", C". Let Ia be the incenter of B'C'A", and define Ib, 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*, 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(66) = ISOGONAL CONJUGATE OF X(22)

Trilinears bc/(b⁴ + c⁴ - a⁴) : :
Barycentrics 1/(b⁴ + c⁴ - a⁴) : :

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

X(67) = ISOGONAL CONJUGATE OF X(23)

Trilinears bc/(b⁴ + c⁴ - a⁴ - b²c²) : :
Barycentrics 1/(b⁴ + c⁴ - a⁴ - b²c²) : :

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(68) PRASOLOV POINT

Trilinears        cos A sec 2A : cos B sec 2B : cos C sec 2C
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(69) = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE

Trilinears    (cos A)/a² : (cos B)/b² : (cos C)/c²
Trilinears    bc(b² + c² - a²) : ca(c² + a² - b²) : ab(a² + b² - c²)
Trilinears    sec²(A/2) - csc²(A/2) : :
Barycentrics    cot A : cot B : cot C
Barycentrics    b² + c² - a² : c² + a² - b² : a² + b² - c²

Barycentrics    cot B + cot C - cot ω : :
Barycentrics    cot B + cot C - cot A - cot ω : :
X(69) = 2(r² + 2rR + s²)*X(1) + 3(r² - s²)*X(2) - 4r²*X(3)    (Peter Moses, April 2, 2013)
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) 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 74,99 110,206 125,895 144,190 150,668 189,309 192,742 194,695 200,269 248,287 263,308 265,328 274,443 290,670 297,393 347,664 350,497 404,1014 478,651 485,639 486,640 520,879

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.

If you have The Geometer's Sketchpad, you can view X(69).

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(70) = ISOGONAL CONJUGATE OF X(26)

Trilinears bc/[b² cos 2B + c² cos 2C - a² cos 2A]
X(70) = (S cot ω- 2R²)/(S cot ω-3 R²)*X(3) - X(8907) (Peter Moses, December 28, 2015)

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) = 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)

X(71) = ISOGONAL CONJUGATE OF X(27)

Trilinears (b + c) cos A : (c + a) cos B : (a + b) cos C
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 {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(72) = ISOGONAL CONJUGATE OF X(28)

Trilinears    (b + c) cot A : (c + a) cot B : (a + b) cot C
Trilinears    (b + c)(b² + c² - a²) : (c + a)(c² + a² - b²) : (a + b)(a² + b² - c²)
Barycentrics    (b + c) cos A : (c + a) cos B : (a + b) cos C
X(72) = (r + 2R)*X(1) - 3R*X(2) - r*X(3)    (Peter Moses, April 2, 2013)

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 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), (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(73) CROSSPOINT OF INCENTER AND CIRCUMCENTER

Trilinears    sec B + sec C : sec C + sec A : sec A + sec B
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) = (r² - 4R² + s²)*X(1) - 6rR*X(2) + 4rR*X(3)    (Peter Moses, April 2, 2013)

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)

X(74) ISOGONAL CONJUGATE OF EULER INFINITY POINT

Trilinears    1/(cos A - 2 cos B cos C) : 1/(cos B - 2 cos C cos A) : 1/(cos C - 2 cos A cos B)
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/[2a⁴ - (b² - c²)² - a²(b² + c²)] : :
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) = (r² + 2rR + s²)*X(1) - R(6r + 9R)*X(2) + (r² + 12rR + 18R² - 3s²)*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 fromed 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 B' and C' cyclically. Let L' be the reflection of L in sideline BC, and define B' 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)

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(75) ISOTOMIC CONJUGATE OF INCENTER

Trilinears    1/a² : 1/b² : 1/c²
Trilinears    1/(1 - cos 2A) : 1/(1 - cos 2B) : 1/(1 - cos 2C)
Trilinears    S_A + S² : S_B + S² : S_C + S²
Trilinears    sec²(A/2) + csc²(A/2) : :
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)²
Barycentrics   1/a : 1/b : 1/c
Barycentrics  csc A : csc B : csc C
X(75) = (csc A)*[A] + (csc B)*[B] + (csc C)*[C], where A, B, C denote angles and [A], [B], [C] denote vertices

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)

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; http://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(76) = 3rd BROCARD POINT

Trilinears    1/a³ : 1/b³ : 1/c³
Trilinears    csc(A - ω) : csc(B - ω) : csc(C - ω)
Barycentrics    1/a² : 1/b² : 1/c²
X(76) = 3*X(2) - 2*X(39) = 3*X(2) - P(1) - U(1)

Let A' be the perspector of the A-McCay circle, and define B', C' cyclically. The lines AA', BB', CC' concur in X(76). (Randy Hutson, April 9, 2016)

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
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(77) = ISOGONAL CONJUGATE OF X(33)

Trilinears       1/(1 + sec A) : 1/(1 + sec B) : 1/(1 + sec C)
                        = cos A sec²(A/2) : cos B sec²(B/2) : cos C sec²(C/2)
                        = (b² + c² - a²)/(b + c - a) : (c² + a² - b²)/(c + a - b) : (a² + b² - c²)/(a + b - c)
                        = S_A(S_A - bc) : S_B(S_B - ca) : S_C(S_C - ab)

Barycentrics a/(1 + sec A) : b/(1 + sec B) : c/(1 + sec C)

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) = 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)

X(78) = ISOGONAL CONJUGATE OF X(34)

Trilinears       1/(1 - sec A) : 1/(1 - sec B) : 1/(1 - sec C)
                        = cos A csc²(A/2) : cos B csc²(B/2) : cos C csc²(C/2)
                        = (b + c - a)(b² + c² - a²) : (c + a - b)(c² + a² - b²) : (a + b - c)(a² + b² - c²)
                        = S_A(S_A + bc) : S_B(S_B + ca) : S_C(S_C + ab)

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).

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(79) = ISOGONAL CONJUGATE OF X(35)

Trilinears    1/(1 + 2 cos A) : :
Trilinears    bc/(b² + c² - a² + bc)
Trilinears    (sin A/2)(sin 3B/2)(sin 3C/2) : :
Trilinears    sin(A/2) csc(3A/2) : :
Barycentrics    1/(b² + c² - a² + bc) : :

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', 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)

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

X(80) REFLECTION OF INCENTER IN FEUERBACH POINT

Trilinears    1/(1 - 2 cos A) : 1/(1 - 2 cos B) : 1/(1 - 2 cos C)
Trilinears    bc/(b² + c² - a² - bc) : :
Trilinears    cos(A/2) sec(3A/2) : :
Barycentrics    1/(b² + c² - a² - bc) : :
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

X(81) CEVAPOINT OF INCENTER AND SYMMEDIAN POINT

Trilinears 1/(b + c) : 1/(c + a) : 1/(a + b)
Barycentrics a/(b + c) : b/(c + a) : c/(a + b)

X(81) = (r² + 2rR + s²)*X(1) - 3rR*X(2) - 2r²*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)

If you have The Geometer's Sketchpad, you can view X(81).

If you have GeoGebra, you can view X(81).

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(82) = ISOGONAL CONJUGATE OF X(38)

Trilinears 1/(b² + c²) : 1/(c² + a²) : 1/(a² + b²)
= sin A csc(A + ω) : sin B csc(B + ω) : sin C csc(C + ω)

Barycentrics a/(b² + c²) : b/(c² + a²) : c/(a² + b²)

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(83) CEVAPOINT OF CENTROID AND SYMMEDIAN POINT

Trilinears bc/(b² + c²) : ca/(c² + a²) : ab/(a² + b²)
= csc(A + ω) : csc(B + ω) : csc(C + ω)

Barycentrics 1/(b² + c²) : 1/(c² + a²) : 1/(a² + b²)

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(84) = ISOGONAL CONJUGATE OF X(40)

Trilinears    1/(cos B + cos C - cos A - 1) : :
Trilinears    a^2[a^2 - (b - c)^2]^2 - (b - c)^2[a^2 - (b + c)^2]^2 : :
X(84) = (r + 2R)*X(1) - 6R*X(2) + (4R - 2r)*X(3)    (Peter Moses, April 2, 2013)

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 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(85) ISOTOMIC CONJUGATE OF X(9)

Trilinears b²c²/(b + c - a) : c²a²/(c + a - b) : a²b²/(a + b - c)
= tan(A/2) csc²A : tan(B/2) csc²B : tan(C/2) csc²C

Barycentrics bc/(b + c - a) : ca/(c + a - b) : ab/(a + b - c)

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(86) CEVAPOINT OF INCENTER AND CENTROID

Trilinears bc/(b + c) : ca/(c + a) : ab/(a + b)
Barycentrics 1/(b + c) : 1/(c + a) : 1/(a + b)

X(86) = 2(r² + 2rR + s²)*X(1) + 3(r² + s²)*X(2) - 4r²*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)

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(87) X(2)-CROSS CONJUGATE OF X(1)

Trilinears 1/(ab + ac - bc) : 1/(bc + ba - ca) : 1/(ca + cb - ab)
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) = 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)

X(88) = ISOGONAL CONJUGATE OF X(44)

Trilinears    1/(b + c - 2a) : 1/(c + a - 2b) : 1/(a + b - 2c)
Barycentrics  a/(b + c - 2a) : b/(c + a - 2b) : c/(a + b - 2c)
X(88) = (3r² + 6rR - s²)*X(1) + 9rR*X(2) - 6r²*X(3)    (Peter Moses, April 2, 2013)

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) = 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) = trilinear product of PU(50)
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(89) = ISOGONAL CONJUGATE OF X(45)

Trilinears 1/(2b + 2c - a) : 1/(2c + 2a - b) : 1/(2a + 2b - c)
Barycentrics a/(2b + 2c - a) : b/(2c + 2a - b) : c/(2a + 2b - c)

X(89) lies on these lines: 1,902 2,44 6,88 649,1022

X(89) = isogonal conjugate of X(45)

X(90) = X(3)-CROSS CONJUGATE OF X(1)

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

X(91) = ISOGONAL CONJUGATE OF X(47)

Trilinears sec 2A : sec 2B : sec 2C
Barycentrics sin A sec 2A : sin B sec 2B : sin C sec 2C

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(92) CEVAPOINT OF INCENTER AND CLAWSON POINT

Trilinears    csc 2A : csc 2B : csc 2C
Trilinears    cot A + tan A : :
Barycentrics    sec A : sec B : sec C
X(92) = X(4) - ((r + 2R)² - s²)*X(8)
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 A₁ be the midpoint of the arc BC of the circumcircle that passes through A, and let A₂ be the point, other than A, in which the A-altitude meets the circumcircle. Let A" = A₁A₂∩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(93) = ISOGONAL CONJUGATE OF X(49)

Trilinears sec 3A : sec 3B : sec 3C
Barycentrics sin A sec 3A : sin B sec 3B : sin C sec 3C

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(94) = ISOGONAL CONJUGATE OF X(50)

Trilinears        csc 3A : csc 3B : csc 3C
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', 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(95) CEVAPOINT OF CENTROID AND CIRCUMCENTER

Trilinears b²c²sec(B - C) : :

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 http://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)

X(96) = ISOGONAL CONJUGATE OF X(52)

Trilinears sec 2A sec(B - C) : sec 2B sec(C - A) : sec 2C sec(A - B)
Barycentrics a sec 2A sec(B - C) : b sec 2B sec(C - A) : c sec 2C sec(A - B)

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)

X(97) = ISOGONAL CONJUGATE OF X(53)

Trilinears cot A sec(B - C) : cot B sec(C - A) : cot C sec(A - B)
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)

Centers 74, 98 - 112,
and others lie on the circumcircle. Mappings Λ and Ψ derived from such a point P for application to points X, are defined here:

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.

underbar

X(98) = TARRY POINT

Trilinears    sec(A + ω) : sec(B + ω) : sec(C + ω)
Trilinears    bc/(b⁴ + c⁴ - a²b² - a²c²) : :
Barycentrics    1/(b⁴ + c⁴ - a²b² - a²c²) : :
X(98) = 2(r² + 4rR - s²)(r² + 2rR + s²)*X(1) + 3(r⁴ + 4Rr³ + 2r²s² - 4rRs² + s⁴)*X(2) - 2(r² + 4rR - s²)(3r² + 4rR - s²)    (Peter Moses, April 2, 2013)

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', 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)

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) = 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(99) = STEINER POINT

Trilinears    bc/(b² - c²) : ca/(c² - a²) : ab/(a² - b²)
Trilinears    b²c² csc(B - C) : c²a² csc(C - A) : a²b² csc(A - B)
Barycentrics    1/(b² - c²) : 1/(c² - a²) : 1/(a² - b²)
Barycentrics    d(A,L) : : , where d = directed distance and L = X(115)X(125)
X(99) = 2(r² + 4rR - s²)(r² + 2rR + s²)*X(1) - 3(r⁴ + 4Rr³ + 2r²s² - 4rRs² + s⁴)*X(2) - 4r²(r² + 4rR + 5s²)*X(3)    (Peter Moses, April 2, 2013)

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', 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', 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)

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) = 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(100) ANTICOMPLEMENT OF FEUERBACH POINT

Trilinears 1/(b - c) : 1/(c - a) : 1/(a - b)
Trilinears (a - b)(a - c) : (b - c)(b - a) : (c - a)(c - b)

Barycentrics a/(b - c) : b/(c - a) : c/(a - b)

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), (48,3), (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(101) = Ψ(INCENTER, SYMMEDIAN POINT)

Trilinears    a/(b - c) : b/(c - a) : c/(a - b)
Trilinears    a(a - b)(a - c) : b(b - c)(b - a) : c(c - a)(c - b)
Barycentrics    a²/(b - c) : b²/(c - a) : c²/(a - b)
X(101) = (r ² + 6rR + 8R² + s²)*X(1) - 6R(r + 4R)*X(2) - 2(r² + 4rR - s²)*X(3)    (Peter Moses, April 2, 2013)

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_aI_bI_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), (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) = trilinear product of circumcircle intercepts of line X(1)X(6)

X(102) = Λ(INCENTER, ORTHOCENTER)

Trilinears    1/[sin B (sec A - sec B) + sin C (sec A - sec C)]
Trilinears    a/[2a⁵ + (b + c)a⁴ - 2(b² + c²)a³ - (b + c)(b² - c²)²]
Barycentrics    (sin A)/[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) = 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(103) = ANTIPODE OF X(101)

Trilinears    a/[(a - b) cot C + (a - c) cot B] : b/[(b - c) cot A + (b - a) cot C] : c/[(c - a) cot B + (c - b) cot A]
Trilinears    1/(a² - b²cos C - c² cos B) : 1/(b² - c²cos A - a² cos C) : 1/(c² - a²cos B - c² cos A)
Barycentrics    a²/[(a - b) cot C + (a - c) cot B] : :
Barycentrics    a^2 / (2 a^3 - a^2 (b + c) - (b - c)^2 (b + c)) : :

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) = circumcircle-antipode of X(101)
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) = SR(P,U), where P and U are the circumcircle intercepts of the Soddy line

X(104) = ANTIPODE OF X(100)

Trilinears    1/(-1 + cos B + cos C) : 1/(-1 + cos C + cos A) : 1/(-1 + cos C + cos B)
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)
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) = 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(105) = Λ(INCENTER, SYMMEDIAN POINT)

Trilinears 1/[b² + c² - a(b + c)] : 1/[c² + a² - b(c + a)] : 1/[a² + b² - c(a + b)]
Barycentrics a/[b² + c² - a(b + c)] : b/[c² + a² - b(c + a)] : c/[a² + b² - c(a + b)]

X(105) = Λ(X(1), X(6))
X(105) = Ψ(X(101), X(1))

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): (6,513), (101,1), (190,9)
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(106) = Λ(INCENTER, CENTROID)

Trilinears a/(2a - b - c) : b/(2b - c - a) : c/(2c - a -b)
Barycentrics a²/(2a - b - c) : b²/(2b - c - a) : c²/(2c - a - b)

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) = 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) = Ψ(X(101), X(6))
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): (6,649), (190,2)
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(107) = Ψ(SYMMEDIAN POINT, ORTHOCENTER)

Trilinears    1/[cos A (sin 2B - sin 2C)] : :
Trilinears    (sec A)/(tan B - tan C) : :
Trilinears    bc/[(b² - c²)(b² + c² - a²)²] : :
Barycentrics   1/[(b² - c²)(b² + c² - a²)²] : 1/[(c² - a²)(c² + a² - b²)²] : 1/[(a² - b²)(a² + b² - c²)²]

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) = anticomplement of X(122)
X(107) = cevapoint of X(4) and X(523)
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(108) = Ψ(CIRCUMCENTER, INCENTER)

Trilinears a/(sec B - sec C) : b/(sec C - sec A): c/(sec A - sec B)
= g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = 1/[(b - c)(b + c - a)(b² + c² - a²)]

Barycentrics a²/(sec B - sec C) : b²/(sec C - sec A): c²/(sec A - sec B)

X(108) = Ψ(X(3), X(1))
X(108) = Ψ(X(1), X(4))

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) = 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): (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(109) = Ψ(INCENTER, CIRCUMCENTER)

Trilinears    a/(cos B - cos C) : b/(cos C - cos A) : c/(cos A - cos B)
Trilinears    a/[(b - c)(b + c - a)] : : :
Barycentrics  a²/(cos B - cos C) : b²/(cos C - cos A): c²/(cos A - cos B)

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) = 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) = 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) = intersection of antipedal lines of circumcircle intercepts of line X(3)X(10)

X(110) = FOCUS OF KIEPERT PARABOLA

Trilinears    csc(B - C) : csc(C - A) : csc(A -B)
Trilinears    cos(B - C) - cos(C - A)cos(A - B) : :
Trilinears    a/(b² - c²) : :
Barycentrics    a²/(b² - c²) : :

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) 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', 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 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) = crosssum of X(i) and X(j) for these (i,j): (2,148), (512,647), (520,647)
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
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), (19,1), (53,5), (54,3), (64,3), (66,3), (67,3), (68,3), (69,3), (73,3), (74,3), (75,1), (76,2), (115,125), (190,99)
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(111) = PARRY POINT

Trilinears a/(2a² - b² - c²) : b/(2b² - c² - a²) : c/(2c² - a² - b²)
Barycentrics a²/(2a² - b² - c²) : b²/(2b² - c² - a²) : c²/(2c² - a² - b²)

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) = perspector of ABC and the triangle formed by the reflections of line PU(7) in the sides of ABC
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), (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(112) = Ψ(ORTHOCENTER, SYMMEDIAN POINT)

Trilinears    a/(sin 2B - sin 2C) : b/(sin 2C - sin 2A) : c/(sin 2A - sin 2B)
Trilinears    a/[(b² - c²)(b² + c² - a²)]
Trilinears    tan A csc(B - C) : :
Barycentrics    a²/(sin 2B - sin 2C) : b²/(sin 2C - sin 2A) : c²/(sin 2A - sin 2B)

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)

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) = 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) = 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)

Centers 113-139
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_AH_BH_C is homothetic to and congruent to the cyclic quadrilateral ABCX', and X is the center of homothety. (Randy Hutson, 9/23/2011)

underbar

X(113) = JERABEK ANTIPODE

Trilinears    sin B sin C [(sin C)/(cos C - 2 cos A cos B) + (sin B)/(cos B - 2 cos A cos C)] : :
Barycentrics    (sin C)/(cos C - 2 cos A cos B) + (sin B)/(cos B - 2 cos A cos C) : :
Barycentrics    b²/(b²S_B - 2S_AS_C) + c²/(c²S_C - 2S_AS_B) : : (Peter J. C. Moses, 3/2003)
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(114) = KIEPERT ANTIPODE

Trilinears   bc[b sec(B + ω) + c sec(C + ω)] : :
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

X(115) = CENTER OF KIEPERT HYPERBOLA

Trilinears bc(b² - c²)² : ca(c² - a²)² : ab(a² - b²)²
Trilinears    cos A - 2 cos(B - C) + cot ω sin A : : (Peter J. C. Moses, 9/12/03)
Trilinears    sin A sin²(B - C) : :
Barycentrics    (b² - c²)² : (c² - a²)² : (a² - b²)²
Barycentrics    (SB - SC)^2 : :
Barycentrics    (SA - SW) (SB + SC) + 4 SB SC : :
X(115) = 2(tan ω sin 2ω)*X(5) - X(39)
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(116) = MIDPOINT OF X(4) AND X(103)

Trilinears bc[(b - c)²(b² + bc + c² - ab - ac)] : :
Barycentrics (b - c)²(b² + bc + c² - ab - ac) : :

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) = 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.

X(117) = MIDPOINT OF X(4) AND X(109)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = g(b,c,a) + g(c,b,a), and
g(b,c,a) = b²c/[c(sec B - sec C) + a(sec B - sec A)]

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) = complementary conjugate of X(515)
X(117) = X(4)-Ceva conjugate of X(515)
X(117) = complement of X(102)
X(117) = anticenter of cyclic quadrilateral ABCX(109)
X(117) = Λ(X(1), X(4)), wrt orthic triangle

X(118) = MIDPOINT OF X(4) AND X(101)

Trilinears g(b,c,a) + g(c,b,a) : : ,where g(b,c,a) = b³c/[(b - c) cot A + (b - a) cot C]

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

X(119) = FEUERBACH ANTIPODE

Trilinears (csc A)(-1 + cos B + cos C)[sin 2B + sin 2C + 2(-1 + cos A)(sin B + sin C)] : :
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) = 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)

X(120) = X(105)-OF-MEDIAL-TRIANGLE

Trilinears bc[2abc - (b + c)(a² + (b - c)²)](b² + c² - ab -ac) : :
Barycentrics [2abc - (b + c)(a² + (b - c)²)](b² + c² - ab -ac) : :

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) = X(105)-of-medial triangle.

X(120) = complement of X(105)
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))

X(121) = X(106)-OF-MEDIAL-TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = bc(b + c - 2a)[b³ + c³ + a(b² + c²) - 2bc(b + c)]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = (b + c - 2a)[b³ + c³ + a(b² + c²) - 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(122) = X(107)-OF-MEDIAL-TRIANGLE

Trilinears (b² - c²)²(cos A - cos B cos C) cot²A : :
Barycentrics a(b² - c²)²(cos A - cos B cos C) cot²A : :

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) = 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) = X(2)-Ceva conjugate of X(6587)

X(123) = X(108)-OF-MEDIAL-TRIANGLE

Trilinears    (csc A)(sec B - sec C)[(sec A)(sin²B - sin²C) + sin C tan C - sin B tan B] : :
Barycentrics    (sec B - sec C)[(sec A)(sin²B - sin²C) + sin C tan C - sin B tan B]
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

X(124) = X(109)-OF-MEDIAL-TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = bc(b + c - a)(b - c)²[(b + c)(b² + c² - a² - bc) + abc]

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)(b² + c² - a² - 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) = 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)

X(125) = CENTER OF JERABEK HYPERBOLA

Trilinears    cos A sin²(B - C) : cos B sin²(C - A) : cos C sin²(A - B)
Trilinears    (sec A)(c cos C - b cos B)² : (sec B)(a cos A - c cos C)² : (sec C)(b cos B - a cos A)²
Trilinears    bc(b² + c² - a²)(b² - c²)² : :
Barycentrics    (sin 2A)[sin(B - C)]² : (sin 2B)[sin(C - A)]² : (sin 2C)[sin(A - B)]²

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) = 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(126) = X(111)-OF-MEDIAL-TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = bc(2a² - b² - c²)[b⁴ + c⁴ + a²(b² + c²) - 4b²c²]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = (2a² - b² - c²)[b⁴ + c⁴ + a²(b² + c²) - 4b²c²]

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

X(127) = X(112)-OF-MEDIAL-TRIANGLE

Trilinears    bc(sin 2B - sin 2C)[(b² - c²)sin 2A - b²sin 2B + c²sin 2C] : :
Barycentrics   (sin 2B - sin 2C)[(b² - c²)sin 2A - b²sin 2B + c²sin 2C] : :
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 and 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)

X(128) = X(74)-OF-ORTHIC-TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where
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) = X(2)-Ceva conjugate of X(231)
X(128) = orthojoin of X(231)

X(129) = X(98)-OF-ORTHIC-TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sec A)(sin 2A)(sin 2B + sin 2C) s(A,B,C) t(A,B,C),
                        s(A,B,C) = sin⁴(2B) + sin⁴(2C) - sin²(2A) sin²(2B) - sin²(2A) sin²(2C),
                        t(A,B,C) = sin⁴(2A) + sin²(2A) u(A,B,C) + v(A,B,C),
                        u(A,B,C) = sin 2B sin 2C - sin²(2B) - sin²(2C),
                        v(A,B,C) = (sin 2B sin 2C)(sin 2B - sin 2C)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(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(130) = X(99)-OF-ORTHIC-TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where
f(A,B,C) = (sin A)(sin 2B + sin 2C)[(sin 2B - sin 2C)²][sin²(2A) + 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(131) = INTERSECTION OF LINES X(3)X(125) AND X(4)X(135)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        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(132) = X(2)X(107)∩X(4)X(32)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sec A) u(A,B,C) v(A,B,C),
                        u(A,B,C) = [sin²(2A) + (sin 2B - sin 2C)² + (sin 2A)(sin 2A - sin 2B - sin 2C)],
                        v(A,B,C) = [sin²(2B) + sin²(2C) - (sin 2A sin 2B) - (sin 2A 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(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) = 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(133) = INTERSECTION OF LINES X(4)X(74) AND X(5)X(122)

Trilinears (sec A)[(sin 2B - sin 2C)² + (sin 2A)(sin 2B) + (sin 2A)(sin 2C) - 2(sin 2B)(sin 2C)](2 sin 2A - sin 2B - sin 2C) : :
Barycentrics (tan A)[(sin 2B - sin 2C)² + (sin 2A)(sin 2B) + (sin 2A)(sin 2C) - 2(sin 2B)(sin 2C)](2 sin 2A - 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: 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(134) = X(107)-OF-ORTHIC-TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sec A) u(A,B,C) [v(B,C,A) - v(C,B,A)],
                        u(A,B,C) = (sin 2A)[sin²(2B) - sin²(2C)][sin²(2B) + sin²(2C) - sin²(2A)]²,
                        v(B,C,A) = sin 2C [sin²(2A) - sin²(2B)][sin²(2A) + sin²(2B) - sin²(2C)]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(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(135) = INTERSECTION OF LINE X(4)X(131) AND X(25)X(114)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where
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(136) = INTERSECTION OF LINE X(4)X(110) AND X(25)X(132)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sec A)[(sin 2B - sin 2C)²](sin 2B + sin 2C - sin 2A) u(A,B,C),
                        u(A,B,C) = [sin²(2B) + sin²(2C) - sin²(2A)]

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(137) = X(110)-OF-ORTHIC-TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sec A)(sin 2B + sin 2C)[(sin 2B - sin 2C)²] u(A,B,C),
                        u(A,B,C) = [sin²(2A) - sin²(2B) - sin²(2C) - sin 2B sin 2C]

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 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(138) = X(111)-OF-ORTHIC-TRIANGLE

Trilinears (v + w) sec A : (w + u) sec B : (u + v) sec C, where
u = u(A,B,C) = (sin 2A)/(2 sin²2A - sin²2B - sin²2C), v = u(B,C,A), w = u(C,A,B)

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(139) = X(112)-OF-ORTHIC-TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sec A)(sin 2B + sin 2C)[(sin 2B - sin 2C)²][sin²(2B) + sin²(2C) - sin²(2A)] u(A,B,C),
                        u(A,B,C) = (sin 2B)⁴ + (sin 2C)⁴ - (sin 2A)⁴ + (sin 2B sin 2C)[sin²(2B) + sin²(2C) - sin²(2A)]

Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(139) lies on the nine-point circle
X(139) = X(112) of the orthic triangle

X(139) lies on these lines: 52,129 53,128

Centers 140- 170
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
underbar

X(140) = MIDPOINT OF X(3) AND X(5)

Trilinears    2 cos A + cos(B - C) : 2 cos B + cos(C - A) : 2 cos C + cos(A - B)
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 : :
X(140) = 3*X(2) + X(3) = 3*X(2) - X(5) = 3*X(3) + X(4) = X(4) - 3*X(5)

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) 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(141) = COMPLEMENT OF SYMMEDIAN POINT

Trilinears    bc(b² + c²) : :
Trilinears    csc²A sin(A + ω) : :
Barycentrics    b² + c² : :
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) = 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

X(142) = COMPLEMENT OF X(9)

Trilinears b + c - [(b - c)²]/a : c + a - [(c - a)²]/b : a + b - [(a - b)²]/c
Barycentrics ab + ac - (b - c)² : bc + ba - (c - a)² : ca + cb - (a - b)²
X(142) = 3*X(2) + X(7) = 3*X(2) - X(9)

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) = 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

X(143) = NINE-POINT CENTER OF ORTHIC TRIANGLE

Trilinears    (sec A)[cos(2C - 2A) + cos(2A - 2B)] : :
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)] : :
X(143) = X[5] - 3 X[51] = 3 X[51] + X[52] = X[4] + 3 X[568] = X[3] + 3 X[3060] = 3 X[2979] - 7 X[3526] = X[3] - 5 X[3567] = 3 X[3060] + 5 X[3567] = 5 X[632] - 3 X[3917] = 3 X[140] - 2 X[5447] = X[5447] - 3 X[5462] = 3 X[5] - X[5562] = 9 X[51] - X[5562] = 3 X[52] + X[5562] = 5 X[1656] - 9 X[5640] = 3 X[381] - X[5876] = 3 X[381] + X[5889] = X[382] + 3 X[5890] = 5 X[5562] - 9 X[5891] = 5 X[5] - 3 X[5891] = 5 X[51] - X[5891] = 5 X[52] + 3 X[5891] = 2 X[3530] - 3 X[5892] = X[1216] - 3 X[5943] = 2 X[3628] - 3 X[5943] = X[3] - 3 X[5946] = 5 X[3567] - 3 X[5946] = 3 X[568] - X[6102] = 3 X[3830] + X[6241] = 3 X[2] + X[6243] = 3 X[5093] + X[6403] = X[195] + 3 X[7730] = 11 X[5070] - 7 X[7999]

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) = 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) = {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}

X(144) = ANTICOMPLEMENT OF X(7)

Trilinears    (csc A)(tan B/2 + tan C/2 - tan A/2) : :
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 : :

X(144) = X(7)-of-anticomplementary triangle

X(144) lies on these lines:
2,7 8,516 20,72 21,954 69,190 75,391 100,480 145,192 219,347 220,279 320,344

X(144) is the {X(7),X(9)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(144), click Tables at the top of this page.

X(144) = reflection of X(i) in X(j) for these (i,j): (7,9), (145,390), (149,1156)
X(144) = anticomplement of X(7)
X(144) = anticomplementary conjugate of X(3434)
X(144) = X(8)-Ceva conjugate of X(2)
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(145) = ANTICOMPLEMENT OF NAGEL POINT

Trilinears    bc(3a - b - c) : :
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(146) = REFLECTION OF X(20) IN X(110)

Trilinears bc(-avw + bwu + cuv) : ca(-bwu + cuv + avw) : ab(-cuv + avw + bwu), where
u = u(A,B,C) = cos A - 2 cos B cos C, v = u(B,C,A), w = u(C,A,B)

Barycentrics -avw + bwu + cuv : -bwu + cuv + avw : -cuv + avw + bwu

X(146) = X(74)-of-anticomplementary triangle

X(146) lies on 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) = anticomplementary conjugate of X(30)

X(147) = TARRY POINT OF ANTICOMPLEMENTARY TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = bc[a⁸ + (b² + c²)a⁶ - (2b⁴ + 3b²c² + 2c⁴)a⁴
                        + (b⁶ + b⁴c² + b²c⁴ + c⁶)a² - b⁸ + b⁶c² + b²c⁶ - c⁸]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(147) = X(98)-of-anticomplementary triangle

X(147) lies on 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(148) = STEINER POINT OF ANTICOMPLEMENTARY TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = bc[a⁴ - (b² - c²)² + b²c² - a²b² - a²c²]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a⁴ - (b² - c²)² + b²c² - a²b² - a²c²

X(148) = X(99)-of-anticomplementary triangle

X(148) lies on 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

X(149) = REFLECTION OF X(20) IN X(104)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = bc[b³ + c³ - a³ + (a² - bc)(b + c) + a(bc - b² - c²)]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = b³ + c³ - a³ + (a² - bc)(b + c) + a(bc - b² - c²)

X(149) = X(100)-of-anticomplementary-triangle

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 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) = anticomplementary conjugate of X(513)

X(150) = REFLECTION OF X(20) IN X(103)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = bc[b⁴ + c⁴ - a⁴ + a(bc² +cb² - b³ - c³) - bc(a² + b² + c²) + (b + c)a³]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = b⁴ + c⁴ - a⁴ + a(bc² +cb² - b³ - c³) - bc(a² + b² + c²) + (b + c)a³

X(150) = X(101)-of-anticomplementary triangle

X(150) lies on 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) = anticomplementary conjugate of X(514)

X(151) = REFLECTION OF X(20) IN X(109)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = (by + cz - ax)/a, where x : y : z = X(102)

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(151) = X(102)-of-anticomplementary triangle

X(151) lies on these lines: 2,102 20,109 149,962 152,928

X(151) = reflection of X(i) in X(j) for these (i,j): (20,109), (102,117)
X(151) = anticomplementary conjugate of X(515)

X(152) = REFLECTION OF X(20) IN X(101)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = (by + cz - ax)/a, where x : y : z = X(103)

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(152) = X(103)-of-anticomplementary triangle

X(152) lies on 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) = anticomplementary conjugate of X(516)

X(153) = REFLECTION OF X(20) IN X(100)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = (by + cz - ax)/a, where x : y : z = X(104)

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(153) = X(104)-of-anticomplementary triangle

X(153) lies on 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) = anticomplementary conjugate of X(517)

X(154) = X(3)-CEVA CONJUGATE OF X(6)

Trilinears    (cos A - cos B cos C)a² : (cos B - cos C cos A)b² : (cos C - cos A cos B)c²
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    (sin² A)(tan B + tan C - tan A) : :

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) = 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(155) EIGENCENTER OF ORTHIC TRIANGLE

Trilinears (cos A)[cos²B + cos²C - cos²A] : :
Barycentrics a^2 SA [a^2 (SA^2 - SB SC) - SA (SB^2 + SC^2)] : :

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 these lines:
1,90 3,49 4,254 5,6 20,323 24,110 25,52 26,154 30,1498 159,511 195,381 382,399 450,1075 648,1093 651,1068

X(155) is the {X(26),X(156)}-harmonic conjugate of X(154). For a list of harmonic conjugates of X(155), click Tables at the top of this page.

X(155) = reflection of X(i) in X(j) for these (i,j): (3,1147), (26,156), (68,5)
X(155) = isogonal conjugate of X(254)
X(155) = X(4)-Ceva conjugate of X(3)
X(155) = crosssum of X(136) and X(523)
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) = orthic-isogonal conjugate of X(3)
X(155) = tangential-isogonal conjugate of X(26)
X(155) = crossdifference of every pair of points on line X(924)X(2501)
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(156) = X(5)-OF-TANGENTIAL-TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b²y/v + c²z/w - a²x/u],
                        u = u(A,B,C) = sin 2A, v = u(B,C,A), w = u(C,A,B);
                        x = x(A,B,C) = u²(v² + w²) - (v² - w²)², y = x(B,C,A), z = x(C,A,B)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

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

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(157) = X(6)-OF-TANGENTIAL-TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b³cos B + c³cos C - a³cos A]
                        = g(a,b,c) : g(b,c,a) : g(c,a,b), where
                        g(a,b,c) = a[a⁶ - b⁶ - c⁶ - a⁴(b² + c²) + b⁴(c² + a²) + c⁴(a² + b²)]

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(158) = X(19)-CROSS CONJUGATE OF X(92)

Trilinears sec²A : sec²B : sec²C
= 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

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

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(159) = X(9)-OF-TANGENTIAL-TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a[(a² + b² + c²)sin 2A + (c² - b² - a²)sin 2B + (b² - c² - a²)sin 2C]

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(160) = X(37)-OF-TANGENTIAL-TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a[(b² + c²)sin 2A + (c² - a²)sin 2B + (b² - a²)sin 2C]

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)

X(161) = X(63)-OF-TANGENTIAL-TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a[(a² + b² + c²)sin²(2A) + (c² - b² - a²)sin²(2B) + (b² - c² - a²)sin²(2C)]

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)

X(162) = CEVAPOINT OF X(108) AND X(109)

Trilinears    1/(sin 2B - sin 2C) : 1/(sin 2C - sin 2A) : 1/(sin 2A - sin 2B)
Trilinears    1/[(b² - c²)(b² + c² - a²)]
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) = 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(163) = TRILINEAR PRODUCT X(6)*X(110)

Trilinears    (sin 2A)/(sin 2B - sin 2C) : (sin 2B)/(sin 2C - sin 2A) : (sin 2C)/(sin 2A - sin 2B)
Trilinears    a²/(b² - c²) : :
Barycentrics  a³/(b² - c²) : b³/(c² - a²) : c³/(a² - b²)

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) = 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) = isogonal conjugate of X(1577)
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(164) INCENTER OF EXCENTRAL TRIANGLE

Trilinears sin B/2 + sin C/2 - sin A/2 : sin C/2 + sin A/2 - sin B/2 : sin A/2 + sin B/2 - sin C/2
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 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(165) = CENTROID OF THE EXCENTRAL TRIANGLE

Trilinears    tan(B/2) + tan(C/2) - tan(A/2) : tan(C/2) + tan(A/2) - tan(B/2) : tan(A/2) + tan(B/2) - tan(C/2)
Trilinears    3a² - 2a(b + c) - (b - c)² : :
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(166) GERGONNE POINT OF EXCENTRAL TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where
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)

X(167) NAGEL POINT OF EXCENTRAL TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        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)

X(168) MITTENPUNKT OF EXCENTRAL TRIANGLE

Trilinears sin A - sin B - sin C + 2[cos A/2 + sin(B/2 - A/2) + sin(C/2 - A/2)] : :
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 triangle
X(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

X(169) = X(85)-CEVA CONJUGATE OF X(1)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where
f(A,B,C) = - (sin A)cos²(A/2) + (sin B)cos²(B/2) + (sin C)cos²(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(169) = X(32)-of-excentral triangle.

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)

X(170) = X(9)-ALEPH CONJUGATE OF X(9)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where
f(A,B,C) = - (tan A/2)sec²(A/2) + (tan B/2)sec²(B/2) + (tan C/2)sec²(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(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)

X(171) = {X(2), X(31)}-HARMONIC CONJUGATE OF X(238)

Trilinears a² + bc : b² + ca : c² + ab
Barycentrics a³ + abc : b³ + abc : c³ + abc

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) = 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)}}

X(172) = TRILINEAR PRODUCT X(6)*X(171)

Trilinears a³ + abc : b³ + abc : c³ + abc
Barycentrics a⁴ + bca² : b⁴ + cab² : c⁴ + abc²

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

X(173) = CONGRUENT ISOSCELIZERS POINT

Trilinears    cos B/2 + cos C/2 - cos A/2 : :
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_BQ_C is an isosceles triangle. The line P_BQ_C is called an isoscelizer. The lines P_BQ_C, P_CQ_A, P_AQ_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)

X(174) = YFF CENTER OF CONGRUENCE

Trilinears    sec A/2 : sec B/2 : sec C/2
Trilinears    [bc/(b + c - a)]^1/2 : [ca(c + a - b)]^1/2 : [ab(a + b - c)]^1/2
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_BQ_C, P_CQ_A, P_AQ_B, be constructed so that, on putting

R_A = P_AQ_B∩P_BQ_C, R_B = P_BQ_C∩P_CQ_A, R_C = P_CQ_A∩P_AQ_B,

the following four triangles are congruent:

P_AQ_AR_A, P_BQ_BR_B, P_CQ_CR_C, R_AR_BR_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_AR_BR_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) = (q² + r² + 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(175) = ISOPERIMETRIC POINT

Trilinears    -1 + sec A/2 cos B/2 cos C/2 : :
Barycentrics    (sin A)(-1 + sec A/2 cos B/2 cos C/2) : :
Barycentrics    -2a + (a + b + c) tan(A/2) : :
X(175) = 2s*X(1) - (r + 4R)*X(7)

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.

X(175) lies on these lines: 1,7 8,1270 174,483 226,1131 490,664 651,1335

X(175) = X(8)-Ceva conjugate of X(176)
X(175) = X(664)-beth conjugate of X(175)
X(175) = {X(1),X(7)}-harmonic conjugate of X(176)
X(175) = anticomplement of X(14121)
X(175) = {X(481),X(482)}-harmonic conjugate of X(1374)
X(175) = X(6406)-of-excentral-triangle

X(176) = EQUAL DETOUR POINT

Trilinears    1 + sec A/2 cos B/2 cos C/2 : :
Barycentrics    (sin A)(1 + sec A/2 cos B/2 cos C/2)

Barycentrics    2a + (a + b + c) tan(A/2) : :
X(176) = 2s*X(1) + (r + 4R)*X(7)

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 these lines: 1,7 8,1271 174,1143 226,1132 489,664 651,1124

X(176) = X(8)-Ceva conjugate of X(175)
X(176) = X(664)-beth conjugate of X(176)
X(176) = {X(1),X(7)}-harmonic conjugate of X(175)
X(176) = {X(481),X(482)}-harmonic conjugate of X(1373)
X(176) = X(6291)-of-excentral-triangle

X(177) = 1st MID-ARC POINT

Trilinears    (cos B/2 + cos C/2) sec A/2 : (cos C/2 + cos A/2) sec B/2 : (cos A/2 + cos B/2) sec C/2
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

X(178) = 2nd MID-ARC POINT

Trilinears (cos B/2 + cos C/2) csc A : (cos C/2 + cos A/2) csc B : (cos A/2 + cos B/2) csc C
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)

X(179) = 1st AJIMA-MALFATTI POINT

Trilinears sec⁴(A/4) : sec⁴(B/4) : sec⁴(C/4)
Barycentrics sin A sec⁴(A/4) : sin B sec⁴(B/4) : sin C sec⁴(C/4)

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 csc⁴(A/4) : csc⁴(B/4) : csc⁴(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.

If you have The Geometer's Sketchpad, you can view X(179).

X(179) lies on this line: 1,1142

X(180) = 2nd AJIMA-MALFATTI POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/t(B,C,A) + 1/t(C,B,A) - 1/t(A,C,B),
t(A,B,C) = 1 + 2(sec A/4 cos B/4 cos C/4)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)

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

X(181) = APOLLONIUS POINT

Trilinears a(b + c)²/(b + c - a) : b(c + a)²/(c + a - b) : c(a + b)²/(a + b - c)
= a²cos²(B/2 - C/2) : b²cos²(C/2 - A/2) : c²cos²(A/2 - B/2)

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 a³cos²(B/2 - C/2) : b³cos²(C/2 - A/2) : c³cos²(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)

X(182) = MIDPOINT OF BROCARD DIAMETER

Trilinears       cos(A- ω) : cos(B - ω) : cos(C -ω)
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) = X(3) + X(6)
X(182) is the midpoint of the Brocard diameter (the segment X(3)-to-X(6)); also the center of the 1st Lemoine circle, and the center of the Brocard circle. If you have The Geometer's Sketchpad, you can view X(1316), which includes X(182).

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) = 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(183) = TRILINEAR PRODUCT X(75)X(182)

Trilinears    b²c²cos(A- ω) : c²a²cos(B - ω) : a²b²cos(C - ω)
Barycentrics   csc A cos(A - ω) : csc B cos(B - ω) : csc C cos(C - ω)
Barycentrics    cot A + tan ω : :
X(183) = 3*X(2) - 2(cos ω)²*X(6)

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
X(182) = endo-similarity image of reflection triangles of PU(1); the similitude center of these triangles is X(6)

X(184) = INVERSE OF X(125) IN THE BROCARD CIRCLE

Trilinears       a²cos A : :
Trilinears        sin A sin 2A : :
Barycentrics  a³cos A : :

X(184) is the homothetic center of triangles ABC and A'B'C', the latter defined as follows: let B₁ and C₁ be the points where the perpendicular bisector BC meets sidelines CA and AB, and cyclically define C₂, A₂; A₃, B₃. Then A'B'C' is formed by the perpendicular bisectors of segments B₁C₁, C₂A₂, A₃B₃. (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_CC_B, XC_AA_C, XA_BB_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)

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) = 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

X(185) = NAGEL POINT OF THE ORTHIC TRIANGLE

Trilinears    (cos A)[1 - cos A cos(B - C)] : :
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(186) = INVERSE-IN-CIRCUMCIRCLE OF X(4)

Trilinears    4 cos A - sec A : 4 cos B - sec B : 4 cos C - sec C
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) : :
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).

X(186) lies on these lines: 2,3 54,389 93,252 98,935 107,477 112,187 249,250

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(187) = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER)

Trilinears    a(2a² - b² - c²) : :
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    a²(2a² - b² - c²) : :
X(187) = X(15) + X(16)

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)

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) = 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)

X(188) = 2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE

Trilinears    csc A/2 : csc B/2 : csc C/2
Trilinears    [bc(b + c - a)]^1/2 : :
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(189) = CYCLOCEVIAN CONJUGATE OF X(8)

Trilinears bc/(cos B + cos C - cos A - 1) : ca/(cos C + cos A - cos B - 1) : ab/(cos A + cos B - cos C - 1)
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 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(190) = YFF PARABOLIC POINT

Trilinears bc/(b - c) : ca/(c - a) : ab/(a - b)
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)

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)

Centers 191- 236
are Ceva conjugates. The P-Ceva conjugate of Q is the perspector
of the cevian triangle of P and the anticevian triangle of Q.
underbar

X(191) = X(10)-CEVA CONJUGATE OF X(1)

Trilinears (b + c - a)(bc + ca + ab) + b³ + c³ - a³ : :
Trilinears S_A + rR : S_B + rR : S_C + rR

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(192) = X(1)-CEVA CONJUGATE OF X(2)
(CONGRUENT PARALLELIANS POINT)

Trilinears bc(ca + ab - bc) : ca(ab + bc - ca) : ab(bc + ca - ab)
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(193) = X(4)-CEVA CONJUGATE OF X(2)

Trilinears       (csc A)(cot B + cot C - cot A) : (csc B)(cot C + cot A - cot B) : (csc C)(cot A + cot B - cot C)
Trilinears        (S_A - a²)/a : (S_B - b²)/b : (S_C - c²)/c
Barycentrics  cot B + cot C - cot A : cot C + cot A - cot B : cot A + cot B - cot C
                        = 3a² - b² - c² : 3b² - c² - a² : 3c² - a² - b² (Milorad Stevanovic, 5/12/2003)

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) = 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) = 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

X(194) = X(6)-CEVA CONJUGATE OF X(2)

Trilinears    bc[a²b² + a²c² - b²c²] : :
Barycentrics    a²b² + a²c² - b²c² : :
Barycentrics    cot²A - csc²A cos 2ω : :      (M. Iliev, 5/13/07)
X(194) = 2X(39) - X(76) - P(1) + U(1) - X(76)

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) = 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(195) = X(5)-CEVA CONJUGATE OF X(3)

Trilinears    (cos A)(v + w - u) : : , where u = u(A,B,C) = cos A cos(B - A) cos(C - A)
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 + cot²A - cot A cot ω : :(   (M. Iliev, 5/13/07)

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) = 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]

X(196) = X(7)-CEVA CONJUGATE OF X(4)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos B + cos C - cos A - 1) sec A tan A/2
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(197) = X(8)-CEVA CONJUGATE OF X(6)

Trilinears a[-a²tan A/2 + b²tan B/2 + c²tan C/2] : :

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)

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

X(198) = X(9)-CEVA CONJUGATE OF X(6)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a(cos B + cos C - cos A - 1)
Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

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(199) = X(10)-CEVA CONJUGATE OF X(6)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b⁴ + c⁴ - a⁴ + (b² + c² - a²)(bc + ca + ab)]
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 + $bc$).

X(199) lies on these lines: 2,3 42,172 51,572 55,1030 184,573

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(200) = X(8)-CEVA CONJUGATE OF X(9)

Trilinears    cot²(A/2) : :
Trilinears    (b + c - a)² : :
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)² : :

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(201) = X(10)-CEVA CONJUGATE OF X(12)

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

X(202) = X(1)-CEVA CONJUGATE OF X(15)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = u(v + w - u),
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)

X(203) = X(1)-CEVA CONJUGATE OF X(16)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = u(v + w - u),
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)

X(204) = X(1)-CEVA CONJUGATE OF X(19)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan A)(tan B + tan C - tan A)
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)

X(205) = X(9)-CEVA CONJUGATE OF X(31)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²[b²tan B/2 + c²tan C/2 - a²tan A/2]
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)

X(206) = X(2)-CEVA CONJUGATE OF X(32)

Trilinears a³(b⁴ + c⁴ - a⁴) : :
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) = 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

X(207) = X(1)-CEVA CONJUGATE OF X(34)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 - sec A)(sec B + sec C - sec A - 1)
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(208) = X(4)-CEVA CONJUGATE OF X(34)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 - sec A)(cos B + cos C - cos A - 1)
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)

X(209) = X(4)-CEVA CONJUGATE OF X(37)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin B + sin C)[sin A + sin(A - B) + sin(A - C)]
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)

X(210) = X(10)-CEVA CONJUGATE OF X(37)

Trilinears       (b + c)(b + c - a) : (c + a)(c + a - b) : (a + b)(a + b - c)
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) = 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(211) = X(4)-CEVA CONJUGATE OF X(39)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C)
= sin(A + ω)[cos B sin(B + ω) + cos C sin(C + ω) - cos A sin(A + ω)]

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(211) lies on these lines: 5,141 32,184 52,114

X(211) = X(4)-Ceva conjugate of X(39)

X(212) = X(9)-CEVA CONJUGATE OF X(41)

Trilinears       (cos A)(1 + cos A) : (cos B)(1 + cos B) : (cos C)(1 + cos C)
                        = (cos A)cos²(A/2) : (cos B)cos²(B/2) : (cos C)cos²(C/2)
                        = a²(b + c - a)(b² + c² - a²) : b²(c + a - b)(c² + a² - b²) : c²(a + b - c)(a² + b² - c²)

Barycentrics (sin 2A)(1 + cos A) : (sin 2B)(1 + cos B) : (sin 2C)(1 + cos C)

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) = 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)

X(213) = X(6)-CEVA CONJUGATE OF X(42)

Trilinears       (b + c)a² : (c + a)b² : (a + b)c²
Trilinears       a²(ar - S) : b²(br - S) : c²(cr - S)
Barycentrics  (b + c)a³ : (c + a)b³ : (a + b)c³

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) = 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) = X(2)-CEVA CONJUGATE OF X(44)

Trilinears (b + c - 2a)(b² + c² - a² - bc) : :

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) = QA-P3 (Gergonne-Steiner Point) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/quadrangle-objects/10-mathematics/quadrangle-objects/18-qa-p3.html)

X(215) = X(1)-CEVA CONJUGATE OF X(50)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin 3A)(sin 3B + sin 3C - sin 3A)
Trilinears       cos²(3A/2) : cos²(3B/2) : cos²(3C/2)     (M. Iliev, 4/12/07)
Trilinears       1 + cos 3A : 1 + cos 3B : 1+ cos 3C     (M. Iliev, 4/12/07)

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)

X(216) = X(5)-CEVA CONJUGATE OF X(51)

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

X(217) = X(6)-CEVA CONJUGATE OF X(51)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin³A) cos A cos(B - C)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(217) lies on these lines: 4,6 32,184 39,185 54,112 83,287 232,389

X(217) = isogonal conjugate of X(276)
X(217) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,51), (216,418)
X(217) = crosspoint of X(i) and X(j) for these (i,j): (6,184), (51,216)
X(217) = crosssum of X(i) and X(j) for these (i,j): (2,264), (95,275)
X(217) = crossdifference of every pair of points on line X(340)X(520)

X(218) = X(7)-CEVA CONJUGATE OF X(55)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where
f(A,B,C) = cos²(A/2) [cos⁴(B/2) + cos⁴(C/2) - cos⁴(A/2)]

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

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)

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

X(219) = X(8)-CEVA CONJUGATE OF X(55)

Trilinears    cos A cot A/2 : :
Trilinears    (sin A)/(1 - sec A) : :
Trilinears    1/(csc A - 2 csc 2A) : :
Trilinears    a(b + c - a)(b² + c² - a²) : :
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(220) = X(9)-CEVA CONJUGATE OF X(55)

Trilinears    a(b + c - a)² : :
Trilinears    (1 + cos A)²/sin A : :     (M. Iliev, 4/12/07)
Barycentrics    a²(b + c - a)² : :

The trilinear polar of X(220) passes through X(657) (Randy Hutson, July 20, 2016)

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(221) = X(1)-CEVA CONJUGATE OF X(56)

Trilinears (sin²A/2)(cos B + cos C - cos A - 1) : :
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) = trilinear product X(40)*X(56)
X(221) = X(92)-isoconjugate of X(268
) X(221) = perspector of unary cofactor triangles of 1st and 3rd extouch triangles

X(222) = X(7)-CEVA CONJUGATE OF X(56)

Trilinears       cos A tan A/2 : cos B tan B/2 : cos C tan C/2
                        = 1/(csc A + 2 csc 2A) : 1/(csc B + 2 csc 2B) : 1/(csc A + 2 csc 2C)
                        = a(b² + c² - a²)/(b + c - a) : b(c² + a² - b²)/(c + a - b) : c(a² + b² - c²)/(a + b - c)

Barycentrics a²/(1 + sec A) : b²/(1 + sec B) : c²/(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) = 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) = 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)

X(223) = X(2)-CEVA CONJUGATE OF X(57)

Trilinears    (tan A/2)(cos B + cos C - cos A - 1)
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)

X(223) lies on the Thomson cubic and these lines:
1,4 2,77 3,1035 6,57 9,1073 40,221 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)

X(224) = X(7)-CEVA CONJUGATE OF X(63)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where
f(A,B,C) = [cot B cos²(B/2) + cot C cos²(C/2) - cot A cos²(A/2)]cot A

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

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(225) = X(4)-CEVA CONJUGATE OF X(65)

Trilinears (sec A)(cos B + cos C) : (sec B)(cos C + cos A) : (sec C)(cos A + cos B)
Barycentrics (tan A)(cos B + cos C) : (tan B)(cos C + cos A) : (tan C)(cos A + cos B)

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(226) = X(7)-CEVA CONJUGATE OF X(65)

Trilinears    (csc A)(cos B + cos C) : (csc B)(cos C + cos A) : (csc C)(cos A + cos B)
Trilinears    bc(b + c)/(b + c - a) : ca(c + a)/(c + a - b) : ab(a + b)/(a + b - c)
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

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. 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)

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(227) = X(10)-CEVA CONJUGATE OF X(65)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(cos B + cos C - cos A - 1)tan A/2
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)

X(228) = X(3)-CEVA CONJUGATE OF X(71)

Trilinears (sin 2A)(sin B + sin C) : (sin 2B)(sin C + sin A) : (sin 2C)(sin A + sin B)
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(229) = X(7)-CEVA CONJUGATE OF X(81)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (v + w - u)/(b + c),
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(2)-CEVA CONJUGATE OF X(114)

Trilinears bc[a²(2a² - b² - c²) + (b² - c²)²] : :
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

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)

X(231) = X(2)-CEVA CONJUGATE OF X(128)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = u(-au + bv + cw), u : v : w = X(128)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

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(232) = X(2)-CEVA CONJUGATE OF X(132)

Trilinears tan A cos(A + ω) : tan B cos(B + ω) : tan C cos(C + ω)
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) = 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(233) = X(2)-CEVA CONJUGATE OF X(140)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [b cos(C - A) + c cos(B - A)]cos(B - C)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

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) = 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(234) = X(7)-CEVA CONJUGATE OF X(177)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos B/2 + cos C/2)(cos B/2 cos C/2)²
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(235) = X(4)-CEVA CONJUGATE OF X(185)

Trilinears    sec A - cos(B - C) : :
Trilinears    (sec A)(cos²B + cos²C) : :
Barycentrics    (tan A)(cos²B + cos²C) : :

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) = 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)

X(236) = X(2)-CEVA CONJUGATE OF X(188)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A/2)(cos B/2 + cos C/2 - cos A/2)
Trilinears [1 + sin(A/2)]/sin A : [1 + sin(B/2)]/sin B : [1 + sin(C/2)]//sin C (M. Iliev, 4/12/07)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(236) lies on these lines: 2,174 8,178 9,173

X(236) = isogonal conjugate of X(289)
X(236) = X(2)-Ceva conjugate of X(188)

Centers 237- 248
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.
underbar

X(237) = X(3)-LINE CONJUGATE OF X(2)

Trilinears a²cos(A + ω) : b²cos(B + ω) : c²cos(C + ω)
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a³(b⁴ + c⁴ - a²b² - a²c²) (Darij Grinberg, 3/29/03)

Barycentrics a³cos(A + ω) : b³cos(B + ω) : c³cos(C + ω)

As a point on the Euler line, X(237) has Shinagawa coefficients (EF + F² + S², -(E + F)² - S²).

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) = 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)

X(238) = X(1)-LINE CONJUGATE OF X(37)

Trilinears a² - bc : b² - ca : c² - ab
Barycentrics a³ - abc : b³ - abc : c³ - abc

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(239) = X(1)-LINE CONJUGATE OF X(42)

Trilinears bc(a² - bc) : :
Barycentrics a² - bc : :

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) = 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(240) = X(1)-LINE CONJUGATE OF X(48)

Trilinears sec A cos(A + ω) : sec B cos(B + ω) : sec C cos(C + ω)
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)

X(241) = X(1)-LINE CONJUGATE OF X(55)

Trilinears cos⁴B/2 - [cos²(A/2)][cos²(B/2) +cos²(C/2)] + cos⁴(C/2) : :
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)

X(242) = X(4)-LINE CONJUGATE OF X(71)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)(sin²A - sin B sin C)
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) = X(4)-LINE CONJUGATE OF X(73)

Trilinears (sec A)(cos²A - cos B cos C) : :

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)

X(244) = X(1)-LINE CONJUGATE OF X(100)

Trilinears    (b - c)² : (c - a)² : (a - b)²
Trilinears    [1 - cos(B - C)]sin²(A/2) : :
Trilinears    d(a,b,c) : : , where d(a,b,c) = (distance from A to Nagel line)²
Barycentrics    a(b - c)² : :

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) = 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)

X(245) = X(1)-LINE CONJUGATE OF X(110)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where
f(A,B,C) = csc²(C - A) + csc(C - B) [csc(C - A) -csc(B - A)] + csc²(A - B)

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)

X(246) = X(3)-LINE CONJUGATE OF X(110)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        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)

X(247) = X(4)-LINE CONJUGATE OF X(110)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        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)

X(248) = X(4)-LINE CONJUGATE OF X(132)

Trilinears sin 2A sec(A + ω) : sin 2B sec(B + ω) : sin 2C sec(C + ω)
Barycentrics sin A sin 2A sec(A + ω) : sin B sin 2B sec(B + ω) : sin C sin 2C sec(C + ω)

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)

Centers 249- 297
are isogonal conjugates of previously listed centers.
underbar

X(249) = ISOGONAL CONJUGATE OF X(115)

Trilinears (csc A)csc²(B - C) : (csc B)csc²(C - A) : (csc C)csc²(A - B)
= a/(b² - c²)² : b/(c² - a²)² : c/(a² - b²)²

Barycentrics csc²(B - C) : csc²(C - A) : csc²(A - B)

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)
X(249) = X(i)-cross conjugate of X(j) for these (i,j): (3,99), (6,110)

X(250) = ISOGONAL CONJUGATE OF X(125)

Trilinears (sec A)csc²(B - C) : (sec B)csc²(C - A) : (sec C)csc²(A - B)
= (a²sec A)/(b² - c²)² : (b²sec B)/(c² - a²)² : (c²sec C)/(a² - b²)²

Barycentrics (tan A)csc²(B - C) : (tan B)csc²(C - A) : (tan C)csc²(A - B)

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)
X(250) = X(i)-cross conjugate of X(j) for these (i,j): (3,110), (22,99), (24,107), (25,112), (199,101)

X(251) = ISOGONAL CONJUGATE OF X(141)

Trilinears a²csc(A + ω) : b²csc(B + ω) : c²csc(C + ω)
= a/(b² + c²) : b/(c² + a²) : c/(a² + b²)

Barycentrics a³csc(A + ω) : b³csc(B + ω) : c³csc(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(252) = ISOGONAL CONJUGATE OF X(143)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where
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)

X(252) lies on these lines: 3,930 54,140 93,186

X(252) = isogonal conjugate of X(143)

X(253) = X(4)-CROSS CONJUGATE OF X(2)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)/(tan B + tan C - tan A)
= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (csc²A)/(cos A - cos B cos C)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/(tan B + tan C - tan A)

X(253) is the perspector of ABC and the pedal triangle of X(64).

X(253) lies on these lines: 2,1073 7,280 8,307 20,64 193,287 306,329 318,342 322,341

X(253) = isogonal conjugate of X(154)
X(253) = isotomic conjugate of X(20)
X(253) = cyclocevian conjugate of X(69)
X(253) = cevapoint of X(i) and X(j) for these (i,j): (4,459), (122,525)
X(253) = X(i)-cross conjugate of X(j) for these (i,j): (4,2), (122,525)

X(254) = X(3)-CROSS CONJUGATE OF X(4)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)/(cos²B + cos²C - cos²A)
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (tan A)/(cos²B + cos²C - cos²A)

X(254) lies on these lines: 2,847 4,155 24,393 46,225 68,136

X(254) = isogonal conjugate of X(155)
X(254) = cevapoint of X(136) and X(523)
X(254) = X(3)-cross conjugate of X(4)

X(255) = ISOGONAL CONJUGATE OF X(158)

Trilinears cos²A : cos²B : cos²C
= 1 + cos 2A : 1 + cos 2B : 1 + cos 2C

Barycentrics sin A cos²A : sin B cos²B : sin C cos²C

X(255) lies on these lines: 1,21 3,73 35,991 36,1106 40,109 48,563 55,601 56,602 57,580 91,1109 92,1087 158,775 162,1099 165,1103 200,271 201,1060 219,268 293,304 326,1102 411,651 498,750 499,748

X(255) = isogonal conjugate of X(158)
X(255) = X(i)-Ceva conjugate of X(j) for these (i,j): (63,48), (283,3)
X(255) = crosspoint of X(63) and X(326)
X(255) = crosssum of X(i) and X(j) for these (i,j): (1,290), (4,1068), (19,1096)
X(255) = X(i)-aleph conjugate of X(j) for these (i,j): (775,255), (1105,158)

X(256) = 1st SHARYGIN POINT

Trilinears 1/(a² + bc) : 1/(b² + ca) : 1/(c² + ab)
Barycentrics a/(a² + bc) : b/(b² + ca) : c/(c² + ab)

See the description at X(1281). The lines AD, BE, CF defined there concur in X(256).

X(256) lies on these lines: 1,511 3,987 4,240 7,982 8,192 9,43 21,238 37,694 40,989 55,983 84,988 104,1064 291,894 314,350 573,981

X(256) = isogonal conjugate of X(171)
X(256) = isotomic conjugate of X(1909)
X(256) = X(239)-cross conjugate of X(291)
X(256) = crosssum of X(43) and X(846)
X(256) = X(238)-Hirst inverse of X(904)
X(256) = cevapoint of PU(6)
X(256) = trilinear pole of line X(650)X(3250)

X(257) = ISOGONAL CONJUGATE OF X(172)

Trilinears 1/(a³ + abc) : 1/(b³ + abc) : 1/(c³ + abc)
Barycentrics 1/(a² + bc) : 1/(b² + ca) : 1/(c² + ab)

X(257) lies on these lines: 1,385 8,192 29,242 65,894 75,698 92,297 194,986 239,333 330,982 335,694

X(257) = isogonal conjugate of X(172)
X(257) = isotomic conjugate of X(894)
X(257) = X(350)-cross conjugate of X(335)
X(257) = X(239)-Hirst inverse of X(893)
X(257) = trilinear pole of line X(522)X(1491)
X(257) = cevapoint of PU(10)
X(257) = pole wrt polar circle of line X(2533)X(3287)
X(257) = X(48)-isoconjugate (polar conjugate) of X(7009)

X(258) = CONGRUENT INCIRCLES ISOSCELIZER POINT

Trilinears    1/(cos B/2 + cos C/2 - cos A/2) : :
Trilinears    1 + sin(B/2) + sin(C/2) - sin(A/2) : :
Trilinears    tan(A/2) - sec(A/2) : :
Trilinears    tan(B/2) - sec(B/2) : tan(C/2) - sec(C/2) : :
Trilinears    1/(b' + c' - a') : : , where A'B'C' is the excentral triangle
Trilinears    tan A'/2 : : , where A'B'C' is the excentral triangle
Trilinears    cot A' - csc A' : : , where A'B'C' is the excentral triangle
Trilinears    (distance from A to A-excircle) : :

In Yff's isoscelizer configuration, if X = X(258), then the isosceles triangles T_A, T_B, T_C have congruent incircles.

If you have The Geometer's Sketchpad, you can view X(258).

X(258) lies on these lines: 1,164 57,173 259,289

X(258) = isogonal conjugate of X(173)
X(258) = X(259)-cross conjugate of X(1)
X(258) = X(366)-aleph conjugate of X(363)
X(258) = SS(a->a') of X(57), where A'B'C' is the excentral triangle (trilinear substitution)
X(258) = X(33)-of-excentral-triangle
X(258) = homothetic center of ABC and intangents triangle of excentral triangle
X(258) = insimilicenter of incircle and incircle of excentral triangle
X(258) = {X(1),X(164)}-harmonic conjugate of X(8078)
X(258) = perspector of ABC and the extouch triangle of the intouch triangle

X(259) = ISOGONAL CONJUGATE OF X(174)

Trilinears    cos A/2 : :
Trilinears    [a(b + c - a)]^1/2 : :
Trilinears    sin A csc A/2 : :
Trilinears    sin A' : : , where A'B'C' is the excentral triangle
Trilinears    sin(∠BIC) : :
Barycentrics  sin A cos A/2 : :

X(259) lies on these lines: 1,168 258,289 260,266

X(259) = isogonal conjugate of X(174)
X(259) = X(i)-Ceva conjugate of X(j) for these (i,j): (174,266), (260,55)
X(259) = cevapoint of X(1) and X(503)
X(259) = crosspoint of X(i) and X(j) for these (i,j): (1,258), (174,188)
X(259) = crosssum of X(i) and X(j) for these (i,j): (1,173), (259,266)
X(259) = SS(A->A') of X(6), where A'B'C' is the excentral triangle
X(259) = trilinear square root of X(55)
X(259) = perspector of ABC and unary cofactor triangle of tangential mid-arc triangle

X(260) = ISOGONAL CONJUGATE OF X(177)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A/2)/(cos B/2 + cos C/2)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(260) lies on these lines: 1,3 259,266

X(260) = isogonal conjugate of X(177)
X(260) = cevapoint of X(55) and X(259)

X(261) = ISOTOMIC CONJUGATE OF X(12)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [(csc A)(sec(B/2 - C/2))]²
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(261) lies on these lines:
2,593 9,645 21,314 28,242 58,86 75,99 272,310 284,332 317,406 319,502 552,873 572,662

X(261) = isogonal conjugate of X(181)
X(261) = isotomic conjugate of X(12)
X(261) = X(873)-Ceva conjugate of X(1509)
X(261) = cevapoint of X(21) and X(333)

X(262) = ISOGONAL CONJUGATE OF X(182)

Trilinears    sec(A - ω) : sec(B - ω) : sec(C - ω)
Barycentrics    sin A sec(A - ω) : sin B sec(B - ω) : sin C sec(C - ω)
Barycentrics    1/(a^4 - a^2b^2 - a^2c^2 - 2b^2c^2) : :

Let A'B'C' be the orthic triangle. X(262) is the radical center of the Brocard circles of AB'C', BC'A', CA'B'. (Randy Hutson, February 10, 2016)

Let Pa be the parabola with focus A and directrix BC. Let La be the polar of X(3) wrt Pa and define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The lines AA', BB', CC' concur in X(262). Also, X(262) is also the isotomic conjugate, wrt A'B'C', of X(3).

Let A' be the apex of the isosceles triangle BA'C constructed outward on BC such that ∠A'BC = ∠A'CB = ω. Define B', C' cyclically. Let Ha be the orthocenter of BA'C, and define Hb, Hc cyclically. The lines AHa, BHb, CHc concur in X(262). (Randy Hutson, July 20, 2016)

Let A' be the apex of the isosceles triangle BA'C constructed intward on BC such that ∠A'BC = ∠A'CB = ω/2. Define B', C' cyclically. Let Oa be the circumcenter of BA'C, and define Ob, Oc cyclically. The lines AOa, BOb, COc concur in X(262). (Randy Hutson, July 20, 2016)

X(262) lies on these lines: 2,51 3,83 4,39 5,76 6,98 13,383 14,1080 25,275 30,598 226,982 381,671 385,576

X(262) = midpoint of X(4) and X(7709)
X(262) = isogonal conjugate of X(182)
X(262) = isotomic conjugate of X(183)
X(262) = radical center of (Brocard circle reflected in BC, CA, and AB)
X(262) = pole wrt polar circle of trilinear polar of X(458)
X(262) = X(48)-isoconjugate (polar conjugate) of X(458)
X(262) = trilinear pole of line X(523)X(3569)
X(262) = pole of Lemoine axis wrt orthoptic circle of the Steiner inellipse (a.k.a. {circumcircle, nine-point circle}-inverter)
X(262) = perspector of orthoptic circle of the Steiner inellipse (a.k.a. {circumcircle, nine-point circle}-inverter)
X(262) = perspector of ABC and 2nd Neuberg triangle
X(262) = trilinear product of vertices of 2nd Neuberg triangle
X(262) = centroid of X(4)PU(1)
X(262) = complement of X(6194)
X(262) = Cundy-Parry Phi transform of X(83)
X(262) = Cundy-Parry Psi transform of X(39)

X(263) = ISOGONAL CONJUGATE OF X(183)

Trilinears a²sec(A - ω) : :
Barycentrics a³sec(A - ω) : :

Let V = U(2)-of-pedal-triangle-of-P(1), and let W = P(2)-of-pedal-triangle-of-U(1). Then X(263) = trilinear pole of VW. (Randy Hutson, December 26, 2015)

Let A1B1C1 and A2B2C2 be the pedal triangles of PU(1). Then X(263) is the radical center of the circumcircles of AA1A2, BB1B2, CC1C2. (Randy Hutson, July 31 2018)

X(263) lies on these lines: 2,51 6,160 69,308 184,251

X(263) = isogonal conjugate of X(183)

X(264) = ISOTOMIC CONJUGATE OF CIRCUMCENTER

Trilinears    csc A csc 2A : csc B csc 2B : csc C csc 2C
Trilinears    sec A csc²A : sec B csc²B : sec C csc²C
Trilinears    tan A csc(A - ω) : tan B csc(B - ω) : tan C csc(C - ω)
Trilinears    sec A + cot A csc A : :
Barycentrics    csc 2A : csc 2B : csc 2C
Barycentrics    1/[a²(a² - b² - c²)] : :
Barycentrics    tan A + cot A : :
Barycentrics    tan A - cot B - cot C + cot ω : :

Five constructions by Randy Hutson, January 29, 2015:

(1) Let A'B'C' be the tangential triangle. Let A" be the pole, wrt the polar circle, of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(264).

(2) Let A'B'C' be the symmedial triangle. Let A" be the pole, wrt the polar circle, of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(264).

(3) Let A'B'C' be the circumsymmedial triangle. Let A" be the pole, wrt the polar circle, of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(264).

(4) Let A'B'C' be the Lucas(t) central triangle (for any t). Let A" be the pole, wrt the polar circle, of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(264).

(5) X(264) is the trilinear pole of the line X(297)X(525). This line is the isotomic conjugate of the MacBeath circumconic, which is the isogonal conjugate of the orthic axis. The line is also the polar of X(6) wrt the polar circle, and the radical axis of the polar and orthosymmedial circles, and the polar conjugate of the circumcircle)

Let A' be the trilinear product of the vertices of the A-anti-altimedial triangle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(264). (Randy Hutson, November 2, 2017)

X(264) lies on these cubics: K045, K146, K183, K184, K208, K257, K276, K342a, K342b, K494, K504, K647, K674, K675, K677, K718

X(264) lies on these lines:
{2,216}, {3,95}, {4,69}, {5,1093}, {6,287}, {9,1948}, {22,1629}, {24,1078}, {25,183}, {33,350}, {34,1909}, {53,141}, {57,1947}, {75,225}, {85,309}, {92,306}, {93,1273}, {98,3425}, {99,378}, {107,1995}, {112,2367}, {157,1485}, {186,7771}, {193,3087}, {250,1316}, {253,3091}, {254,3541}, {262,2967}, {274,475}, {275,1993}, {281,344}, {298,472}, {299,473}, {300,302}, {301,303}, {305,325}, {310,4196}, {319,5081}, {320,7282}, {328,6344}, {339,381}, {379,823}, {384,1968}, {401,577}, {419,1974}, {450,5651}, {491,1585}, {492,1586}, {524,6748}, {623,6116}, {624,6117}, {801,2063}, {811,5136}, {847,1594}, {850,7703}, {1007,6340}, {1043,7513}, {1105,1593}, {1217,3088}, {1225,7809}, {1238,7796}, {1249,3618}, {1309,2861}, {1441,2476}, {1595,3933}, {1726,7094}, {1785,4357}, {1896,2478}, {1897,4360}, {1969,3262}, {1990,3589}, {2207,7770}, {2419,3267}, {2453,3447}, {2897,6840}, {2970,5094}, {3148,6394}, {3168,5943}, {3199,3934}, {3520,7782}, {3575,7750}, {3629,6749}, {3785,7487}, {5064,7788}, {5117,6374}, {5523,7790}, {6103,7806}, {6240,7802}, {6524,7392}, {6525,7398}, {6756,7767}, {7378,8024}, {7507,7773}, {7576,7811}

X(264) = reflection of X(3164) in X(216)
X(264) = isogonal conjugate of X(184)
X(264) = isotomic conjugate of X(3)
X(264) = complement of X(3164)
X(264) = anticomplement of X(216)
X(264) = X(264) = X(i)-Ceva conjugate of X(j) for these (i,j): (276,2), (1969,7017), (6528,850)
X(264) = cevapoint of X(i) and X(j) for these (i,j): (2,4), (5,324), (6,157), (92,318), (273,342), (338,523), (491,492)
X(264) = X(i)-cross conjugate of X(j) for these (i,)}: (2,76), (3,5392), (4,2052), (5,2), (30,94), (92,331), (235,459), (318,7017), (339,850), (427,4), (442,321), (523,648), (850,6528), (858,671), (1312,2593), (1313,2592), (1368,2996), (1441,75), (1591,5490), (1592,5491), (1594,275), (2072,2986), (2450,98), (2967,297), (2968,4391), (2971,2501), (2972,525), (3007,903), (3134,2394), (3136,10), (3141,4049), (3142,226), (3143,5466), (5133,83), (5169,598), (6530,6330), (6563,99)
X(264) = X(1988)-complementary conjugate of X(10)
X(264) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (54,6360), (92,2888), (95,4329), (275,8), (276,6327), (933,4560), (2148,3164), (2167,20), (2190,2)
X(264) = antipode of X(1972) in hyperbola {}A,B,C,X(2),X(69)}}
X(264) = pole of Lemoine axis wrt polar circle
X(264) = X(48)-isoconjugate (polar conjugate) of X(6)
X(264) = polar-circle inverse of X(5167)
X(264) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,324,2052), (2,3164,216), (4,69,317), (4,1235,76), (4,3186,1843), (53,141,297), (69,311,76), (69,317,340), (273,318,75), (311,3260,69)
X(264) = Brianchon point (perspector) of the MacBeath inconic
X(264) = X(i)-isoconjugate of X(j) for these {i,j}: {1,184}, {3,31}, {6,48}, {19,577}, {25,255}, {28,4055}, {32,63}, {33,7335}, {34,6056}, {41,222}, {42,1437}, {47,2351}, {51,2169}, {55,603}, {56,212}, {58,228}, {69,560}, {71,1333}, {72,2206}, {73,2194}, {77,2175}, {78,1397}, {81,2200}, {97,2179}, {109,1946}, {110,810}, {112,822}, {163,647}, {172,7116}, {213,1790}, {216,2148}, {217,2167}, {219,604}, {220,7099}, {221,2188}, {237,293}, {248,1755}, {251,4020}, {268,2199}, {283,1402}, {284,1409}, {295,2210}, {304,1501}, {305,1917}, {326,1974}, {393,4100}, {394,1973}, {418,2190}, {512,4575}, {563,2165}, {571,1820}, {607,7125}, {608,2289}, {649,906}, {652,1415}, {656,1576}, {662,3049}, {667,1331}, {669,4592}, {692,1459}, {798,4558}, {849,3690}, {895,922}, {904,3955}, {923,3292}, {1092,1096}, {1106,1260}, {1110,3937}, {1176,1964}, {1253,7053}, {1259,1395}, {1332,1919}, {1399,8606}, {1400,2193}, {1407,1802}, {1408,2318}, {1410,2328}, {1433,2187}, {1444,1918}, {1472,7085}, {1473,7084}, {1474,3990}, {1797,2251}, {1798,3725}, {1799,1923}, {1804,2212}, {1813,3063}, {1910,3289}, {1911,7193}, {1914,2196}, {1924,4563}, {1949,1951}, {1950,7107}, {1980,4561}, {2149,7117}, {2150,2197}, {2159,3284}, {2192,7114}, {2203,3682}, {2207,6507}, {2208,7078}, {2300,2359}, {7011,7118}, {7015,7122}
X(264) = X(i)-beth conjugate of X(j) for these (i,j): (264,273), (811,7), (3596,322)
X(264) = trilinear pole of the line (297,525)
X(264) = barycentric product X(i)*X(j) for these {i,j}: {1,1969}, {4,76}, {5,276}, {7,7017}, {8,331}, {19,561}, {25,1502}, {27,313}, {29,349}, {69,2052}, {75,92}, {83,1235}, {85,318}, {93,7769}, {94,340}, {95,324}, {107,3267}, {158,304}, {273,312}, {275,311}, {278,3596}, {281,6063}, {286,321}, {290,297}, {300,470}, {301,471}, {305,393}, {308,427}, {310,1826}, {317,5392}, {326,6521}, {327,458}, {341,1847}, {523,6331}, {525,6528}, {648,850}, {670,2501}, {683,5254}, {693,6335}, {811,1577}, {847,7763}, {1016,2973}, {1088,7101}, {1093,3926}, {1231,1896}, {1240,1848}, {1509,7141}, {1824,6385}, {1897,3261}, {1928,1973}, {1978,7649}, {2489,4609}, {2970,4590}, {3064,4572}, {3114,5117}, {3264,6336}, {6344,7799}, {6386,6591}
X(264) = trilinear product of PU(20) (see Tables: Bicentric Pairs)
X(264) = trilinear product X(i)*X(j) for these {i,j}: {{2,92}, {4,75}, {6,1969}, {7,318}, {8,273}, {9,331}, {10,286}, {19,76}, {25,561}, {27,321}, {28,313}, {29,1441}, {33,6063}, {34,3596}, {57,7017}, {63,2052}, {69,158}, {82,1235}, {85,281}, {91,317}, {162,850}, {225,314}, {240,290}, {242,334}, {253,1895}, {274,1826}, {276,1953}, {278,312}, {279,7101}, {280,342}, {297,1821}, {304,393}, {305,1096}, {307,1896}, {309,7952}, {310,1824}, {311,2190}, {324,2167}, {326,1093}, {336,6530}, {340,2166}, {341,1119}, {346,1847}, {347,7020}, {349,1172}, {394,6521}, {419,1934}, {427,3112}, {514,6335}, {523,811}, {525,823}, {648,1577}, {653,4391}, {656,6528}, {661,6331}, {668,7649}, {693,1897}, {757,7141}, {765,2973}, {799,2501}, {873,7140}, {1088,7046}, {1118,3718}, {1240,1829}, {1446,2322}, {1494,1784}, {1502,1973}, {1748,5392}, {1783,3261}, {1857,7182}, {1861,2481}, {1928,1974}, {1947,7108}, {1948,1952}, {1978,6591}, {2333,6385}, {2489,4602}, {2580,2592}, {2581,2593}, {2969,7035}, {2997,5125}, {3064,4554}, {3113,5117}, {3926,6520}, {4358,6336}, {5342,5936}, {7009,7018}
X(264) = barycentric quotient X(i)/X(j) for these (i,j): (1,48), (2,3), (4,6), (5,216), (6,184), (7,222), (8,219), (9,212), (10,71), (19,31), (25,32), (27,58), (29,284), (33,41), (37,228), (51,217), (63,255), (69,394), (94,265), (95,97), (98,248), (107,112), (162,163), (196,221), (216,418), (232,237), (304,326), (311,343), (445,500)

X(265) = REFLECTION OF X(3) IN X(125)

Trilinears    sin 2A csc 3A : :
Trilinears    1/(4 cos A - sec A) : :
Trilinears    csc(A + π/3) - csc(A - π/3) : :
Trilinears    (cos A)/(1 - 4 cos^2 A) : :
Barycentrics    sin A sin 2A csc 3A : :
Barycentrics    (a^2 - b^2 - c^2)/[(a^2 - b^2 - c^2)^2 - b^2c^2] : :

Let P = X(74), H = X(4), H' =H-of-BCP, H'' = H-of-CAP, and H''' = H-of ABP. Then X(265) is the circumcenter of the cyclic quadrilateral HH'H''H'''. (Randy Hutson, 9/23/2011)

Let A' be the reflection in line BC of the A-vertex of the tangential triangle, and define B' and C' cyclically. The circumcircles of A'BC, B'CA, and C'AB concur in X(265). Let A'' be the reflection in line BC of the A-vertex of the anticevian triangle of X(5), and define B'' and C'' cyclically. The circumcircles of AB''C'', BC''A'', CA''B'' concur in X(265). (Randy Hutson, August 26, 2014)

Let A*B*C* be the Kosnita triangle. Let A' be the orthopole of line B*C*, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(265). (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'' isinversely similar to ABC, with similtude center X(265), and A''B''C'' is perspective to ABC with persepctor X(74). (Randy Hutson, August 26, 2014)

Let A'B'C' be the reflection 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(265). (Randy Hutson, August 26, 2014)

Let A'B'C' be the reflection triangle. Let A" be the trilinear pole of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(5). Let A* be the trilinear pole, wrt A'B'C', of line B"C", and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(265). (Randy Hutson, July 20, 2016)

Let A' be the point such that triangle A'BC is directly similar to the orthic triangle, and define B', C' cyclically. The lines AA', BB', CC' concur in X(265). If 'inversely' is substituted for 'directly', the lines concur in X(3). (Randy Hutson, July 20, 2016)

Let A' be the isogonal conjugate of A wrt the A-altimedial triangle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(265). (Randy Hutson, November 2, 2017)

Let La be the line through A parallel to the Euler line of the A-altimedial triangle, and define Lb and Lc cyclically. Lines La, Lb, Lc concur in X(265). (Randy Hutson, November 2, 2017)

X(265) lies on these lines: 3,125 4,94 5,49 6,13 30,74 64,382 65,79 67,511 69,328 290,316 300,621 301,622

X(265) = midpoint of X(4) and X(3448)
X(265) = reflection of X(i) in X(j) for these (i,j): (3,125), (110,5), (146,1539), (399,113)
X(265) = isogonal conjugate of X(186)
X(265) = isotomic conjugate of X(340)
X(265) = anticomplement of X(1511)
X(265) = cevapoint of X(5) and X(30)
X(265) = crosspoint of X(94) and X(328)
X(265) = inverse-in-circumcircle of X(5961)
X(265) = antigonal image of X(3)
X(265) = syngonal conjugate of X(5)
X(265) = Johnson-circumconic antipode of X(110)
X(265) = perspector of ABC and 2nd isogonal triangle of X(4)
X(265) = perspector of ABC and Ehrmann side-triangle
X(265) = perspector, wrt Ehrmann side-triangle, of Ehrmann conic
X(265) = pole of line X(3)X(523) (the line of the Ehrmann cross-triangle) wrt the Ehrmann conic
X(265) = homothetic center of orthic triangle and cross-triangle of Ehrmann side- and Ehrmann vertex-triangles

X(266) = ISOGONAL CONJUGATE OF X(188)

Trilinears sin A/2 : sin B/2 : sin C/2
= [a/(b + c - a)]^1/2 : [b/(c + a - b)]^1/2 : [c/(a + b - c)]^1/2

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

X(266) lies on these lines:1,164 56,289 174,188 259,260 361,978

X(266) = isogonal conjugate of X(188)
X(266) = eigencenter of cevian triangle of X(174)
X(266) = eigencenter of anticevian triangle of X(259)
X(266) = X(174)-Ceva conjugate of X(259)
X(266) = cevapoint of X(1) and X(361)
X(266) = X(6)-cross conjugate of X(289)
X(266) = crosspoint of X(1) and X(505)
X(266) = crosssum of X(1) and X(164)

X(267) = ISOGONAL CONJUGATE OF X(191)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = 1/[b³ + c³ - a³ + (b + c - a)(bc + ca + ab)]

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(267) lies on these lines: 1,229 10,191 35,37

X(267) = reflection of X(1) in X(229)
X(267) = isogonal conjugate of X(191)
X(267) = cevapoint of X(58) and X(501)
X(267) = X(58)-cross conjugate of X(1)

X(268) = ISOGONAL CONJUGATE OF X(196)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A cot A/2)/(-1 - cos A + cos B + cos C)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(268) lies on these lines: 3,9 21,280 219,255 220,577 222,1073 281,1012

X(268) = isogonal conjugate of X(196)
X(268) = X(i)-cross conjugate of X(j) for these (i,j): (48,219), (55,3)
X(268) = crosssum of X(19) and X(207)

X(269) = ISOGONAL CONJUGATE OF X(200)

Trilinears tan²A/2 : tan²B/2 : tan²C/2

= [a² - (b - c)²]² : [b² - (c - a)²]² : [c² - (a - b)²]² Barycentrics sin A tan²A/2 : sin B tan²B/2 : sin C tan²C/2

X(269) lies on these lines: 1,7 3,939 6,57 9,241 46,1103 56,738 69,200 86,1088 106,934 142,948 273,1111 292,1020 307,936 320,326 479,614

X(269) = isogonal conjugate of X(200)
X(269) = isotomic conjugate of X(341)
X(269) = X(279)-Ceva conjugate of X(57)
X(269) = X(56)-cross conjugate of X(57)
X(269) = crosspoint of X(279) and X(479)
X(269) = crosssum of X(220) and X(480)

X(270) = ISOGONAL CONJUGATE OF X(201)

Trilinears (sec A)/[1 + cos(B - C)] : (sec B)/[1 + cos(C - A)] : (sec C)/[1 + cos(A - B)]
Barycentrics (tan A)/[1 + cos(B - C)] : (tan B)/[1 + cos(C - A)] : (tan C)/[1 + cos(A - B)]

X(270) lies on these lines: 4,162 27,58 28,60 29,283 759,933

X(270) = isogonal conjugate of X(201)
X(270) = X(250)-Ceva conjugate of X(162)
X(270) = cevapoint of X(28) and X(58)
X(270) = X(58)-cross conjugate of X(60)

X(271) = ISOGONAL CONJUGATE OF X(208)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cot A cot A/2)/(-1 - cos A + cos B + cos C)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(271) lies on these lines: 2,1034 8,20 78,394 200,255 282,283

X(271) = isogonal conjugate of X(208)
X(271) = isotomic conjugate of X(342)
X(271) = X(i)-cross conjugate of X(j) for these (i,j): (3,78), (9,63)

X(272) = ISOGONAL CONJUGATE OF X(209)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where
f(A,B,C) = 1/[(sin A + sin(A - B) + sin(A - C))(sin B + sin C)]

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(272) lies on these lines: 2,284 7,58 21,75 28,273 60,86 261,310 1014,1088

X(272) = isogonal conjugate of X(209)
X(272) = X(3)-cross conjugate of X(81)

X(273) = ISOGONAL CONJUGATE OF X(212)

Trilinears sec A sec²(A/2) : sec B sec²(B/2) : sec C sec²(C/2)
= (1- sec A)csc²A : (1 - sec B)csc²B : (1 - sec C)csc²C

Barycentrics tan A sec²(A/2) : tan B sec²(B/2) : tan C sec²(C/2)

X(273) lies on these lines: 2,92 4,7 19,653 27,57 28,272 29,34 53,1086 75,225 78,322 108,675 226,469 269,1111 317,320 458,894

X(273) = isogonal conjugate of X(212)
X(273) = isotomic conjugate of X(78)
X(273) = X(i)-Ceva conjugate of X(j) for these (i,j): (264,342), (286,7), (331,92)
X(273) = cevapoint of X(i) and X(j) for these (i,j): (4,278), (34,57)
X(273) = X(i)-cross conjugate of X(j) for these (i,j): (4,92), (57,85), (225,278)

X(274) = ISOGONAL CONJUGATE OF X(213)

Trilinears    b²c²/(b + c) : :
Trilinears    [a csc(A - ω)]/(b + c) : :
Barycentrics    bc/(b + c) : ca/(c + a) : ab/(a + b)

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 ABaCa, and define Ob and Oc cyclically. Then X(274) is the radical center of Oa, Ob, Oc. see also X(8) and X(21). (Randy Hutson, April 9, 2016)

X(274) lies on these lines:
1,75 2,39 7,959 10,291 21,99 28,242 57,85 58,870 69,443 81,239 88,799 110,767 183,474 213,894 264,475 278,331 315,377 325,442 961,1014

X(274) = isogonal conjugate of X(213)
X(274) = isotomic conjugate of X(37)
X(274) = complement of X(1655)
X(274) = X(310)-Ceva conjugate of X(314)
X(274) = cevapoint of X(i) and X(j) for these (i,j): (2,75), (85,348), (86,333)
X(274) = X(i)-cross conjugate of X(j) for these (i,j): (2,86), (75,310), (81,286), (333,314)
X(274) = crossdifference of every pair of points on line X(669)X(798)
X(274) = trilinear pole of line X(320)X(350) (anticomplement of antiorthic axis)
X(274) = pole wrt polar circle of trilinear polar of X(1824)
X(274) = X(48)-isoconjugate (polar conjugate) of X(1824)

X(275) = CEVAPOINT OF ORTHOCENTER AND SYMMEDIAN POINT

Trilinears csc 2A sec(B - C) : csc 2B sec(C - A) : csc 2C sec(A - B)
Barycentrics sec A sec(B - C) : sec B sec(C - A) : sec C sec(A - B)

X(275) lies on these lines:
2,95 4,54 13,472 14,473 17,471 18,470 25,262 51,107 53,288 76,276 83,297 94,324 98,427

X(275) = isogonal conjugate of X(216)
X(275) = isotomic conjugate of X(343)
X(275) = X(276)-Ceva conjugate of X(95)
X(275) = cevapoint of X(4) and X(6)
X(275) = X(i)-cross conjugate of X(j) for these (i,j): (6,54), (54,95)
X(275) = crosssum of X(217) and X(418)

X(276) = ISOGONAL CONJUGATE OF X(217)

Trilinears csc³A sec A sec(B - C) : csc³B sec B sec(C - A) : csc³C sec C sec(A - B)
Barycentrics csc²A sec A sec(B - C) : csc²B sec B sec(C - A) : csc²C sec C sec(A - B)

X(276) lies on these lines: 3,95 4,327 54,290 76,275 97,401

X(276) = isogonal conjugate of X(217)
X(276) = isotomic conjugate of X(216)
X(276) = cevapoint of X(i) and X(j) for these (i,j): (2,264), (95,275)
X(276) = X(i)-cross conjugate of X(j) for these (i,j): (2,95), (401,290) X(276) = trilinear pole of line X(216)X(647)
X(276) = inverse-in-Kiepert-hyperbola of X(1989)
X(276) = {X(13),X(14)}-harmonic conjugate of X(1989)
X(276) = X(92)-isoconjugate of X(50)
X(276) = Hofstadter -2 point

X(277) = ISOGONAL CONJUGATE OF X(218)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = [sec²(A/2)]/[- cos⁴A/2 + cos⁴B/2 + cos⁴C/2]
Barycentrics  af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(277) lies on these lines: 1,142 3,105 7,218 57,169 220,1086 241,278 942,1002

X(277) = isogonal conjugate of X(218)
X(277) = isotomic conjugate of X(344)
X(277) = X(55)-cross conjugate of X(7)

X(278) = ISOGONAL CONJUGATE OF X(219)

Trilinears    sec A tan A/2 : sec B tan B/2 : sec C tan C/2
Trilinears    csc A - 2 csc 2A : csc B - 2 csc 2B : csc C - 2 csc 2C
Trilinears    (1 - sec A)/a : (1 - sec B)/b : (1 - sec C)/c
Trilinears    bc/[(b + c - a)(b² + c² - a²)]

Barycentrics    tan A tan A/2 : tan B tan B/2 : tan C tan C/2
Barycentrics    1 - sec A : 1 - sec B : 1 - sec C

Let A'B'C' be the extouch 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(278). (Randy Hutson, September 14, 2016)

Let A'B'C' be the 2nd extouch triangle. Let A" be the pole, wrt the polar circle, of line B'C', and define B" and C" cyclically. The triangle A'B'C' is homothetic to ABC at X(278). (Randy Hutson, September 14, 2016)

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(3) with respect to Ha. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Triangle A'B'C' is homothetic to ABC, and the center of homothety is X(278). (Randy Hutson, September 14, 2016)

X(278) lies on these lines:
1,4 2,92 7,27 19,57 25,105 28,56 65,387 88,653 109,917 219,329 240,982 241,277 242,459 274,331 354,955 393,1108 412,962 443,1038 614,1096

X(278) = isogonal conjugate of X(219)
X(278) = isotomic conjugate of X(345)
X(278) = X(i)-Ceva conjugate of X(j) for these (i,j): (27,57), (92,196), (273,4), (331,7)
X(278) = cevapoint of X(19) and X(34)
X(278) = X(i)-cross conjugate of X(j) for these (i,j): (19,4), (56,7), (225,273)
X(278) = trilinear pole of line X(513)X(1835) (the polar of X(8) wrt polar circle, and inverse-in-polar-circle of Fuhrmann circle)
X(278) = pole wrt polar circle of trilinear polar of X(8) (line X(522)X(650))
X(278) = X(48)-isoconjugate (polar conjugate) of X(8)
X(278) = {X(2),X(92)}-harmonic conjugate of X(281)
X(278) = vertex conjugate of foci of inconic that is the polar conjugate of the isogonal conjugate of the incircle
X(278) = Brianchon point (perspector) of inconic that is the polar conjugate of the isotomic conjugate of the incircle

X(279) = ISOGONAL CONJUGATE OF X(220)

Trilinears csc A tan²A/2 : csc B tan²B/2 : csc C tan²C/2
= bc[a² - (b - c)²]² : ca[b² - (c - a)²]² : ab[c² - (a - b)²]²

Barycentrics tan²A/2 : tan²B/2 : tan²C/2

X(279) lies on these lines: 1,7 2,85 28,1014 56,105 57,479 65,1002 144,220 145,664 304,346 942,955 985,1106

X(279) = isogonal conjugate of X(220)
X(279) = isotomic conjugate of X(346)
X(279) = cevapoint of X(57) and X(269)
X(279) = X(i)-cross conjugate of X(j) for these (i,j): (57,7), (269,479)
X(279) = crosssum of X(1) and X(170)

X(280) = X(1)-CROSS CONJUGATE OF X(8)

Trilinears (csc²A/2)/(-1 - cos A + cos B + cos C)

Let Ea be the ellipse passing through A, and with foci at B and C (the A-Soddy ellipse). Let La be the polar of X(20) with respect to Ea. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The lines AA', BB', CC' concur in X(280). (Randy Hutson, September 14, 2016)

X(280) lies on these lines: 2,318 7,253 8,20 21,268 75,309 78,282 285,1043 341,345

X(280) = isogonal conjugate of X(221)
X(280) = isotomic conjugate of X(347)
X(280) = X(309)-Ceva conjugate of X(189)
X(280) = cevapoint of X(1) and X(84)
X(280) = X(i)-cross conjugate of X(j) for these (i,j): (1,8), (281,2), (282,189)
X(280) = pole wrt polar circle of trilinear polar of X(196)
X(280) = X(48)-isoconjugate (polar conjugate) of X(196)
X(280) = trilinear pole of line X(521)X(3239) (the radical axis of circumcircle and Mandart circle, and the Monge line of the nine-point circles of the A-, B- and C-extouch triangles)

X(281) = X(37)-CROSS CONJUGATE OF X(9)

Trilinears    sec A cot A/2 : sec B cot B/2 : sec C cot C/2
Trilinears    csc A + 2 csc 2A : csc B + 2 csc 2B : csc C + 2 csc 2C
Trilinears    (1 + sec A)/a : (1 + sec B)/b : (1 + sec C)/c
Trilinears    (b + c - a) sec A : :
Barycentrics    tan A cot A/2 : tan B cot B/2 : tan C cot C/2
Barycentrics    1 + sec A : 1 + sec B : 1 + sec C
Barycentrics    (b + c - a)/(b^2 + c^2 - a^2) : :

X(281): Let A'B'C' be the intouch triangle. Let A" be the pole, wrt the polar circle, of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(281). (Randy Hutson, September 14, 2016)

Let A'B'C' be the 3rd extouch triangle. Let A" be the pole, wrt the polar circle, of line B'C', and define B", C" cyclically. The triangle A'B'C' is homothetic to ABC at X(281). (Randy Hutson, September 14, 2016)

Let A'B'C' be the 4th extouch triangle. Let A" be the pole, wrt the polar circle, of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(281). (Randy Hutson, September 14, 2016)

Let A'B'C' be the 5th extouch 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(281). (Randy Hutson, September 14, 2016)

Let Ea be the ellipse passing through A, and with foci at B and C (the A-Soddy ellipse). Let La be the polar of X(3) with respect to Ea. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Triangle A'B'C' is homothetic to ABC, and the center of homothety is X(281). (Randy Hutson, September 14, 2016)

X(281) lies on these lines:
1,282 2,92 4,9 6,1146 7,653 8,29 28,958 33,200 37,158 45,53 48,944 100,1013 189,222 196,226 220,594 240,984 264,344 268,1012 318,346 380,950 451,1068 515,610 612,1096

X(281) = isogonal conjugate of X(222)
X(281) = isotomic conjugate of X(348)
X(281) = complement of X(347)
X(281) = X(i)-Ceva conjugate of X(j) for these (i,j): (29,33), (92,4)
X(281) = X(i)-cross conjugate of X(j) for these (i,j): (33,4), (37,9), (55,8)
X(281) = crosspoint of X(i) and X(j) for these (i,j): (2,280), (92,318)
X(281) = crosssum of X(i) and X(j) for these (i,j): (6,221), (48,603), (73,1409), (652,1364)
X(281) = {X(2),X(92)}-harmonic conjugate of X(278)
X(281) = pole, wrt polar circle, of Gergonne line
X(281) = X(48)-isoconjugate (polar conjugate) of X(7)
X(281) = crossdifference of every pair of points on line X(1459)X(1946)
X(281) = trilinear pole of line X(3064)X(3700) (the polar of X(7) wrt polar circle)

X(282) = X(6)-CROSS CONJUGATE OF X(9)

Trilinears (cot A/2)/(-1 - cos A + cos B + cos C) : :

X(282) lies on the Thomson cubic and these lines:
1,281 2,77 3,9 4,3351 6,3341 19,102 48,947 57,3343 78,280 200,219 271,283 380,1036

X(282) = isogonal conjugate of X(223)
X(282) = X(189)-Ceva conjugate of X(84)
X(282) = X(i)-cross conjugate of X(j) for these (i,j): (6,9), (33,1)
X(282) = crosspoint of X(189) and X(280)
X(282) = crosssum of X(i) and X(j) for these (i,j): (6,1035), (198,221)

X(283) = X(3)-CROSS CONJUGATE OF X(21)

Trilinears (cos A)/(cos B + cos C) : (cos B)/(cos C + cos A) : (cos C)/(cos A + cos B)
Barycentrics (sin 2A)/(cos B + cos C) : (sin 2B)/(cos C + cos A) : (sin 2C)/(cos A + cos B)

X(283) lies on these lines: 1,21 2,580 3,49 29,270 60,284 77,603 78,212 86,307 102,110 271,282 474,582 643,1043 859,945 1010,1065

X(283) = isogonal conjugate of X(225)
X(283) = X(333)-Ceva conjugate of X(284)
X(283) = cevapoint of X(i) and X(j) for these (i,j): (3,255), (212,219)
X(283) = X(3)-cross conjugate of X(21)
X(283) = crosspoint of X(332) and X(333)

X(284) = X(55)-CROSS CONJUGATE OF X(21)

Trilinears (sin A)/(cos B + cos C) : (sin B)/(cos C + cos A) : (sin C)/(cos A + cos B)
Barycentrics a²/(cos B + cos C) : b²/(cos C + cos A) : c²/(cos A + cos B)
X(284) = s*X(3) + (r + 2R)*cot(ω)*X(6)

X(284) lies on these lines:
1,19 2,272 3,6 9,21 27,226 29,950 35,71 37,101 55,219 57,77 60,283 73,951 86,142 102,112 109,296 163,909 198,859 261,332 405,965 501,942 515,1065

X(284) = isogonal conjugate of X(226)
X(284) = isotomic conjugate of X(349)
X(284) = inverse-in-Brocard-circle of X(579)
X(284) = X(i)-Ceva conjugate of X(j) for these (i,j): (81,58), (333,283)
X(284) = cevapoint of X(i) and X(j) for these (i,j): (6,48), (41,55)
X(284) = X(55)-cross conjugate of X(21)
X(284) = crosspoint of X(i) and X(j) for these (i,j): (21,81), (29,333)
X(284) = crosssum of X(i) and X(j) for these (i,j): (37,65), (73,1400)
X(284) = crossdifference of every pair of points on line X(523)X(656)
X(284) = trilinear pole of line X(652)X(663)
X(284) = X(92)-isoconjugate of X(73)
X(284) = X(1577)-isoconjugate of X(109)
X(284) = perspector of ABC and unary cofactor triangle of isogonal triangle of X(1) (a.k.a. reflection triangle of X(1))

X(285) = X(58)-CROSS CONJUGATE OF X(21)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where
f(A,B,C) = 1/[(cos B + cos C)(-1 - cos A + cos B + cos C)]

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(285) lies on these lines: 21,84 29,81 271,282 280,1043

X(285) = isogonal conjugate of X(227)
X(285) = X(58)-cross conjugate of X(21)

X(286) = X(4)-CROSS CONJUGATE OF X(27)

Trilinears (csc 2A)/(sin B + sin C) : (csc 2B)/(sin C + sin A) : (csc 2C)/(sin A + sin B)
Barycentrics (sec A)/(sin B + sin C) : (sec B)/(sin C + sin A) : (sec C)/(sin A + sin B)

X(286) lies on these lines: 4,69 7,331 19,27 28,242 29,34 99,915 112,767 158,969 322,1043

X(286) = isogonal conjugate of X(228)
X(286) = isotomic conjugate of X(72)
X(286) = cevapoint of X(i) and X(j) for these (i,j): (4,92), (7,273), (27,29), (28,81)
X(286) = X(i)-cross conjugate of X(j) for these (i,j): (4,27), (7,86), (81,274)

X(287) = X(2)-HIRST INVERSE OF X(98)

Trilinears cot A sec(A + ω) : cot B sec(B + ω) : cot C sec(C + ω)
Barycentrics cos A sec(A + ω) : cos B sec(B + ω) : cos C sec(C + ω)

X(287) lies on the MacBeath circumconic and these lines:
2,98 6,264 69,248 83,217 95,141 185,384 193,253 293,306 297,685 305,394 401,511 651,894 879,895

X(287) = reflection of X(648) in X(6)
X(287) = isogonal conjugate of X(232)
X(287) = isotomic conjugate of X(297)
X(287) = X(290)-Ceva conjugate of X(98)
X(287) = cevapoint of X(2) and X(401)
X(287) = X(248)-cross conjugate of X(98)
X(287) = X(2)-Hirst inverse of X(98)
X(287) = trilinear pole of PU(37) (line X(3)X(525))
X(287) = antipode of X(69) in hyperbola {A,B,C,X(2),X(69)}
X(287) = MacBeath circumconic antipode of X(648)
X(287) = X(92)-isoconjugate of X(237)
X(287) = pole wrt polar circle of trilinear polar of X(6530)
X(287) = X(48)-isoconjugate (polar conjugate) of X(6530)

X(288) = CEVAPOINT OF X(6) AND X(54)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [sec(B - C)]/[b cos(C - A) + c cos(B - A)]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(288) lies on these lines: 51,54 53,275 97,216

X(288) = isogonal conjugate of X(233)
X(288) = cevapoint of X(6) and X(54)

X(289) = ISOGONAL CONJUGATE OF X(236)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin A/2)/(cos B/2 + cos C/2 - cos A/2)
Trilinears [1 - sin(A/2)] tan(A/2) : [1 - sin(B/2)] tan(B/2) : [1 - sin(C/2)] tan(C/2) (M. Iliev, 4/12/07)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(289) lies on these lines: 1,363 56,266 258,259

X(289) = isogonal conjugate of X(236)
X(289) = X(6)-cross conjugate of X(266)
X(289) = crosssum of X(1) and X(363)

X(290) = ISOGONAL CONJUGATE OF X(237)

Trilinears csc²A sec(A + ω) : :
Barycentrics csc A sec(A + ω) : :

If you have The Geometer's Sketchpad, you can view the following dynamic sketch:

X(290).

X(290) lies on the Steiner circumellipse and these lines:
2,327 3,76 6,264 54,276 66,317 67,340 68,315 69,670 71,190 72,668 73,336 248,385 265,316 308,311 892,895

X(290) = isogonal conjugate of X(237)
X(290) = isotomic conjugate of X(511)
X(290) = cevapoint of X(i) and X(j) for these (i,j): (2,511), (98,287)
X(290) = X(i)-cross conjugate of X(j) for these (i,j): (385,308), (401,276), (511,2)
X(290) = point of intersection, other than A, B, C, of Steiner circumellipse and Jerabek hyperbola
X(290) = trilinear pole of line X(2)X(647) (the line through the polar conjugates of PU(39))
X(290) = pole wrt polar circle of trilinear polar of X(232)
X(290) = X(48)-isoconjugate (polar conjugate) of X(232)
X(290) = X(6)-isoconjugate of X(1755)
X(290) = crossdifference of PU(89)

X(291) = 2nd SHARYGIN POINT

Trilinears 1/(a² - bc) : 1/(b² - ca) : 1/(c² - ab)
Barycentrics a/(a² - bc) : b/(b² - ca) : c/(c² - ab)

See the description at X(1281). The lines AD', BE', CF' defined there concur in X(291).

X(291) lies on these lines: 1,39 2,38 6,985 8,330 10,274 42,81 43,57 88,660 105,238 256,894 337,986 350,726 659,897 876,891

X(291) = reflection of X(i) in X(j) for these (i,j): (1,1015), (668,10)
X(291) = isogonal conjugate of X(238)
X(291) = isotomic conjugate of X(350)
X(291) = X(i)-cross conjugate of X(j) for these (i,j): (239,256), (518,1)
X(291) = X(i)-Hirst inverse of X(j) for these (i,j): (1,292), (2,335)
X(291) = trilinear pole of PU(i) for these i: 6, 52, 53
X(291) = antipode of X(1) in hyperbola {A,B,C,X(1),X(2)}
X(291) = point of intersection, other than A, B, C, of 1st and 2nd bicentrics of the circumcircle

X(292) = X(1)-HIRST INVERSE OF X(291)

Trilinears a/(a² - bc) : b/(b² - ca) : c/(c² - ab)
Barycentrics a²/(a² - bc) : b²/(b² - ca) : c²/(c² - ab)

X(292) lies on these lines: 1,39 2,334 6,869 9,87 37,86 44,660 58,101 106,813 171,893 269,1020 659,665

X(292) = isogonal conjugate of X(239)
X(292) = isotomic conjugate of X(1921)
X(292) = X(i)-Ceva conjugate of X(j) for these (i,j): (335,292), (813,3572)
X(292) = cevapoint of X(171) and X(238)
X(292) = crossdifference of every pair of points on line X(659)X(812)
X(292) = crossdifference of PU(134)
X(292) = X(1)-Hirst inverse of X(291)
X(292) = trilinear pole of line X(42)X(649)
X(292) = point of intersection, other than A, B, C, of 1st and 2nd bicentrics of the Steiner circumellipse

X(293) = ISOGONAL CONJUGATE OF X(240)

Trilinears cos A sec(A + ω) : cos B sec(B + ω) : cos C sec(C + ω)
Barycentrics sin 2A sec(A + ω) : sin 2B sec(B + ω) : sin 2C sec(C + ω)

X(293) lies on these lines: 1,163 31,92 72,248 98,109 255,304 287,306

X(293) = isogonal conjugate of X(240)

X(293) = trilinear pole of line X(48)X(656)
X(293) = X(92)-isoconjugate of X(1755)

X(294) = ISOGONAL CONJUGATE OF X(241)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b² + c² - ab - ac)
Barycentrics af(a,b,c) : bf(b,c,a) :cf(c,a,b)

X(294) lies on these lines: 1,41 2,949 4,218 6,7 8,220 19,1041 84,580 104,919 239,666 241,910 314,645

X(294) = isogonal conjugate of X(241)
X(294) = crosssum of X(i) and X(j) for these (i,j): (672,1458), (910,1279)
X(294) = crossdifference of every pair of points on line X(926)X(1362)
X(294) = X(1)-Hirst inverse of X(105)
X(294) = trilinear pole of line X(55)X(650)
X(294) = perspector of conic {A,B,C,PU(96)}
X(294) = intersection of trilinear polars of P(96) and U(96)

X(295) = ISOGONAL CONJUGATE OF X(242)

Trilinears (cos A)/(a² - bc) : (cos B)/(b² - ca) : (cos C)/(c² - ab)
Barycentrics (sin 2A)/(a² - bc) : (sin 2B)/(b² - ca) : (sin 2C)/(c² - ab)

X(295) lies on these lines: 27,335 43,57 58,101 72,337 103,813 150,334 875,926 876,928

X(295) = isogonal conjugate of X(242)
X(295) = X(335)-Ceva conjugate of X(292)
X(295) = crosspoint of X(335) and X(337)

X(296) = ISOGONAL CONJUGATE OF X(243)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (cos A)/[cos²A - cos B cos C]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where
                        g(A,B,C) = (sin 2A)/[cos²A - cos B cos C] X(296) lies on these lines: 1,185   3,820   29,65   109,284

X(296) = isogonal conjugate of X(243)
X(296) = trilinear pole of line X(73)X(652)
X(296) = X(92)-isoconjugate of X(1951)

X(297) = X(2)-HIRST INVERSE OF X(4)

Trilinears    csc 2A cos(A + ω) : csc 2B cos(B + ω) : csc 2C cos(C + ω)
Barycentrics    sec A cos(A + ω) : sec B cos(B + ω) : sec C cos(C + ω)
Barycentrics    tan A - cot ω : :
Barycentrics    SB*SC(-SA^2 + SB*SC) : :
X(297) = X(340) + 2X(1990)

As a point on the Euler line, X(297) has Shinagawa coefficients (EF + F², - S²).

X(297) lies on these lines:
{2, 3}, {6, 317}, {53, 141}, {69, 393}, {76, 343}, {83, 275}, {92, 257}, {107, 2710}, {112, 2857}, {193, 1249}, {230, 2966}, {232, 325}, {239, 5081}, {249, 316}, {273, 3662}, {286, 1865}, {287, 685}, {290, 1987}, {315, 394}, {318, 3661}, {324, 1235}, {340, 524}, {459, 2996}, {525, 850}, {530, 6111}, {531, 6110}, {623, 6117}, {624, 6116}, {626, 3199}, {1515, 1533}, {1654, 2322}, {1785, 3912}, {1915, 1970}, {3087, 3618}, {3978, 6331}, {3981, 5254}, {5032, 5702}

X(297) = midpoint of X(340) and X(648)
X(297) = reflection of X(i) in X(j) for these (i,j): (401,441), (648,1990), (2966,230)
X(297) = isogonal conjugate of X(248)
X(297) = isotomic conjugate of X(287)
X(297) = inverse-in-orthocentroidal-circle of X(458)
X(297) = complement of X(401)
X(297) = anticomplement of X(441)
X(297) = X(i)-Ceva conjugate of X(j) for these (i,j): (264,2967), (6330,2)
X(297) = cevapoint of X(232) and X(511)
X(297) = X(i)-cross conjugate of X(j) for these (i,j): (511,325), (2967,264), (3569,4230)
X(297) = crossdifference of every pair of points on line X(184)X(647)
X(297) = X(i)-Hirst inverse of X(j) for (i,j) = (2,4), (193,1249)
X(297) = X(i)-complementary conjugate of X(j) for these (i,j): (1953,129), (1956,141), (1972,2887), (1987,10)
X(297) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1297,4329), (6330,6327}
X(297) = perspector of conic {A,B,C,PU(45)}
X(297) = intersection of trilinear polars of P(45) and U(45)
X(297) = trilinear pole of line X(114)X(132) (the polar of X(98) wrt polar circle)
X(297) = pole wrt polar circle of trilinear polar of X(98) (line X(6)X(523))
X(297) = X(48)-isoconjugate (polar conjugate) of X(98)
X(297) = inverse-in-polar-circle of X(1316)
X(297) = inverse-in-Steiner-circumellipse of X(4)
X(297) = {X(2479),X(2480)}-harmonic conjugate of X(4)
X(297) = inverse-in-Steiner-inellipse of X(5)
X(297) = {X(2454),X(2455)}-harmonic conjugate of X(5)
X(297) = midpoint of polar conjugates of PU(4)
X(297) = X(i)-isoconjugate of X(j) for these (i,j): {1,248}, {3,1910}, {6,293}, {31,287}, {32,336}, {48,98}, {63,1976}, {163,879}, {184,1821}, {656,2715}, {662,878}, {685,822}, {810,2966}, {1176,3404}, {1973,6394}, {2395,4575}, {2422,4592}
X(297) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,458), (2,401,441), (4,419,460), (4,420,419), (4,5117,427), (53,141,264), (237,868,2450), (237,2450,1513), (419,420,468), (460,468,419), (470,471,468), (472,473,428), (868,5112,1513), (1585,1586,25), (2450,5112,237), (2454,2455,5), (2479,2480,4)

Centers 298- 350
are isotomic conjugates of previously listed centers.
underbar

X(298) ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER

Trilinears csc²A sin(A + π/3) : csc²B sin(B + π/3) : csc²C sin(C + π/3)
Barycentrics csc A sin(A + π/3) : csc B sin(B + π/3) : csc C sin(C + π/3)

X(298) lies on these lines:
2,6 3,617 5,634 13,532 14,76 15,533 18,636 99,531 140,628 264,472 316,530 317,473 319,1082 340,470 381,622 511,1080

X(298) = midpoint of X(616) and X(621)
X(298) = reflection of X(i) in X(j) for these (i,j): (13,623), (15,618), (299,325), (385,395)
X(298) = isogonal conjugate of X(3457)
X(298) = isotomic conjugate of X(13)
X(298) = complement of X(3180)
X(298) = anticomplement of X(396)
X(298) = X(300)-Ceva conjugate of X(303)
X(298) = X(15)-cross conjugate of X(470)
X(298) = X(2)-Hirst inverse of X(299)

X(299) = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER

Trilinears csc²A sin(A - π/3) : csc²B sin(B - π/3) : csc²C sin(C - π/3)
Barycentrics csc A sin(A - π/3) : csc B sin(B - π/3) : csc C sin(C - π/3)

X(299) lies on these lines:
2,6 3,616 5,633 13,76 14,533 16,532 17,635 30,617 75,554 99,530 140,627 264,473 316,531 317,472 319,559 340,471 381,621 383,511

X(299) = midpoint of X(617) and X(622)
X(299) = reflection of X(i) in X(j) for these (i,j): (14,624), (16,619), (298,325), (385,396)
X(299) = isogonal conjugate of X(3458)
X(299) = isotomic conjugate of X(14)
X(299) = complement of X(3181)
X(299) = anticomplement of X(395)
X(299) = X(301)-Ceva conjugate of X(302)
X(299) = X(16)-cross conjugate of X(471)
X(299) = X(2)-Hirst inverse of X(298)

X(300) ISOTOMIC CONJUGATE OF 1st ISODYNAMIC CENTER

Trilinears csc²A csc(A + π/3) : csc²B csc(B + π/3) : csc²C csc(C + π/3)
Barycentrics csc A csc(A + π/3) : csc B csc(B + π/3) : csc C csc(C + π/3)

X(300) lies on these lines: 2,94 13,76 264,302 265,621 303,311

X(300) = isotomic conjugate of X(15)
X(300) = cevapoint of X(298) and X(303)
X(300) = X(94)-Hirst inverse of X(301)

X(301) ISOTOMIC CONJUGATE OF 2nd ISODYNAMIC CENTER

Trilinears csc²A csc(A - π/3) : csc²B csc(B - π/3) : csc²C csc(C - π/3)
Barycentrics csc A csc(A - π/3) : csc B csc(B - π/3) : csc C csc(C - π/3)

X(301) lies on these lines: 2,94 14,76 264,303 265,622 302,311

X(301) = isotomic conjugate of X(16)
X(301) = cevapoint of X(299) and X(302)
X(301) = X(94)-Hirst inverse of X(300)

X(302) ISOTOMIC CONJUGATE OF 1st NAPOLEON POINT

Trilinears csc²A sin(A + π/6) : csc²B sin(B + π/6) : csc²C sin(C + π/6)
Barycentrics csc A sin(A + π/6) : csc B sin(B + π/6) : csc C sin(C + π/6)

If you have The Geometer's Sketchpad, you can view X(302).

X(302) lies on these lines:
2,6 3,621 5,622 14,99 16,316 18,76 61,629 140,633 264,300 301,311 317,470 381,616 549,617

X(302) = isotomic conjugate of X(17)
X(302) = X(301)-Ceva conjugate of X(299)
X(302) = X(61)-cross conjugate of X(473)

X(303) ISOTOMIC CONJUGATE OF 2nd NAPOLEON POINT

Trilinears csc²A sin(A - π/6) : csc²B sin(B - π/6) : csc²C sin(C - π/6)
Barycentrics csc A sin(A - π/6) : csc B sin(B - π/6) : csc C sin(C - π/6)

If you have The Geometer's Sketchpad, you can view X(303).

X(303) lies on these lines:
2,6 3,622 5,621 13,99 15,316 17,76 62,630 140,634 264,301 300,311 317,471 381,617 549,616

X(303) = isotomic conjugate of X(18)
X(303) = X(300)-Ceva conjugate of X(298)
X(303) = X(62)-cross conjugate of X(472)

X(304) = ISOTOMIC CONJUGATE OF X(19)

Trilinears    (cot A)csc²A : (cot B)csc²B : (cot C)csc²C
Trilinears    cos A csc(A - ω) : cos B csc(B - ω) : cos C csc(C - ω)
Barycentrics    (cos A)csc²A : (cos B)csc²B : (cos C)csc²C

Let AaBaCa, AbBbCb, AcBcCc be the A-, B-, and C-anti-altimedial triangles, resp. X(304) is the trilinear product Ba*Ca*Cb*Ab*Ac*Bc. (Randy Hutson, November 2, 2017)

X(304) lies on these lines:
1,75 8,3263 63,1102 69,72 76,85 92,561 255,293 279,346 305,306 309,322 337,1565 341,668 345,348 662,2172 742,2176 799,2349 811,1895 1921,3061 1958,1973

X(304) = isogonal conjugate of X(1973)
X(304) = isotomic conjugate of X(19)
X(304) = cevapoint of X(i) and X(j) for these (i,j): (63,326), (69,345), (312,322)
X(304) = X(i)-cross conjugate of X(j) for these (i,j): (63,75), (306,69)
X(304) = X(i)-isoconjugate of X(j) for these (i,j): (6,25), (48,1096), (92,560)
X(304) = polar conjugate of X(1096)

X(305) = ISOTOMIC CONJUGATE OF X(25)

Trilinears b⁴c⁴cos A : c⁴a⁴cos B : a⁴b⁴cos C
= cot A csc(A - ω) : cot B csc(B - ω) : cot C csc(C - ω)

Barycentrics b³c³cos A : c³a³cos B : a³b³cos C

X(305) = trilinear-pole-of-line-X(525)X(3267) = pole-with-respect-to-polar-circle-of trilinear-polar-of-X(2207) = X(48)-isoconjugate-of-X(2207) = X(92)-isoconjugate-of-X(1501) Randy Hutson, August 15, 2013

X(305) lies on these lines:
2,39 22,99 25,683 95,183 264,325 287,394 304,306 311,1007 341,1088 350,614 561,1441

X(305) = isogonal conjugate of X(1974)
X(305) = isotomic conjugate of X(25)
X(305) = anticomplement of X(1196)
X(305) = X(i)-cross conjugate of X(j) for these (i,j): (69,76), (339, (3267)
X(305) = cevapont of X(339) and X(3267)

X(306) = ISOTOMIC CONJUGATE OF X(27)

Trilinears (b²c²)(b + c)cos A : (c²a²)(c + a)cos B : (a²b²)(a + b)cos C
Barycentrics bc(b + c)cos A : ca(c + a)cos B : ab(a + b)cos C

X(306) lies on these lines:
1,2 27,1043 63,69 72,440 92,264 209,518 226,321 253,329 287,293 304,305 319,333

X(306) = isogonal conjugate of X(1474)
X(306) = isotomic conjugate of X(27)
X(306) = complement of X(3187)
X(306) = X(i)-Ceva conjugate of X(j) for these (i,j): (69, 72), (312,321), (313,10)
X(306) = X(i)-cross conjugate of X(j) for these (i,j): (71,10), (72,307), (440,2)
X(306) = crosspoint of X(i) and X(j) for these (i,j): (69,304), (312,345)
X(306) = crosssum of X(604) and X(608)

X(307) = ISOTOMIC CONJUGATE OF X(29)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b²c²(b + c)(cos A)/(b + c - a)
Barycentrics af(a,b,c) : bf(b,c,a) :cf(c,a,b)

X(307) lies on these lines: 2,7 8,253 69,73 75,225 86,283 95,320 141,241 269,936 319,664 948,966

X(307) = isogonal conjugate of X(2299)
X(307) = isotomic conjugate of X(29)
X(307) = X(349)-Ceva conjugate of X(226)
X(307) = X(i)-cross conjugate of X(j) for these (i,j): (72,306), (73,226)
X(307) = crosspoint of X(69) and X(75)
X(307) = crosssum of X(25) and X(31)

X(308) = ISOTOMIC CONJUGATE OF X(39)

Trilinears       b³c³/(b² + c²) : c³a³/(c² + a²) : a³b³/(a² + b²)
                        = csc²A csc(A + ω) : csc²B csc(B + ω) : csc²C csc(C + ω)
                        = [csc(A - ω)]/(b² + c²) : [csc(B - ω)]/(c² + a²) : [csc(C - ω)]/(a² + b²)

Barycentrics (b²c²)/(b² + c²) : (c²a²)/(c² + a²) : (a²b²)/(a² + b²)
= csc A csc(A + ω) : csc B csc(B + ω) : csc C csc(C + ω)

X(308) lies on these lines: 2,702 6,76 25,183 42,313 69,263 111,689 141,670 251,385 290,311

X(308) = isogonal conjugate of X(3051)
X(308) = isotomic conjugate of X(39)
X(308) = cevapoint of X(2) and X(76)
X(308) = X(i)-cross conjugate of X(j) for these (i,j): (2,83), (385,290)

X(309) = ISOTOMIC CONJUGATE OF X(40)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc²A)/(-1 - cos A + cos B + cos C)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(309) lies on these lines: 69,189 75,280 77,318 84,314 85,264 304,322

X(309) = isogonal conjugate of X(2187)
X(309) = isotomic conjugate of X(40)
X(309) = cevapoint of X(189) and X(280)
X(309) = X(i)-cross conjugate of X(j) for these (i,j): (7,75), (92,85)

X(310) = ISOTOMIC CONJUGATE OF X(42)

Trilinears b³c³/(b + c) : c³a³/(c + a) : a³b³/(a + b)
Barycentrics b²c²/(b + c) : c²a²/(c + a) : a²b²/(a + b)

X(310) lies on these lines: 2,39 7,314 38,75 86,350 99,675 261,272 321,335 333,673 670,903 871,982

X(310) = isogonal conjugate of X(1918)
X(310) = isotomic conjugate of X(42)
X(310) = cevapoint of X(i) and X(j) for these (i,j): (75,76), (274,314)
X(310) = X(75)-cross conjugate of X(274)

X(311) = ISOTOMIC CONJUGATE OF X(54)

Trilinears csc²A cos(B - C) : csc²B cos(C - A) : csc²C cos(A - B)
Barycentrics csc A cos(B - C) : csc B cos(C - A) : csc C cos(A - B)

X(311) lies on these lines: 2,570 4,69 22,157 53,324 95,99 141,338 290,308 300,303 301,302 305,1007

X(311) = isotomic conjugate of X(54)
X(311) = anticomplement of X(570)
X(311) = X(76)-Ceva conjugate of X(343)
X(311) = cevapoint of X(5) and X(343)
X(311) = X(5)-cross conjugate of X(324)

X(312) = ISOTOMIC CONJUGATE OF X(57)

Trilinears    (b + c - a)b²c² : (c + a - b)c²a² : (a + b - c)a²b²
Trilinears    (1 + cos A)csc(A - ω) : (1 + cos B)csc(B - ω) : (1 + cos C)csc(C - ω)
Trilinears    (csc A)/(1 - cos A) : (csc B)/(1 - cos B) : (csc C)/(1 - cos C)     (M. Iliev, 4/12/07)
Trilinears    bc tan A' : ca tan B' : ab tan C', where A'B'C' is the excentral triangle
Barycentrics    bc(b + c - a) : ca(c + a - b) : ab(a + b - c)

X(312) lies on these lines: 1,1089 2,37 8,210 9,314 29,33 63,190 69,189 76,85 92,264 212,643 223,664 726,982 894,940 975,1010

X(312) = isogonal conjugate of X(604)
X(312) = isotomic conjugate of X(57)
X(312) = complement of X(3210)
X(312) = anticomplement of X(3752)
X(312) = X(i)-Ceva conjugate of X(j) for these (i,j): (76,75), (304,322), (314,8)
X(312) = cevapoint of X(i) and X(j) for these (i,j): (2,329), (8,346), (9,78), (306,321)
X(312) = X(i)-cross conjugate of X(j) for these (i,j): (8,75), (9,318), (306,345), (346,341)
X(312) = crosssum of X(i) and X(j) for these (i,j): (32,1397), (56,1403), (57,1424)
X(312) = polar conjugate of X(34)
X(312) = perspector of ABC and extraversion triangle of X(85)
X(312) = trilinear pole of line X(522)X(3717) (the polar of X(34) wrt polar circle, and the radical axis of the circumcircles of the outer and inner Garcia triangles)

X(313) = ISOTOMIC CONJUGATE OF X(58)

Trilinears (b + c)b³c³ : (c + a)c³a³ : (a + b)a³b³
= (b + c)csc(A - ω) : (c + a)csc(B - ω) : (a + b)csc(C - ω)

Barycentrics (b + c)b²c² : (c + a)c²a² : (a + b)a²b²

X(313) lies on these lines: 10,75 12,349 42,308 71,190 80,314 92,264 321,594 561,696

X(313) = isogonal conjugate of X(2206)
X(313) = isotomic conjugate of X(58)
X(313) = X(76)-Ceva conjugate of X(321)
X(313) = cevapoint of X(10) and X(306)
X(313) = X(321)-cross conjugate of X(349)
X(313) = crosssum of X(32) and X(560)

X(314) = ISOTOMIC CONJUGATE OF X(65)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b²c²(b + c - a)/(b + c)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc(b + c - a)/(b + c)

Let D and E be the intersections of line X(1)X(3) with lines PU(3) and PU(6), respectively. Let D' and E' be the isogonal conjugates of D and E, respectively. Let D" and E" be the isotomic conjugates of D and E, respectively. Then X(314) = D'D" ∩E'E". (Randy Hutson, December 26, 2015)

Let A4B4C4 be the 4th Conway triangle. Let A' be the cevapoint of B4 and C4, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(314). (Randy Hutson, December 10, 2016)

Let A5B5C5 be the 5th Conway triangle. Let A' be the cevapoint of B5 and C5, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(314). (Randy Hutson, December 10, 2016)

X(314) lies on these lines:
1,75 2,941 4,69 6,981 7,310 9,312 21,261 29,1039 58,987 79,320 80,313 81,321 84,309 99,104 256,350 294,645

X(314) = isogonal conjugate of X(1402)
X(314) = isotomic conjugate of X(65)
X(314) = anticomplement of X(2092)
X(314) = X(310)-Ceva conjugate of X(274)
X(314) = cevapoint of X(i) and X(j) for these (i,j): (8,312), (69,75)
X(314) = X(i)-cross conjugate of X(j) for these (i,j): (8,333), (69,332), (333,274), (497,29)

X(314) = crosspoint of X(1) and X(1764) wrt excentral triangle
X(314) = crosspoint of X(1) and X(1764) wrt anticomplementary triangle
X(314) = pole wrt polar circle of trilinear polar of X(1880)
X(314) = X(48)-isoconjugate (polar conjugate) of X(1880)
X(314) = perspector of 4th Conway triangle and cross-triangle of ABC and 4th Conway triangle
X(314) = perspector of 5th Conway triangle and cross-triangle of ABC and 5th Conway triangle
X(314) = perspector of inverse-in-Conway-circle triangle and cross-triangle of ABC and inverse-in-Conway-circle triangle

X(315) = ISOTOMIC CONJUGATE OF X(66)

Trilinears bc(b⁴ + c⁴ - a⁴) : ca(c⁴ + a⁴ - b⁴) : ab(a⁴ + b⁴ - c⁴)
Barycentrics b⁴ + c⁴ - a⁴ : c⁴ + a⁴ - b⁴ : a⁴ + b⁴ - c⁴

X(315) lies on these lines:
2,32 3,325 4,69 5,183 8,760 20,99 68,290 192,746 194,736 274,377 297,394 343,458 371,491 372,492 631,1007

X(315) = midpoint of X(637) and X(638)
X(315) = reflection of X(i) in X(j) for these (i,j): (32,626), (371,640, (372,639)
X(315) = isogonal conjugate of X(2353)
X(315) = isotomic conjugate of X(66)
X(315) = anticomplement of X(32)
X(315) = anticomplementary conjugate of X(194)
X(315) = X(i)-cross conjugate of X(j) for these (i,j): (206,2)

X(316) = DROUSSENT PIVOT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b⁴ + c⁴ - a⁴ - b²c²)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b⁴ + c⁴ - a⁴ - b²c²

Let A'B'C' be the antipedal triangle of X(4) (the anticomplementary triangle). The circumcircles of AA'X(4), BB'X(4), CC'X(4) concur, other than X(4), in X(316). (Randy Hutson, June 27, 2018)

See Lucien Droussent, "Cubiques circulaires anallagmatiques par points réciproques ou isogonaux," Mathesis 62 (1953) 204-215.

X(316) lies on these lines:
2,187 4,69 15,303 16,302 30,99 115,385 148,538 183,381 249,297 265,290 298,530 299,531 376,1007 384,626 512,850 524,671 691,858

X(316) = midpoint of X(621) and X(622)
X(316) = reflection of X(i) in X(j) for these (i,j): (15,624), (16,623), (99,325), (385,115), (691,858)
X(316) = isogonal conjugate of X(3455)
X(316) = isotomic conjugate of X(67)
X(316) = anticomplement of X(187)
X(316) = crosssum of X(39) and X(187)
X(316) = reflection of X(99) in the polar of X(76)
X(316) = antigonal conjugate of X(23)
X(316) = reflection of X(99) in the de Longchamps line
X(316) = inverse-in-polar-circle of X(1843)
X(316) = trilinear pole of line X(2492)X(7664)
X(316) = pole wrt polar circle of trilinear polar of X(8791) (line X(512)X(1843))
X(316) = polar conjugate of X(8791)
X(316) = intersection of Lemoine axes of 1st and 2nd Ehrmann circumscribing triangles
X(316) = intersection of Lemoine axes of anticevian triangles of PU(4)

X(317) = ISOTOMIC CONJUGATE OF X(68)

Trilinears    sec A cos 2A csc²A : :
Trilinears    bc cot 2A : ca cot 2B : ab cot 2C
Trilinears    sec A - cot A csc A : :
Barycentrics    cot 2A : cot 2B : cot 2C
Barycentrics   tan A cos 2A csc²A
Barycentrics   tan A - cot A : :
Barycentrics   tan A + cot B + cot C - cot ω : :

X(317) lies on these lines:
2,95 4,69 6,297 25,325 53,524 66,290 141,458 183,427 193,393 261,406 273,320 298,473 299,472 302,470 303,471 318,319 1007,6353

X(317) = isogonal conjugate of X(2351)
X(317) = isotomic conjugate of X(68)
X(317) = anticomplement of X(577)
X(317) = cevapoint of X(52) and X(467)
X(317) = polar conjugate of X(2165)
X(317) = X(1721)-of-orthic-triangle if ABC is acute

X(318) = ISOTOMIC CONJUGATE OF X(77)

Trilinears (1 + sec A)/a² : (1 + sec B)/b² : (1 + sec C)/c²
= sec A csc²A/2 : sec B csc²B/2 : sec C csc²C/2

Barycentrics (1 + sec A)/a : (1 + sec B)/b : (1 + sec C)/c

X(318) lies on these lines:
2,280 4,8 10,158 29,33 53,594 63,412 75,225 77,309 108,404 200,1089 208,653 239,458 243,958 253,342 281,346 317,319 475,1068

X(318) = isogonal conjugate of X(603)
X(318) = isotomic conjugate of X(77)
X(318) = X(264)-Ceva conjugate of X(92)
X(318) = cevapoint of X(9) and X(33)
X(318) = X(i)-cross conjugate of X(j) for these (i,j): (9,312), (10,8), (281,92)

X(319) = ISOTOMIC CONJUGATE OF X(79)

Trilinears (1 + 2 cos A)/a² : (1 + 2 cos B)/b² : (1 + 2 cos C)/c²
Barycentrics (1 + 2 cos A)/a : (1 + 2 cos B)/b : (1 + 2 cos C)/c

X(319) lies on these lines: 2,1100 7,8 10,86 80,313 141,239 171,757 200,326 261,502 298,1082 299,559 306,333 307,664 317,318 321,1029 344,391 524,594

X(319) = reflection of X(894) in X(594)
X(319) = isotomic conjugate of X(79)
X(319) = anticomplement of X(1100)

X(320) = ISOTOMIC CONJUGATE OF X(80)

Trilinears (1 - 2 cos A)/a² : (1 - 2 cos B)/b² : (1 - 2 cos C)/c²
Barycentrics (1 - 2 cos A)/a : (1 - 2 cos B)/b : (1 - 2 cos C)/c

X(320) lies on these lines:
1,752 2,44 7,8 58,86 79,314 95,307 141,894 144,344 190,527 239,524 269,326 273,317 334,660 350,513 369,3232 519,679

X(320) = reflection of X(239) in X(1086)
X(320) = isotomic conjugate of X(80)
X(320) = X(214)-cross conjugate of X(1)
X(320) = crosssum of X(42) and X(902)

X(321) = ISOTOMIC CONJUGATE OF X(81)

Trilinears    (b + c)b²c² : :
Trilinears    a(b + c)csc(A - ω) : :
Barycentrics   bc(b + c) : :
Barycentrics   |AP(1)| + |AU(1)| : :

X(321) lies on the Kipert hyperbola and these lines:
1,964 2,37 4,8 10,756 38,726 76,561 81,314 83,213 98,100 190,333 226,306 310,335 313,594 319,1029 668,671 693,824

X(321) = reflection of X(42) in X(1215)
X(321) = isogonal conjugate of X(1333)
X(321) = isotomic conjugate of X(81)
X(321) = X(i)-Ceva conjugate of X(j) for (i,j) = (75,10), (76,313), (312,306)
X(321) = cevapoint of X(37) and X(72)
X(321) = X(442)-cross conjugate of X(264)
X(321) = crosspoint of X(i) and X(j) for these (i,j): (75,76), (313,349)
X(321) = crosssum of X(31) and X(32)
X(321) = crossdifference of every pair of points on line X(667)X(838)
X(321) = anticomplement of X(3666)
X(321) = pole wrt polar circle of trilinear polar of X(28) (line X(513)X(1430))
X(321) = polar conjugate of X(28)
X(321) = Danneels point of X(75)
X(321) = inverse-in-Fuhrmann-circle of X(5016)
X(321) = barycentric product X(100)*X(850)

X(322) = ISOTOMIC CONJUGATE OF X(84)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (-1 - cos A + cos B + cos C)csc²A
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (-1 - cos A + cos B + cos C)csc A

X(322) lies on these lines: 2,1108 7,8 78,273 92,264 227,347 253,341 286,1043 304,309 326,664

X(322) = isogonal conjugate of X(2208)
X(322) = isotomic conjugate of X(84)
X(322) = anticomplement of X(1108)
X(322) = X(304)-Ceva conjugate of X(312)
X(322) = X(347)-cross conjugate of X(75)

X(323) = ISOTOMIC CONJUGATE OF X(94)

Trilinears    sin 3A csc²A : sin 3B csc²B : sin 3C csc²C
Trilinears    4 sin A - 3 csc A :
Barycentrics    sin 3A csc A : sin 3B csc B : sin 3C csc C
Barycentrics    a^2[(a^2 - b^2 - c^2)^2 - b^2 c^2] : :
Barycentrics    directed distance of A to Hatzipolakis axis : :

Let A'B'C' be the Trinh 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(323). (Randy Hutson, October 13, 2015)

X(323) lies on these lines: 2,6 20,155 23,110 30,146 140,195 187,249 401,525

X(323) = reflection of X(23) in X(110)
X(323) = isogonal conjugate of X(1989)
X(323) = isotomic conjugate of X(94)
X(323) = anticomplement of X(3580)
X(323) = X(340)-Ceva conjugate of X(186)
X(323) = cevapoint of X(6) and X(399)
X(323) = X(50)-cross conjugate of X(186)
X(323) = crosssum of X(395) and X(396)
X(323) = crossdifference of every pair of points on line X(51)X(512)
X(323) = crosspoint of X(6) and X(399) wrt both the excentral and tangential triangles
X(323) = inverse-in-MacBeath-circumconic of X(2)
X(323) = orthocentroidal-to-ABC similarity image of X(110)
X(323) = 4th-Brocard-to-circumsymmedial similarity image of X(110)
X(323) = X(13192)-of-circumsymmedial-triangle

X(324) = ISOTOMIC CONJUGATE OF X(97)

Trilinears bc sec A cos(B - C) : ca sec B cos(C - A) : ab sec C cos(A - B)
Barycentrics sec A cos(B - C) : sec B cos(C - A) : sec C cos(A - B)

X(324) lies on these lines: 2,216 4,52 53,311 94,275 110,436 143,565

X(324) = isotomic conjugate of X(97)
X(324) = X(264)-Ceva conjugate of X(5)
X(324) = cevapoint of X(i) and X(j) for these (i,j): (5,53), (52,216)
X(324) = X(5)-cross conjugate of X(311)
X(324) = trilinear pole of polar of X(54) wrt polar circle
X(324) = pole wrt polar circle of trilinear polar of X(54) (line X(50)X(647))
X(324) = X(48)-isoconjugate (polar conjugate) of X(54)
X(324) = Danneels point of X(264)

X(325) = X(2)-HIRST INVERSE OF X(69)

Trilinears    csc²A : :
Trilinears    bc(b⁴ + c⁴ - a²b² - a²c²) : :
Barycentrics    cot A - tan ω : :
>Barycentrics    b⁴ + c⁴ - a²b² - a²c² : :

Let La be the line through A parallel to the Lemoine axis, and define Lb and Lc cyclically. Let Ma be the reflection of BC in La, and define Mb and Mc cyclically. Let A' = Mb∩Mc, and define 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 the Lemoine axis. The triangle A"B"C" is homothetic to ABC, with center of homothety X(325); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, November 30, 2015)

X(325) lies on the Simson quartic Q101 and these lines:
{2,6}, {3,315}, {4,1975}, {5,76}, {11,350}, {12,1909}, {20,6337}, {22,160}, {23,3447}, {25,317}, {30,99}, {32,3788}, {39,626}, {51,4121}, {74,2855}, {75,2886}, {95,1799}, {98,2065}, {100,2856}, {110,2857}, {111,2858}, {114,511}, {115,538}, {140,1078}, {147,1503}, {187,620}, {194,5025}, {232,297}, {250,340}, {253,6340}, {264,305}, {274,442}, {290,3978}, {310,3136}, {311,1238}, {320,1447}, {332,2893}, {339,1236}, {383,622}, {441,2966}, {523,684}, {532,6108}, {533,6109}, {542,5939}, {621,1080}, {623,6115}, {624,6114}, {631,3785}, {637,6289}, {638,6290}, {639,3103}, {640,3102}, {671,5503}, {672,4766}, {698,1916}, {732,2023}, {740,5988}, {758,5977}, {868,2396}, {892,1494}, {1235,1594}, {1329,6377}, {1368,6375}, {1444,4220}, {1506,3934}, {2071,5866}, {2482,3849}, {3233,6148}, {3734,5475}, {3814,6382}, {3829,4479}, {3847,4357}, {3932,4518}, {5149,5162}, {5965,6036}

X(325) = midpoint of X(i) and X(j) for these (i,j): (99,316), (147,5999), (298,299), (5078,5979)
X(325) = reflection of X(i) in X(j) for these (i,j): (115,625), (187,620), (385,230), (1513,114), (2966,441)
X(325) = isogonal conjugate of X(1976)
X(325) = isotomic conjugate of X(98)
X(325) = complement of X(385)
X(325) = anticomplement of X(230)
X(325) = inverse-in-orthoptic-circle-of-Steiner-inellipe of X(5108)
X(325) = cevapoint of X(2) and X(147)
X(325) = X(i)-cross conjugate of X(j) for these (i,j): (114,2), (511,297)
X(325) = crossdifference of every pair of points on line X(32)X(512)
X(325) = X(2)-Hirst inverse of X(69)
X(325) = {X(2),X(69)}-harmonic conjugate of X(183)
X(325) = perspector of hyperbola {{A,B,C,X(99),PU(37)}}
X(325) = intersection of trilinear polars of X(99), P(37), and U(37)
X(325) = crosspoint of X(2) and X(147) wrt both the excentral and anticomplementary triangles
X(325) = trilinear pole of line X(2799)X(3569)
X(325) = X(115)-of-1st-anti-Brocard-triangle
X(325) = X(114)-of-anti-McCay-triangle
X(325) = intersection of Simson line of X(99) (line X(114)X(325)) and trilinear polar of X(99) (line X(2)X(6))
X(325) = pole wrt polar circle of trilinear polar of X(6531) (line X(25)X(669))
X(325) = X(48)-isoconjugate (polar conjugate) of X(6531)

X(326) = ISOTOMIC CONJUGATE OF X(158)

Trilinears    cot²A : :
Trilinears    b² + c² - S² : :
Trilinears    (distance from A to orthic axis)² : :
Barycentrics    csc A - sin A : :

X(326) lies on these lines: 1,75 48,63 69,73 200,319 255,1102 269,320 322,664 610,662

X(326) = isogonal conjugate of X(1096)
X(326) = isotomic conjugate of X(158)
X(326) = X(i)-Ceva conjugate of X(j) for these (i,j): (304,63), (332,69)
X(326) = X(255)-cross conjugate of X(63)
X(326) = trilinear square of X(63)
X(326) = pole wrt polar circle of trilinear polar of X(6520)
X(326) = X(48)-isoconjugate (polar conjugate) of X(6520)
X(326) = X(92)-isoconjugate of X(1973)
X(326) = trilinear pole of line X(822)X(4131)

X(327) = ISOTOMIC CONJUGATE OF X(182)

Trilinears csc²A sec(A - ω) : csc²B sec(B - ω) : csc²C sec(C - ω)
= sin A csc(2A - 2 ω): sin B csc(2B - 2 ω) : sin C csc(2C - 2 ω)

Barycentrics csc A sec(A - ω) : csc B sec(B - ω) : csc C sec(C - ω)

X(327) lies on these lines: 2,290 4,276 5,76 53,141 69,263 95,160

X(327) = isotomic conjugate of X(182)

X(328) = ISOTOMIC CONJUGATE OF X(186)

Trilinears cot A csc 3A : cot B csc 3B : cot C csc 3C
Barycentrics cos A csc 3A : cos B csc 3B : cos C csc 3C

X(328) lies on these lines: 2,94 69,265 95,99

X(328) = isotomic conjugate of X(186)
X(328) = X(265)-cross conjugate of X(94)

X(329) = ISOTOMIC CONJUGATE OF X(189)

Trilinears    ( -1 - cos A + cos B + cos C)(csc A) : :
Barycentrics    -1 - cos A + cos B + cos C : :
Barycentrics    a^3 + a^2(b + c) - a(b + c)^2 - (b - c)^2(b + c) : :
X(329) = 3X(4) - 4X(10) = 2X(10) - 3X(40) = 3X(20) - X(145) = 2X(1) - 3X(376)

X(329) lies on the Lucas cubic and these lines:
{1,452}, {2,7}, {3,2096}, {4,8}, {5,2095}, {6,4415}, {10,2093}, {11,5825}, {20,78}, {69,189}, {75,14555}, {100,972}, {165,6745}, {193,1999}, {198,1817}, {220,948}, {440,3161}, {971,10430}, {5687,6361}, {5817,7956}.

X(329) = isogonal conjugate of X(1436)
X(329) = isotomic conjugate of X(189)
X(329) = cyclocevian conjugate of X(1034)
X(329) = anticomplement of X(57)
X(329) = anticomplementary conjugate of X(7)
X(329) = X(i)-Ceva conjugate of X(j) for (i,j) = (69,8), (312,2)
X(329) = X(i)-cross conjugate of X(j) for these (i,j): (40,347), (223,2)
X(329) = perspector of triangle ABC and the pedal triangle of X(1490)
X(329) = X(25)-of-2nd-extouch-triangle
X(329) = perspector of 2nd extouch triangle and anticevian triangle of X(8)
X(329) = perspector of ABC and the reflection in X(9) of the pedal triangle of X(9)
X(329) = inverse-in-Fuhrmann-circle of X(5175)
X(329) = orthologic center of 2nd extouch triangle and 1st (or 2nd) mixtilinear triangle
X(329) = trilinear pole of line X(6129)X(8058)
X(329) = perspector of 2nd extouch triangle and cross-triangle of ABC and 2nd extouch triangle

X(330) = ISOTOMIC CONJUGATE OF X(192)

Trilinears bc/(ab + ac - bc) : ca/(bc + ba - ca) : ab/(ca + cb - ab)
Barycentrics 1/(ab + ac - bc) : 1/(bc + ba - ca) : 1/(ca + cb - ab)

X(330) lies on these lines: 1,87 2,1107 8,291 56,385 57,239 76,1015 105,932 145,1002 193,959 257,982

X(330) = isogonal conjugate of X(2176)
X(330) = isotomic conjugate of X(192)
X(330) = X(87)-Ceva conjugate of X(2)
X(330) = X(75)-cross conjugate of X(2)

X(331) = ISOTOMIC CONJUGATE OF X(219)

Trilinears sec²(A/2) csc(2A) : sec²(B/2) csc(2B) : sec²(C/2) csc(2C)
= (1 - sec A)csc(A - ω) : (1 - sec B)csc(B - ω) : (1 - sec C)csc(C - ω)

Barycentrics sec²(A/2) sec A : sec²(B/2) sec B : sec²(C/2) sec C

X(331) lies on these lines: 4,150 7,286 34,870 75,225 85,92 108,767 274,278

X(331) = isotomic conjugate of X(219)
X(331) = cevapoint of X(i) and X(j) for these (i,j): (7,278), (92,273)
X(331) = X(92)-cross conjugate of X(264)

X(332) = ISOTOMIC CONJUGATE OF X(225)

Trilinears (cot A csc A)/(cos B + cos C) : (cot B csc B)/(cos C + cos A) : (cot C csc C)/(cos A + cos B)
Barycentrics (cot A)/(cos B + cos C) : (cot B)/(cos C + cos A) : (cot C)/(cos A + cos B)

X(332) lies on these lines: 1,75 3,69 21,1036 99,102 219,345 261,284 1014,1037

X(332) = isotomic conjugate of X(225)
X(332) = cevapoint of X(i) and X(j) for these (i,j): (69,326), (78,345)
X(332) = X(i)-cross conjugate of X(j) for these (i,j): (69,314), (283,333)

X(333) CEVAPOINT OF X(8) AND X(9)

Trilinears bc(b + c - a)/(b + c) : ca(c + a - b)/(c + a) : ab(a + b - c)/(a + b)
Barycentrics (b + c - a)/(b + c) : (c + a - b)/(c + a) : (a + b - c)/(a + b)

X(333) lies on these lines:
2,6 8,21 9,312 10,58 19,27 29,270 57,85 190,321 239,257 261,284 306,319 310,673 662,909 740,846 859,956 1021,1024

X(333) = isogonal conjugate of X(1400)
X(333) = isotomic conjugate of X(226)
X(333) = X(i)-Ceva conjugate of X(j) for these (i,j): (261,21), (274,86)
X(333) = cevapoint of X(i) and X(j) for these (i,j): (2,63), (8,9), (283,284)
X(333) = X(i)-cross conjugate of X(j) for these (i,j): (8,314), (9,21), (21,86), (283,332), (284,29)
X(333) = crosspoint of X(274) and X(314)
X(333) = crosssum of X(213) and X(1402)
X(333) = crossdifference of every pair of points on line X(512)X(810)

X(334) = ISOTOMIC CONJUGATE OF X(238)

Trilinears b²c²/(a² - bc) : c²a²/(b² - ca) : a²b²/(c² - ab)
Barycentrics bc/(a² - bc) : ca/(b² - ca) : ab/(c² - ab)

X(334) lies on these lines: 2,292 10,274 12,85 75,141 76,1089 150,295 320,660 741,839 767,813

X(334) = isogonal conjugate of X(2210)
X(334) = isotomic conjugate of X(238)
X(334) = X(75)-Hirst inverse of X(335)

X(335) = ISOTOMIC CONJUGATE OF X(239)

Trilinears bc/(a² - bc) : ca/(b² - ca) : ab/(c² - ab)
Barycentrics 1/(a² - bc) : 1/(b² - ca) : 1/(c² - ab)

X(335) lies on these lines: 1,384 2,38 7,192 27,295 37,86 75,141 76,871 239,518 257,694 310,321 320,742 536,903 675,813 741,835 876,900

X(335) = reflection of X(i) in X(j) for these (i,j): (75,1086), (190,37)
X(335) = isogonal conjugate of X(1914)
X(335) = isotomic conjugate of X(239)
X(335) = cevapoint of X(i) and X(j) for these (i,j): (37,518), (292,295)
X(335) = X(i)-cross conjugate of X(j) for these (i,j): (295,337), (350,257)
X(335) = X(i)-Hirst inverse of X(j) for these (i,j): (2,291), (75,334), (292,894)
X(335) = trilinear pole of line X(10)X(514)
X(335) = pole wrt polar circle of trilinear polar of X(242)
X(335) = X(48)-isoconjugate (polar conjugate) of X(242)

X(336) = ISOTOMIC CONJUGATE OF X(240)

Trilinears csc A cot A sec(A + ω) : csc B cot B sec(B + ω) : csc C cot C sec(C + ω)
Barycentrics cot A sec(A + ω) : cot B sec(B + ω) : cot C sec(C + ω)

X(336) lies on these lines: 1,811 48,75 73,290 255,293

X(336) = isotomic conjugate of X(240)

X(337) = ISOTOMIC CONJUGATE OF X(242)

Trilinears (csc A cot A)/(a² - bc) : (csc B cot B)/(b² - ca) : (csc C cot C)/(c² - ab)
Barycentrics (cot A)/(a² - bc) : (cot B)/(b² - ca) : (cot C)/(c² - ab)

X(337) lies on these lines: 12,85 37,86 72,295 201,348 291,986

X(337) = isotomic conjugate of X(242)
X(337) = X(295)-cross conjugate of X(335)

X(338) CEVAPOINT OF X(115) AND X(125)

Trilinears (b² - c²)²/a³ : (c² - a²)²/b³ : (a² - b²)²/c³
= csc A sin²(B - C) : csc B sin²(C - A) : csc C sin²(A - B)

Barycentrics (b² - c²)²/a² : (c² - a²)²/b² : (a² - b²)²/c²
= sin²(B - C) : sin²(C - A) : sin²(A - B)

X(338) lies on these lines:
2,94 4,67 6,264 50,401 76,599 115,127 125,136 141,311

X(338) = isotomic conjugate of X(249)
X(338) = X(264)-Ceva conjugate of X(523)
X(338) = cevapoint of X(115) and X(125)
X(338) = X(125)-cross conjugate of X(339)

X(339) = ISOTOMIC CONJUGATE OF X(250)

Trilinears (b² - c²)²(cos A)/a⁴ : (c² - a²)²(cos B)/b⁴ : (a² - b²)²(cos C)/c⁴
= csc A cot A sin²(B - C) : csc B cot B sin²(C - A) : csc C cot C sin²(A - B)

Barycentrics (b² - c²)²(cos A)/a³ : (c² - a²)²(cos B)/b³ : (a² - b²)²(cos C)/c³
= cot A sin²(B - C) : cot B sin²(C - A) : cot C sin²(A - B)

X(339) lies on the MacBeath inconic and these lines: 3,76 69,265 115,127 264,381

X(339) = isotomic conjugate of X(250)
X(339) = X(i)-Ceva conjugate of X(j) for these (i,j): (76,525), (305,3267)
X(339) = X(125)-cross conjugate of X(338)
X(339) = crosspoint of X(305) and X(3267)

X(340) = ISOTOMIC CONJUGATE OF X(265)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where
f(A,B,C) = sec A sin 3A csc³A

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = sec A sin 3A csc²A

X(340) lies on these lines: 4,69 67,290 95,140 250,325 297,524 298,470 299,471 447,540 458,599 520,850

X(340) = reflection of X(648) in X(297)
X(340) = isotomic conjugate of X(265)
X(340) = anticomplement of X(3284)
X(340) = cevapoint of X(186) and X(323)

X(341) = ISOTOMIC CONJUGATE OF X(269)

Trilinears b²c²(b + c - a)² : c²a²(c + a - b)² : a²b²(a + b - c)²
= csc⁴A/2 : csc⁴B/2 : csc⁴C/2

Barycentrics bc(b + c - a)² : ca(c + a - b)² : ab(a + b - c)²

X(341) lies on these lines: 1,1050 8,210 10,75 40,190 200,1043 253,322 280,345 304,668 305,1088

X(341) = isogonal conjugate of X(1106)
X(341) = isotomic conjugate of X(269)
X(341) = X(346)-cross conjugate of X(312)

X(342) = ISOTOMIC CONJUGATE OF X(271)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc 2A tan A/2)(1 + cos A - cos B - cos C)
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sec A tan A/2)(1 + cos A - cos B - cos C)

X(342) lies on these lines: 4,7 9,653 85,264 92,226 108,1005 196,329 253,318 393,948

X(342) = isogonal conjugate of X(2188)
X(342) = isotomic conjugate of X(271)
X(342) = X(i)-Ceva conjugate of X(j) for these (i,j): (85,92), (264,273)
X(342) = cevapoint of X(208) and X(223)

X(343) = ISOTOMIC CONJUGATE OF X(275)

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

X(343) lies on these lines:
2,6 3,68 5,51 22,161 53,311 76,297 140,569 315,458 427,511 470,634 471,633 472,621 473,622

X(343) = isotomic conjugate of X(275)
X(343) = complement of X(1993)
X(343) = X(i)-Ceva conjugate of X(j) for these (i,j): (76,311), (311,5)
X(343) = X(216)-cross conjugate of X(5)
X(343) = crosspoint of X(69) and X(76)
X(343) = crosssum of X(i) and X(j) for these (i,j): (6,571), (25,32)

X(344) = ISOTOMIC CONJUGATE OF X(277)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where
f(A,B,C) = (csc²A/2)[cos⁴(B/2) + cos⁴(C/2) - cos⁴(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(344) lies on these lines:
2,37 7,190 8,480 9,69 44,193 45,141 144,320 264,281 319,391

X(344) = isotomic conjugate of X(277)

X(345) = ISOTOMIC CONJUGATE OF X(278)

Trilinears (csc A)/(1 - sec A) : (csc B)/(1 - sec B) : (csc C)/(1 - sec C)
= bc(b + c - a)(b² + c² - a²) : ca(c + a - b)(c² + a² - b²) : ab(a + b - c)(a² + b² - c²)

Barycentrics 1/(1 - sec A) : 1/(1 - sec B) : 1/(1 - sec C)

X(345) lies on these lines:
2,37 8,21 22,100 57,728 63,69 78,1040 190,329 219,332 280,341 304,348 498,1089

X(345) = isogonal conjugate of X(608)
X(345) = isotomic conjugate of X(278)
X(345) = X(i)-Ceva conjugate of X(j) for these (i,j): (304,69), (332,78)
X(345) = X(i)-cross conjugate of X(j) for these (i,j): (78,69), (219,8), (306,312)

X(346) = ISOTOMIC CONJUGATE OF X(279)

Trilinears    bc(b + c - a)² : :
Trilinears    cos(A/2) csc³(A/2) : :
Barycentrics   (b + c - a)² : :
Barycentrics   (1 + cos A)(1 - cos A) : :
X(346) = 2 X[2] + 19 X[345] = X[1743] + 2 X[4072] = 17 X[4686] - 2 X[4688] = 19 X[4688] - 4 X[4788] = 5 X[37] - 14 X[4821]

The cevian triangle of X(346) is perspective to the Ayme triangle; see X(3610).

X(346) lies on the cubics K605 and K697 and these lines:
{1, 1219}, {2, 37}, {4, 3695}, {6, 145}, {7, 3729}, {8, 9}, {10, 3731}, {19, 3610}, {20, 1766}, {44, 3621}, {45, 594}, {55, 3974}, {69, 144}, {78, 280}, {100, 198}, {101, 2370}, {141, 4419}, {142, 4659}, {193, 6542}, {200, 4082}, {210, 11997}, {219, 644}, {220, 1043}, {253, 306}, {279, 304}, {281, 318}, {341, 3965}, {347, 4552}, {387, 2901}, {480, 4012}, {497, 3703}, {519, 1743}, {527, 4488}, {545, 7232}, {573, 1018}, {646, 3596}, {672, 10453}, {726, 4310}, {894, 3945}, {941, 1500}, {962, 10445}, {1023, 6790}, {1086, 4373}, {1089, 3085}, {1100, 3623}, {1125, 4098}, {1146, 6558}, {1212, 4673}, {1249, 1897}, {1260, 2322}, {1376, 1696}, {1400, 3501}, {1441, 8232}, {1449, 3241}, {1698, 3986}, {1761, 4427}, {1901, 3936}, {2178, 4188}, {2256, 5782}, {2264, 3189}, {2268, 2329}, {2285, 3600}, {2298, 4195}, {2303, 11115}, {2310, 4073}, {2550, 3932}, {2551, 3704}, {2899, 8165}, {2951, 9950}, {3008, 4402}, {3039, 8168}, {3062, 10324}, {3146, 7270}, {3217, 3684}, {3219, 3719}, {3244, 4898}, {3247, 3616}, {3294, 9534}, {3416, 5698}, {3452, 8055}, {3618, 4360}, {3619, 4389}, {3620, 6646}, {3632, 3973}, {3633, 4856}, {3661, 5232}, {3662, 4346}, {3679, 4058}, {3705, 5274}, {3712, 5218}, {3715, 4046}, {3730, 10449}, {3767, 7230}, {3771, 4135}, {3836, 7613}, {3871, 4254}, {3875, 5222}, {3923, 4307}, {3949, 3952}, {3969, 5739}, {3970, 11036}, {3975, 4110}, {3977, 5744}, {3985, 4451}, {3991, 4385}, {3996, 6605}, {4030, 10385}, {4066, 10198}, {4123, 9539}, {4130, 4397}, {4171, 4529}, {4222, 5687}, {4293, 5525}, {4294, 7206}, {4361, 4422}, {4363, 4648}, {4366, 4473}, {4384, 4431}, {4416, 6172}, {4470, 7227}, {4644, 4851}, {4675, 7222}, {4872, 10513}, {4876, 7155}, {4936, 6737}, {5257, 9780}, {5273, 11679}, {5281, 7081}, {5308, 7229}, {5430, 7027}, {5552, 7110}, {6556, 6736}, {7991, 10443}, {8557, 12649}, {8609, 10529}, {9025, 9309}

X(346) = reflection of X(i) in X(j) for these {i,j}: {8, 4901}, {4452, 4000}, {5838, 9}
X(346) = isogonal conjugate of X(1407)
X(346) = isotomic conjugate of X(279)
X(346) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 6552}, {8, 6555}, {304, 10327}, {312, 8}, {341, 5423}, {345, 7080}, {646, 4397}, {1016, 3699}, {1043, 200}, {3699, 4163}, {4076, 4578}, {6557, 6556}, {7258, 7253}
X(346) = X(i)-cross conjugate of X(j) for these (i,j): {200, 8}, {220, 7046}, {728, 5423}, {1260, 1265}, {2310, 7253}, {2321, 7101}, {3239, 6558}, {3694, 3692}, {3965, 2287}, {4081, 4397}, {4082, 341}, {4130, 4578}, {4163, 3699}, {4515, 200}, {4529, 7258}, {4907, 7}, {4953, 522}, {5423, 6556}, {6554, 2}
X(346) = cevapoint of X(i) and X(j) for these (i,j): {1, 10860}, {6, 1604}, {9, 2324}, {37, 8804}, {200, 728}, {220, 1260}, {650, 4534}, {1146, 3239}, {2310, 4171}, {2321, 3694}, {4081, 4130}, {4082, 4515}
X(346) = crosspoint of X(i) and X(j) for these (i,j): {2, 6553}, {8, 6557}, {312, 341}, {646, 4076}, {1016, 3699}
X(346) = crosssum of X(i) and X(j) for these (i,j): {6, 1616}, {604, 1106}, {649, 3937}
X(346) = crossdifference of every pair of points on line {667, 6363, 7250}
X(346) = complement of X(4452)
X(346) = anticomplement of X(4000)
X(346) = pole wrt polar circle of trilinear polar of X(1119)
X(346) = X(48)-isoconjugate (polar conjugate) of X(1119)
X(346) = centroid of the set consisting of the interiors (with or without boundaries) of the 3 Soddy circles
X(346) = polar conjugate of X(1119)
X(346) = X(i)-beth conjugate of X(j) for these (i,j): {190, 347}, {346, 9}, {644, 573}, {645, 144}, {646, 346}, {1043, 390}, {6558, 346}, {7258, 3596}, {7259, 219}
X(346) = X(i)-gimel conjugate of X(j) for these (i,j): {4462, 346}, {8712, 346}
X(346) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 1407}, {905, 649}
X(346) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1037, 7}, {7084, 2}, {7123, 8}, {7131, 3434}
X(346) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 6552}, {2137, 2886}, {6553, 2887}
X(346) = X(i)-Hirst inverse of X(j) for these (i,j): {8, 3717}, {3975, 4110}
X(346) = trilinear pole of line {3239, 3900, 4148, 4163, 4524, 4528, 4546, 4990}
X(346) = isoconjugate of X(j) and X(j) for these (i,j): {1, 1407}, {2, 1106}, {3, 1435}, {4, 7099}, {6, 269}, {7, 604}, {8, 7366}, {9, 7023}, {19, 7053}, {25, 7177}, {27, 1410}, {31, 279}, {32, 1088}, {34, 222}, {40, 6612}, {41, 479}, {48, 1119}, {55, 738}, {56, 57}, {58, 1427}, {60, 7147}, {63, 1398}, {65, 1412}, {73, 1396}, {77, 608}, {81, 1042}, {84, 6611}, {85, 1397}, {109, 3669}, {110, 7216}, {184, 1847}, {221, 1422}, {223, 1413}, {226, 1408}, {241, 1416}, {244, 1262}, {266, 7370}, {278, 603}, {348, 1395}, {512, 4637}, {513, 1461}, {593, 1254}, {649, 934}, {650, 6614}, {658, 667}, {662, 7250}, {663, 4617}, {669, 4635}, {764, 4619}, {798, 4616}, {849, 6354}, {1014, 1400}, {1015, 7045}, {1020, 3733}, {1118, 7125}, {1122, 3451}, {1275, 3248}, {1333, 3668}, {1357, 4564}, {1358, 2149}, {1402, 1434}, {1403, 7153}, {1414, 7180}, {1415, 3676}, {1417, 3911}, {1426, 1790}, {1431, 7175}, {1439, 1474}, {1440, 2199}, {1446, 2206}, {1458, 1462}, {1472, 7365}, {1919, 4569}, {1973, 7056}, {2150, 6046}, {2170, 7339}, {2171, 7341}, {2185, 7143}, {2221, 4320}, {3063, 4626}, {3937, 7128}, {4017, 4565}, {4559, 7203}, {6358, 7342}, {7132, 7248}, {7183, 7337}
X(346) = barycentric product X(i)*X(j) for these {i,j}: {1, 341}, {4, 1265}, {7, 5423}, {8, 8}, {9, 312}, {10, 1043}, {11, 4076}, {21, 3701}, {29, 3710}, {33, 3718}, {55, 3596}, {63, 7101}, {69, 7046}, {75, 200}, {76, 220}, {78, 318}, {85, 728}, {86, 4082}, {92, 3692}, {100, 4397}, {145, 6556}, {179, 400}, {188, 7027}, {190, 3239}, {210, 314}, {219, 7017}, {261, 6057}, {264, 1260}, {274, 4515}, {280, 7080}, {281, 345}, {304, 7079}, {305, 7071}, {306, 2322}, {313, 2328}, {321, 2287}, {333, 2321}, {480, 6063}, {514, 6558}, {522, 3699}, {523, 7256}, {556, 6731}, {561, 1253}, {594, 7058}, {643, 4086}, {644, 4391}, {645, 3700}, {646, 650}, {657, 1978}, {661, 7258}, {664, 4163}, {668, 3900}, {670, 4524}, {693, 4578}, {799, 4171}, {1016, 1146}, {1021, 4033}, {1089, 1098}, {1222, 6736}, {1229, 6605}, {1264, 1857}, {1320, 4723}, {1577, 7259}, {1639, 4582}, {1802, 1969}, {2310, 7035}, {2319, 4110}, {2325, 4997}, {3161, 6557}, {3685, 4518}, {3686, 4102}, {3912, 6559}, {3952, 7253}, {3975, 4876}, {4012, 8817}, {4041, 7257}, {4073, 7033}, {4081, 4998}, {4087, 7077}, {4092, 6064}, {4105, 4572}, {4130, 4554}, {4148, 4562}, {4373, 6555}, {4451, 7081}, {4528, 4555}, {4673, 4866}, {4990, 6540}, {6386, 8641}, {6552, 6553}
X(346) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 269}, {2, 279}, {3, 7053}, {4, 1119}, {6, 1407}, {7, 479}, {8, 7}, {9, 57}, {10, 3668}, {11, 1358}, {12, 6046}, {19, 1435}, {21, 1014}, {25, 1398}, {31, 1106}, {33, 34}, {37, 1427}, {41, 604}, {42, 1042}, {48, 7099}, {55, 56}, {56, 7023}, {57, 738}, {59, 7339}, {60, 7341}, {63, 7177}, {69, 7056}, {72, 1439}, {75, 1088}, {78, 77}, {92, 1847}, {99, 4616}, {100, 934}, {101, 1461}, {109, 6614}, {144, 9533}, {181, 7143}, {188, 7371}, {190, 658}, {198, 6611}, {200, 1}, {210, 65}, {212, 603}, {219, 222}, {220, 6}, {228, 1410}, {259, 7370}, {261, 552}, {280, 1440}, {281, 278}, {282, 1422}, {284, 1412}, {294, 1462}, {312, 85}, {318, 273}, {321, 1446}, {333, 1434}, {341, 75}, {345, 348}, {388, 7197}, {390, 3598}, {480, 55}, {497, 7195}, {512, 7250}, {522, 3676}, {556, 555}, {594, 6354}, {604, 7366}, {607, 608}, {612, 4320}, {643, 1414}, {644, 651}, {645, 4573}, {646, 4554}, {650, 3669}, {651, 4617}, {657, 649}, {661, 7216}, {662, 4637}, {664, 4626}, {668, 4569}, {728, 9}, {756, 1254}, {765, 7045}, {799, 4635}, {984, 7204}, {1016, 1275}, {1018, 1020}, {1021, 1019}, {1043, 86}, {1098, 757}, {1146, 1086}, {1172, 1396}, {1212, 1418}, {1252, 1262}, {1253, 31}, {1259, 1804}, {1260, 3}, {1261, 1476}, {1264, 7055}, {1265, 69}, {1334, 1400}, {1436, 6612}, {1604, 6609}, {1792, 1444}, {1802, 48}, {1824, 1426}, {1857, 1118}, {1863, 1851}, {2171, 7147}, {2175, 1397}, {2192, 1413}, {2194, 1408}, {2195, 1416}, {2212, 1395}, {2287, 81}, {2289, 7125}, {2310, 244}, {2318, 73}, {2319, 7153}, {2321, 226}, {2322, 27}, {2324, 223}, {2325, 3911}, {2327, 1790}, {2328, 58}, {2329, 7175}, {2332, 1474}, {2340, 1458}, {2345, 7365}, {2968, 1565}, {3022, 3271}, {3056, 7248}, {3057, 1122}, {3059, 354}, {3119, 2170}, {3158, 1420}, {3161, 5435}, {3190, 4306}, {3208, 1423}, {3239, 514}, {3270, 3937}, {3271, 1357}, {3596, 6063}, {3684, 1429}, {3685, 1447}, {3686, 553}, {3687, 3674}, {3688, 1401}, {3689, 1319}, {3690, 1425}, {3692, 63}, {3693, 241}, {3694, 1214}, {3695, 6356}, {3699, 664}, {3700, 7178}, {3701, 1441}, {3703, 3665}, {3705, 7185}, {3706, 4059}, {3707, 4031}, {3709, 7180}, {3710, 307}, {3711, 2099}, {3712, 7181}, {3713, 940}, {3715, 5221}, {3717, 9436}, {3718, 7182}, {3719, 7183}, {3737, 7203}, {3790, 7179}, {3811, 4341}, {3870, 4350}, {3900, 513}, {3939, 109}, {3952, 4566}, {3965, 3666}, {3974, 388}, {3975, 10030}, {4007, 4654}, {4012, 497}, {4030, 7198}, {4041, 4017}, {4046, 3649}, {4060, 3982}, {4061, 3671}, {4069, 4551}, {4073, 982}, {4076, 4998}, {4081, 11}, {4082, 10}, {4086, 4077}, {4092, 1365}, {4095, 4032}, {4103, 4605}, {4105, 663}, {4113, 4955}, {4123, 7210}, {4130, 650}, {4148, 812}, {4152, 1317}, {4157, 7214}, {4163, 522}, {4171, 661}, {4178, 7217}, {4183, 28}, {4319, 614}, {4397, 693}, {4420, 1442}, {4433, 1284}, {4451, 7249}, {4477, 4367}, {4511, 1443}, {4512, 3361}, {4513, 6180}, {4515, 37}, {4517, 1469}, {4518, 7233}, {4524, 512}, {4528, 900}, {4529, 4369}, {4546, 3667}, {4548, 7251}, {4571, 6516}, {4578, 100}, {4587, 1813}, {4827, 4790}, {4847, 10481}, {4853, 7271}, {4873, 5219}, {4882, 4328}, {4907, 5573}, {4936, 1743}, {4953, 3756}, {4990, 4977}, {5423, 8}, {5532, 7336}, {5546, 4565}, {5547, 7316}, {6056, 7335}, {6057, 12}, {6058, 7314}, {6059, 7337}, {6060, 7338}, {6061, 60}, {6062, 1354}, {6064, 7340}, {6065, 59}, {6068, 3321}, {6335, 13149}, {6552, 4452}, {6554, 4000}, {6555, 145}, {6556, 4373}, {6558, 190}, {6559, 673}, {6600, 1617}, {6602, 41}, {6603, 6610}, {6605, 1170}, {6607, 2488}, {6726, 266}, {6731, 174}, {6736, 3663}, {6737, 3664}, {6745, 1323}, {7017, 331}, {7027, 4146}, {7046, 4}, {7054, 593}, {7058, 1509}, {7062, 1355}, {7063, 1356}, {7064, 181}, {7065, 1363}, {7067, 1366}, {7068, 1367}, {7069, 1393}, {7070, 1394}, {7071, 25}, {7074, 221}, {7075, 1424}, {7079, 19}, {7080, 347}, {7081, 7176}, {7101, 92}, {7102, 7103}, {7156, 3213}, {7172, 3600}, {7253, 7192}, {7256, 99}, {7257, 4625}, {7258, 799}, {7259, 662}, {7283, 6359}, {7322, 7273}, {7359, 6357}, {7360, 5088}, {7367, 1436}, {7368, 198}, {8012, 1475}, {8641, 667}, {8706, 6613}, {8805, 8810}, {10382, 1467}

X(347) = ISOTOMIC CONJUGATE OF X(280)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (-1 - cos A + cos B + cos C) sec²(A/2)
Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(347) lies on these lines:
1,7 2,92 8,253 34,452 37,948 69,664 75,280 144,219 223,329 227,322 241,1108 573,1020

X(347) = isogonal conjugate of X(2192)
X(347) = isotomic conjugate of X(280)
X(347) = anticomplement of X(281)
X(347) = X(i)-Ceva conjugate of X(j) for these (i,j): (75,7), (348,2)
X(347) = cevapoint of X(40) and X(223)
X(347) = X(i)-cross conjugate of X(j) for these (i,j): (40,329), (221,196), (227,223)
X(347) = crosspoint of X(75) and X(322)

X(348) = ISOTOMIC CONJUGATE OF X(281)

Trilinears cot A sec²(A/2) : cot B sec²(B/2) : cot C sec²(C/2)
= (csc A)/(1 + sec A) : (csc B)/(1 + sec B) : (csc C)/(1 + sec C)

Barycentrics 1/(1 + sec A) : 1/(1 + sec B) : 1/(1 + sec C)

X(348) lies on these lines: 2,85 7,21 8,664 69,73 75,280 150,944 201,337 274,278 304,345 499,1111

X(348) = isogonal conjugate of X(607)
X(348) = isotomic conjugate of X(281)
X(348) = X(274)-Ceva conjugate of X(85)
X(348) = cevapoint of X(i) and X(j) for these (i,j): (2,347), (63,77)
X(348) = X(222)-cross conjugate of X(7)

X(349) = ISOTOMIC CONJUGATE OF X(284)

Trilinears (cos B + cos C)csc³A : (cos C + cos A)csc³B : (cos A + cos B) csc³C
= (cos B + cos C)csc(A - ω) : (cos C + cos A)csc(B - ω) : (cos A + cos B)csc(C - ω)

Barycentrics (cos B + cos C)csc²A : (cos C + cos A)csc²B : (cos A + cos B)(csc C/2)²

X(349) lies on these lines: 12,313 73,290 75,225 76,85

X(349) = isotomic conjugate of X(284)
X(349) = cevapoint of X(226) and X(307)
X(349) = X(321)-cross conjugate of X(313)

X(350) = X(2)-HIRST INVERSE OF X(75)

Trilinears (a² - bc)b²c² : (b² - ca)c²a² : (c² - ab)a²b²
Barycentrics bc(a² - bc) : ca(b² - ca) : ab(c² - ab)

X(350) is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(10) and U(10) of bicentric points (see the notes just before X(1908). (Randy Hutson, 9/23/2011)

X(350) lies on these lines:
1,76 2,37 11,325 33,264 36,99 42,308 55,183 69,497 86,310 172,384 190,672 256,314 291,726 305,614 320,513 447,811 519,668 538,1015 889,903

X(350) = isogonal conjugate of X(1911)
X(350) = isotomic conjugate of X(291)
X(350) = crosspoint of X(257) and X(335)
X(350) = crossdifference of every pair of points on line X(213)X(667)
X(350) = X(2)-Hirst inverse of X(75)
X(350) = intersection of trilinear polars of P(10) and U(10)

X(351) = CENTER OF THE PARRY CIRCLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b² - c²)(b² + c² - 2a²)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a²(b² - c²)(b² + c² - 2a²)

X(351) is the center of the Parry circle introduced in TCCT (Art. 8.13) as the circle that passes through X(i) for I = 2, 15, 16, 23, 110, 111, 352, 353.

X(351) lies on these lines: 2,804 110,526 184,686 187,237 694,881 865,888

X(351) = midpoint of X(5638) and X(5639)
X(351) = isogonal conjugate of X(892)
X(351) = crosspoint of X(110) and X(111)
X(351) = crosssum of X(i) and X(j) for these (i,j): (2,690), (523,524), (850,1236)
X(351) = crossdifference of every pair of points on line X(2)X(99)
X(351) = Lemoine axis intercept of trilinear polar of X(512)
X(351) = centroid of the triangle X(3)X(5607)X(5608)
X(351) = tripolar centroid of X(6)
X(351) = centroid of Lemoine axis intercepts with sidelines of ABC
X(351) = X(351)-of-circumsymmedial-triangle
X(351) = intersection of tangents to Steiner inellipse at X(1084) and X(2482)
X(351) = crosspoint wrt medial triangle of X(1084) and X(2482)
X(351) = vertex conjugate of PU(62)
X(351) = radical center of (circumcircle, Brocard circle, McCay circumcircle)
X(351) = harmonic center of circles {{X(14),X(15),X(16)}} and {{X(13),X(15),X(16)}}
X(351) = bicentric difference of PU(i) for these i: 62, 63, 64, 65, 66, 67
X(351) = PU(62)-harmonic conjugate of X(6)
X(351) = PU(63)-harmonic conjugate of X(3)
X(351) = PU(64)-harmonic conjugate of X(1)
X(351) = PU(65)-harmonic conjugate of X(5)
X(351) = PU(66)-harmonic conjugate of X(10)
X(351) = PU(67)-harmonic conjugate of X(39)
X(351) = X(3)-of-1st-Parry-triangle
X(351) = X(3)-of-2nd-Parry-triangle
X(351) = X(3)-of-3rd-Parry-triangle
X(351) = inverse-in-Parry-isodynamic-circle of X(187); see X(2)
X(351) = pole of Brocard axis wrt Parry isodynamic circle
X(351) = bicentric sum of PU(105)
X(351) = PU(105)-harmonic conjugate of X(3124)
X(351) = centroid of (degenerate) cross-triangle of 1st and 3rd Parry triangles
X(351) = Lemoine axis (or line PU(2)) intercept of line connecting P(2)-Ceva conjugate of U(2) and U(2)-Ceva conjugate of P(2)

X(352) = INVERSE-IN-CIRCUMCIRCLE OF X(353)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(-a⁴ - b⁴ - c⁴ - 5b²c² + 4a²b² + 4a²c²)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(352) lies on the Parry circle and these lines: 2,6 3,353 23,2502 110,187 111,511

X(352) = reflection of X(843) in X(187)
X(352) = isogonal conjugate of X(6094)
X(352) = inverse-in-circumcircle of X(353)
X(352) = crossdifference of every pair of points on line X(373)X(512)
X(352) = X(23)-of-circumsymmedial-triangle
X(352) = inverse-in-{circumcircle, nine-point circle}-inverter of X(3815)
X(352) = orthocentroidal-to-ABC similarity image of X(111)
X(352) = 4th-Brocard-to-circumsymmedial similarity image of X(111)
X(352) = X(2709)-of-1st-Parry-triangle
X(352) = X(843)-of-2nd-Parry-triangle
X(352) = X(110)-of-3rd-Parry-triangle
X(352) = inverse-in-Parry-isodynamic-circle of X(23); see X(2)
X(352) = radical trace of circumcircles of Artzt and anti-Artzt triangles

X(353) = INVERSE-IN-BROCARD-CIRCLE OF X(111)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(4a⁴ - 2b⁴ - 2c⁴ - b²c² - 4a²b² -4a²c²)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

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' and C' cyclically. Triangle A'B'C' is the anticomplementary triangle of the 1st Brocard triangle, and is similar to the anti-orthocentroidal triangle, with similitude center X(353). (Randy Hutson, November 2, 2017)

X(353) lies on the Parry circle and these lines: 3,352 6,23 110,574 111,182 187,3117

X(353) = inverse-in-circumcircle of X(352)
X(353) = inverse-in-Brocard-circle of X(111)
X(353) = centroid of circumsymmedial triangle
X(353) = circumcevian isogonal conjugate of X(6)
X(353) = X(5640)-of-4th-anti-Brocard triangle
X(353) = isogonal conjugate, wrt 2nd Brocard triangle, of X(2)
X(353) = inverse-in-Parry-isodynamic-circle of X(2); see X(2)

X(354) = WEILL POINT

Trilinears    (b - c)² - ab - ac : :
Trilinears    2 + cos B + cos C : :
Trilinears    cos^2(B/2) + cos^2(C/2) : :
Trilinears    b(cot B/2) + c(cot C/2) : :
Barycentrics  a[(b - c)² - ab - ac] : :

X(354) is the perspector of the intangents triangle and the triangle Q_AQ_BQ_C described at X(3598). (Peter Moses, Nov. 4, 2010)

Let A', B', C' be the inverse-in-{circumcircle, incircle}-inverter of A, B, C. Let A"B"C" be the tangential triangle of A'B'C'. Then A"B"C" is perspective to ABC at X(354). (Randy Hutson, December 26, 2015)

William Gallatly, The Modern Geometry of the Triangle, 2nd edition, Hodgson, London, 1913, page 16.

Let Ab be the intersection, other than B, of circle {{X(1),B,C}} and line AB. Let Ac be the intersection, other than C, of circle {{X(1),B,C}} and line AC. Define Bc, Ba, Ca, Cb cyclically. X(354) is the centroid of AbAcBcBaCaCb. Note that the lines AbAc, BcBa, CaCb bound the intangents triangle. (Randy Hutson, December 26, 2015)

X(354) lies on these lines: 1,3 2,210 6,374 7,479 11,118 37,38 42,244 44,748 48,584 63,1001 81,105 278,955 373,375 388,938 392,551 516,553

X(354) = isogonal conjugate of X(2346)
X(354) = inverse-in-incircle of X(1155)
X(354) = reflection of X(i) in X(j) for these (i,j): (210,2), (392,551)
X(354) = X(101)-Ceva conjugate of X(513)
X(354) = crosspoint of X(1) and X(7)
X(354) = crosssum of X(1) and X(55)
X(354) = X(2)-of-intouch-triangle
X(354) = homothetic center of intouch triangle and inverse-in-incircle triangle
X(354) = centroid of inverse-in-incircle triangle
X(354) = pole of antiorthic axis wrt incircle
X(354) = {X(1),X(40)}-harmonic conjugate of X(3303)
X(354) = {X(1),X(65)}-harmonic conjugate of X(3057)
X(354) = {X(3513),X(3514)}-harmonic conjugate of X(65)
X(354) = centroid of the six intersections of the Conway circle and the sidelines of ABC
X(354) = centroid of the six intersections of the sidelines of ABC and the antiparallels to sidelines through X(1)
X(354) = inverse-in-{circumcircle, incircle}-inverter of X(36)
X(354) = cevian isogonal conjugate of X(7) = intouch isogonal conjugate of X(7)
X(354) = bicentric sum of PU(94)
X(354) = PU(94)-harmonic conjugate of X(650)
X(354) = X(381)-of-incircle-circles-triangle

X(355) = FUHRMANN CENTER

Trilinears    a cos A - (b + c)cos(B - C) : :
Trilinears    bc(b + c)[a²(b² + c²) - (b² - c²)²] - a³bc(b² + c² - a²) : :   (Michel Garitte, 4/3/03)
Trilinears    cos B + cos C - 2 cos B cos C - 1 : :
Trilinears    r - R cos(B - C) : :

X(355) = the center of the Fuhrmann circle, defined as the circumcircle of the Fuhrmann triangle A"B"C", where A" is obtained as follows: let A' be the midpoint of the circumcircle-arc having endpoints B and C and not containing A; then A" is the reflection of A' in line BC. Vertices B" and C" are obtained cyclically. (Other constructions of A'', and hence the Fuhrmann triangle, follow: (1) Let I_A be the reflection of X(1) in BC; then A'' is the circumcenter of I_ABC. (2) Let J_A be the reflection of the A-excenter in BC; then A'' is the circumcenter of J_ABC.

X(355) is the homothetic center of the triangles formed by internal and external tangents of circumcircles of BCX(4), CAX(4), ABX(4). Equivalently, the incenter of the trianglar hull of circumcircles of BCX(4), CAX(4), ABX(4). (Randy Hutson, December 2, 2017)

Let Na = X(5)-of-BCX(1), Nb = X(5)-of-CAX(1), Nc = X(5)- of-ABX(1). Then X(355) = X(20)-of-NaNbNc. (Randy Hutson, December 2, 2017)

Let A'B'C' be the anticomplementary triangle. Then X(355) is the radical center of the incircles of A'BC, B'CA, C'AB. (Randy Hutson, December 2, 2017)

Let A'B'C' be the outer Garcia triangle and A"B"C" the inner Garcia 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(355); see also X(952). (Randy Hutson, December 2, 2017)

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 6: The Fuhrmann Circle.

X(355) lies on these lines:
1,5 2,944 3,10 4,8 30,40 65,68 85,150 104,404 165,550 381,519 382,516 388,942 938,1056

X(355) = midpoint of X(4) and X(8)
X(355) = isogonal conjugate of X(3417)
X(355) = reflection of X(i) in X(j) for these (i,j): (1,5), (3,10), (944,1385), (1482,946)
X(355) = anticomplement of X(1385)
X(355) = complement of X(944)
X(355) = inverse-in-Feuerbach-hyperbola of X(1837)
X(355) = {X(1),X(80)}-harmonic conjugate of X(1837)
X(355) = outer-Garcia-isogonal conjugate of X(40)
X(355) = X(3)-of-outer-Garcia-triangle
X(355) = X(6265)-of-inner-Garcia-triangle
X(355) = X(6102)-of-excentral-triangle
X(355) = inner-Garcia-to-outer-Garcia similarity image of X(3)
X(355) = X(12844)-of-orthic-triangle if ABC is acute
X(355) = endo-homothetic center of Ehrmann side-triangle and 4th anti-Euler triangle; the homothetic center is X(12111)
X(355) = reflection of X(12699) in X(4)
X(355) = {X(1),X(1837)}-harmonic conjugate of X(5722)
X(355) = Ursa-minor-to-Ursa-major similarity image of X(1)

X(356) = MORLEY CENTER

Trilinears cos A/3 + 2 cos B/3 cos C/3 : :
Trilinears = cos(B/3 - C/3) + sqrt(3)sin(A/3 + π/3) (M. Stevanovic, 12/25/2007)
X(356) = X(3276) + X(3277) - X(3)

X(356) is the centroid of the Morley equilateral triangle. For a discussion of the theorem and extensive list of references, see

C. O. Oakley and J. C. Baker, "The Morley trisector theorem," American Mathematical Monthly 85 (1978) 737-745.

For a sketch of the Morley cubic and list of centers on it, including X(356), X(357), X(358), see

Bernard Gibert's site.

Let A'B'C' be the 1st Morley triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(356). (Randy Hutson, September 14, 2016)

If you have The Geometer's Sketchpad, you can view X(356).

For a biographical sketch, including details about Morley's famous theorem on angle trisectors, with history and references, see

Frank Morley (1860-1937) geometer.

X(356) lies on these lines: 2,5455 3,3278 357,358 1134,1135 3605,5456.

X(356) = isogonal conjugate of X(3605)
X(356) = orthology center of 1st Morley adjunct triangle and 1st Morley triangle

X(357) = 1st MORLEY-TAYLOR-MARR CENTER

Trilinears sec A/3 : sec B/3 : sec C/3
Barycentrics sin A sec A/3 : sin B sec B/3 : sin C sec C/3

X(357) is the perspector of Morley triangle and ABC, and also the Hofstadter 1/3 point. See

F. Glanville Taylor and W. L. Marr, "The six trisectors of each of the angles of a triangle," Proceedings of the Edinburgh Mathematical Society 33 (1913-14) 119-131, especially item 9, page 127.

If you have The Geometer's Sketchpad, you can view X(357).

X(357) lies on these lines: 356,358 1134,3275

X(357) = isogonal conjugate of X(358)

X(358) = 2nd MORLEY-TAYLOR-MARR CENTER

Trilinears cos A/3 : cos B/3 : cos C/3
Barycentrics sin A cos A/3 : sin B cos B/3 : sin C cos C/3

X(358) is the perspector of the adjunct Morley triangle and ABC, and also the Hofstadter 2/3 point.

If you have The Geometer's Sketchpad, you can view X(358).

X(358) lies on these lines: 356,357 16,1135

X(358) = isogonal conjugate of X(357)

X(359) = HOFSTADTER ONE POINT

Trilinears a/A : b/B : c/C
Barycentrics a²/A : b²/B : c²/C

This point is the limit as r approaches 1 of the Hofstadter r point. See X(360) for details.

If you have The Geometer's Sketchpad, you can view X(359) and X(360) and Hofstadter Triangles. These sketches include the Hofstadter ellipse (actually a family of ellipses, indexed by a parameter r) introduced (February 4, 2005) by Peter J. C. Moses. The ellipse E(r) is given for 0 < r < 1 by the following equation in trilinears:

x² + y² + z² + yz(D + 1/D) + zx(E + 1/E) + xy(F + 1/F) = 0, where D = cos A - sin A cot rA, E = cos B - sin B cot rB, F = cos C - sin C cot rC.

The Hofstadter ellipse E(1/2), given by x² + y² + z² - 2yz - 2xz - 2xy = 0, has center X(37) and passes through X(i) for these i: 244, 678, 2310, 2632, 2638, 2643, 3248, 4094, 4117.

Taking the limit of E(r) as r tends to 0 gives information about the circumellipse, E(0) (which is also E(1)):

Equation:     ayz/A + bzx/B + cxy/C = 0
Center:     X(5945) = (a/A)(b²/B + c²/C - a²/A) : (b/B)(c²/C + a²/A - b²/B) : (c/C)(a²/A + b²/B - c²/C)
Intersection with circumcircle (other than A, B, C):     X(3067) = a/[A(B - C)] : b/[B(C - A)] : c/[C(A - B)]

The ellipse E(1/2) is also the incentral inellipse, tangent to ABC at the traces of X(1); this ellipse is the incentral isotomic conjugate of the line X(512)X(4895). The perspector of the ellipse E(0) is X(359). (Randy Hutson, August 8, 2014)

See Hofstadter Ellipse at MathWorld.

X(359) lies on this line: {6,9036}

X(359) = isogonal conjugate of X(360)
X(359) = X(2)-Ceva conjugate of X(5945)
X(359) = X(i)-isoconjugate of X(j) for these (i,j): {1, 360}, {2, 1049}
X(359) = barycentric product X(i)*X(j) for these {i,j}: {1, 1077}, {1028, 1049}
X(359) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 360}, {31, 1049}, {1077, 75}

X(360) = HOFSTADTER ZERO POINT

Trilinears A/a : B/b : C/c
Barycentrics A : B : C
X(360) = A*[A] + B*[B] + C*[C], where A, B, C are the angles and [A], [B], [C] are the vertices

This point is obtained as a limit of perspectors. Let r denote a real number, but not 0 or 1. Using vertex B as a pivot, swing line BC toward vertex A through angle rB and swing line BC about C through angle rC. Let A(r) be the point in which the two swung lines meet. Obtain B(r) and C(r) cyclically. Triangle A(r)B(r)C(r) is the r-Hofstadter triangle; its perspector with ABC, called the Hofstadter r point, is the point given by trilinears

(sin rA)/sin(A - rA) : (sin rB)/sin(B - rB) : (sin rC)/sin(C - rC).

The limit of this point as r approaches 0 is X(360). The two Hofstadter points, X(359) and X(360) are examples of transcendental triangle centers, since they have no trilinear or barycentric representation using only algebraic functions of a,b,c (or sin A, sin B, sin C).

Clark Kimberling, "Hofstadter points," Nieuw Archief voor Wiskunde 12 (1994) 109-114.

Conjecture and corollary (Randy Hutson, August 10, 2014):
If r is an integer other than 0, 1, or 2, then the inverse-in-circumcircle of the Hofstadter r point is the Hofstadter (2-r) point;
thus, since the isogonal conjugate of the Hofstadter r point is the Hofstadter (1-r) point, if r is not -1, 0 or 1,
then the antigonal image of the Hofstadter r point is the Hofstadter -r point.

X(1) = Hofstadter 1/2 point = antigonal image of X(80)
X(3) = Hofstadter 2 point = antigonal image of X(265)
X(4) = Hofstadter -1 point
X(35) = Hofstadter 3/2 point
X(79) = Hofstadter -1/2 point = antigonal image of X(3065)
X(186) = Hofstadter 3 point = antigonal image of X(5962)
X(265) = Hofstadter -2 point = antigonal image of X(3)
X(357) = Hofstadter 1/3 point
X(358) = Hofstadter 2/3 point
X(359) = Hofstadter 0 point
X(360) = Hofstadter 1 point
X(1127) = Hofstadter 1/4 point
X(1129) = Hofstadter 3/4 point
X(5457) = Hofstadter -1/3 point
X(5458) = Hofstadter -2/3 point
X(5961) = Hofstadter 4 point = antigonal image of X(5964)
X(5962) = Hofstadter -3 point = antigonal image of X(186)
X(5963) = Hofstadter 5 point
X(5964) = Hofstadter -4 point = antigonal image of X(5961)

X(360) lies on this line: {2,1115}

X(360) = isogonal conjugate of X(359)
X(360) = anticomplement of X(1115)
X(360) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {7021, 8}, {7041, 69}
X(360) = X(i)-isoconjugate of X(j) for these (i,j): {1, 359}, {6, 1077}
X(360) = barycentric product X(75)*X(1049)
X(360) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1077}, {6, 359}, {1049, 1}, {1077, 1028}, {1085, 1049}

X(361) = X(266)-CEVA CONJUGATE OF X(1)

Trilinears csc B/2 + csc C/2 - csc A/2 : csc C/2 + csc A/2 - csc B/2 : csc A/2 + csc B/2 - csc C/2
Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin A)(csc B/2 + csc C/2 - csc A/2)

The isoscelizer equation au(X) = bv(X) = cw(X) has solution X = X(361).

X(361) lies on these lines: 1,188 164,503 266,978

X(361) = X(266)-Ceva conjugate of X(1)

X(362) = CONGRUENT CIRCUMCIRCLES ISOSCELIZER POINT

Trilinears b cos B/2 + c cos C/2 - a cos A/2

The isoscelizer equations u(X)/a = v(X)/b = w(X)/c have solution X = X(362).

If you have The Geometer's Sketchpad, you can view X(362).

X(362) lies on this line: 57,234

X(362) = X(31)-of-excentral triangle

X(363) = EQUAL PERIMETERS ISOSCELIZER POINT

Trilinears b/(1 + sin B/2) + c/(1 + sin C/2) - a/(1 + sin A/2) : :

If X = X(363), the isoscelizer triangles of X have equal perimeters.

X(363) is the homothetic center of the excentral triangle and the inner Hutson triangle. A construction of the latter follows. The internal bisector of angle A meets the A-excircle in two points. Let P_A be the point closer to line BC and let Q_A be the other point. Define P_B and P_C cyclically, and define Q_B and Q_C cyclically. Let L_A be the line tangent to the A-excircle at P_A, and define L_B and L_C cyclically. Let M_A be the line tangent to the A-excircle at Q_A, and define M_B and M_C cyclically. The inner Hutson triangle is the triangle A'B'C' given by A' = L_B∩L_C, B' = L_C∩L_A, C' = L_A∩L_B; the outer Hutson triangle is given by A'' = M_B∩M_C, B'' = M_C∩M_A, C' = M_A∩M_B. (Based on a description of A'B'C' by Randy Hutson, September 23, 2011)

Peter Moses (November 10, 2011) found trilinears for A'B'C' and A''B''C''. As these are central triangles, trilinears for A' and A'' suffice:

A' = aU_A(a² + b² + c² - 2bc - 2ca - 2ab) + bU_B(b + c - a)(c + a - b) + cU_C(b + c - a)(a + b - c)
: aU_A(a² + b² - 3c² + 2bc + 2ca - 2ab) + U_B(c - a)(-a + b + c)(a - b + c) - cU_C(-a + b + c)(a + b - c)
: aU_A(a² - 3b² + c² + 2bc - 2ca + 2ab) - bU_B(-a + b + c)(a - b + c) + U_C(b - a)(-a + b + c)(a + b - c)

A'' = aU_A(a² + b² + c² - 2bc - 2ca - 2ab) - bU_B(b + c - a)(c + a - b) - cU_C(b + c - a)(a + b - c)
: aU_A(a² + b² - 3c² + 2bc + 2ca - 2ab) - U_B(c - a)(-a + b + c)(a - b + c) + cU_C(-a + b + c)(a + b - c)
: aU_A(a² - 3b² + c² + 2bc - 2ca + 2ab) + bU_B(-a + b + c)(a - b + c) - U_C(b - a)(-a + b + c)(a + b - c),

where U_A = sqrt[bc/((a - b + c)(a + b - c)] = (1/2)csc(A/2), and U_B and U_C are defined cyclically.

A'B'C' is perspective to the following triangles: intouch, hexyl,Yff, and the 1st and 2nd circumperp triangles. A''B''C'' is perspective to these: ABC, intouch, hexyl, Yff, and the 1st and 2nd circumperp triangles. A'B'C' is homothetic to A''B''C'', which is perspective to the excentral triangle at X(168). (Peter Moses, 11/10/11)

If you have The Geometer's Sketchpad, you can view X(363).

Let UVW and U'V'W' be the tangential-midarc triangles introduced in the preamble to X(8075). The inner and outer Hutson triangles introduced at X(363) are the excircles-version of UVW and U'V'W'. (Randy Hutson and César Lozada, August 29, 2015)

X(363) lies on these lines: 1,289 40,164 165,166

X(363) = X(57) of excentral triangle
X(363) = homothetic center of ABC and orthic triangle of inner Hutson triangle

X(364) = WABASH CENTER (EQUAL AREAS ISOSCELIZER POINT)

Trilinears b^1/2 + c^1/2 - a^1/2 : :

If X = X(364), the isoscelizer triangles T(X,a), T(X,b), T(X,c) have equal areas.

If you have The Geometer's Sketchpad, you can view X(364).

X(364) lies on these lines: 1,365 9,366

X(364) = X(366)-Ceva conjugate of X(1)

X(365) = SQUARE ROOT POINT

Trilinears a^1/2 : b^1/2 : c^1/2
Barycentrics a^3/2 : b^3/2 : c^3/2

For a construction of X(365), see the note at X(2), which provides for a construction barycentric square roots which one can easily extend to a construction for trilinear square roots.

X(365) lies on these lines: 1,364 6,2118 43,2068 292,2146 2110,2119 2144,2147

X(365) = isogonal conjugate of X(366)
X(365) = crosssum of X(1) and X(364)

X(366) = ISOGONAL CONJUGATE OF X(365)

Trilinears a^-1/2 : b^-1/2 : c^-1/2
Barycentrics a^1/2 : b^1/2 : c^1/2

See the note at X(365).

X(366) lies on these lines: 2,367 6,2068 9,364

X(366) = isogonal conjugate of X(365)
X(366) = cevapoint of X(1) and X(364)
X(366) = X(367)-cross conjugate of X(1)

X(367) = CROSSPOINT OF X(1) and X(366)

Trilinears b^1/2 + c^1/2 : c^1/2 + a^1/2 : a^1/2 + b^1/2
Barycentrics a(b^1/2 + c^1/2) : b(c^1/2 + a^1/2) : c(a^1/2 + b^1/2)

X(367) lies on these lines: 1,364 2,366

X(367) = crosspoint of X(1) and X(366)
X(367) = crosssum of X(1) and X(365)

X(368) = EQUI-BROCARD CENTER

Trilinears (reasonable trilinears are sought)
Barycentrics (reasonable barycentrics are sought)

The center X for which the triangle XBC, XCA, XAB have equal Brocard angles. Peter Yff proved that X(368) lies on the self-isogonal conjugate cubic with trilinear equation f(a,b,c)u + f(b,c,a)v + f(c,a,b)w = 0, where f(a,b,c) = bc(b² - c²) and, for variable x : y : z, the cubics u, v, w are given by u(x,y,z) = x(y² + z²), v = u(y,z,x), w = u(z,x,y).

Cyril Parry proved that X(368) lies on the anticomplement of the Kiepert hyperbola, this anticomplement being given by the trilinear equation a²(b² - c²)x² + b²(c² - a²)y² + c²(a² - b²)z² = 0.

If you have The Geometer's Sketchpad, you can view X(368) and X(368) With Curves.

X(369) = 1st TRISECTED PERIMETER POINT

Trilinears x : y : z (see below)
Barycentrics ax : by : cz

If you have The Geometer's Sketchpad, you can view X(369).

There exist points A', B', C' on segments BC, CA, AB, respectively, such that AB' + AC' = BC' + BA' = CA' + CB' = (a + b + c)/3, and the lines AA', BB', CC' concur in X(369). Near the end of the 20th century, Yff found trilinears for X(369) in terms of the unique real root, r, of the cubic polynomial

2t³ - 3(a + b + c)t² + (a² + b² + c² + 8bc + 8ca + 8ab)t - (cb² + ac² + ba² + 5bc² +

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) -

This is PART 1: Introduction and Centers X(1) - X(1000)

HOW TO USE ETC

NOTATION AND COORDINATES

Introduced on December 28, 2016: Index of Triangles Referenced in ETC

Introduced on March 21, 2015: Shinagawa coefficients for triangle centers on the Euler line

Introduced on November 1, 2011: Combos

X(1) = INCENTER

X(2) = CENTROID

X(3) = CIRCUMCENTER

X(4) = ORTHOCENTER

X(5) = NINE-POINT CENTER

X(6) = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT)

X(7) = GERGONNE POINT

X(8) = NAGEL POINT

X(9) = MITTENPUNKT

X(10) = SPIEKER CENTER

X(11) = FEUERBACH POINT

X(12) = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11)

X(13) = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)

X(14) = 2nd ISOGONIC CENTER

X(15) = 1st ISODYNAMIC POINT

X(16) = 2nd ISODYNAMIC POINT

X(17) = 1st NAPOLEON POINT

X(18) = 2nd NAPOLEON POINT

X(19) = CLAWSON POINT

X(20) = DE LONGCHAMPS POINT

X(21) = SCHIFFLER POINT

X(22) = EXETER POINT

X(23) = FAR-OUT POINT

X(24) = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE

X(25) = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES

X(26) = CIRCUMCENTER OF THE TANGENTIAL TRIANGLE

X(27) = CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER

X(28) = CEVAPOINT OF X(19) AND X(25)

X(29) = CEVAPOINT OF INCENTER AND ORTHOCENTER

X(30) = EULER INFINITY POINT

X(31) = 2nd POWER POINT

X(32) = 3rd POWER POINT

X(33) = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES

X(34) = X(4)-BETH CONJUGATE OF X(4)

X(35) = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36)

X(36) = INVERSE-IN-CIRCUMCIRCLE OF INCENTER

X(37) = CROSSPOINT OF INCENTER AND CENTROID

X(38) = CROSSPOINT OF X(1) AND X(75)

X(39) = BROCARD MIDPOINT

X(40) = BEVAN POINT

X(41) = X(6)-CEVA CONJUGATE OF X(31)

X(42) CROSSPOINT OF INCENTER AND SYMMEDIAN POINT

X(43) = X(6)-CEVA CONJUGATE OF X(1)

X(44) = X(6)-LINE CONJUGATE OF X(1)

X(45) = X(9)-BETH CONJUGATE OF X(1)

X(46) X(4)-CEVA CONJUGATE OF X(1)

X(47) = X(110)-BETH CONJUGATE OF X(34)

X(48) = CROSSPOINT OF X(1) AND X(63)

X(49) = CENTER OF SINE-TRIPLE-ANGLE CIRCLE

X(50) = X(74)-CEVA CONJUGATE OF X(184)

X(51) = CENTROID OF ORTHIC TRIANGLE

X(52) = ORTHOCENTER OF ORTHIC TRIANGLE

X(53) = SYMMEDIAN POINT OF ORTHIC TRIANGLE

X(54) = KOSNITA POINT

X(55) = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)

X(56) = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)

X(57) ISOGONAL CONJUGATE OF X(9)

X(58) ISOGONAL CONJUGATE OF X(10)

X(59) ISOGONAL CONJUGATE OF X(11)

X(60) = ISOGONAL CONJUGATE OF X(12)

X(61) = ISOGONAL CONJUGATE OF X(17)

X(62) = ISOGONAL CONJUGATE OF X(18)

X(63) = ISOGONAL CONJUGATE OF X(19)

X(64) = ISOGONAL CONJUGATE OF X(20)

X(65) = ORTHOCENTER OF THE INTOUCH TRIANGLE

X(66) = ISOGONAL CONJUGATE OF X(22)

X(67) = ISOGONAL CONJUGATE OF X(23)

X(68) PRASOLOV POINT

X(69) = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE

X(70) = ISOGONAL CONJUGATE OF X(26)

X(71) = ISOGONAL CONJUGATE OF X(27)

X(72) = ISOGONAL CONJUGATE OF X(28)

X(73) CROSSPOINT OF INCENTER AND CIRCUMCENTER

X(74) ISOGONAL CONJUGATE OF EULER INFINITY POINT