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, the web's most extensive mathematics resource, covers much of classical and modern triangle geometry, including sketches and references. For an interactive dynamic version of ETC, visit Triangle Centers with C.a.R..

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).

Another site in which triangle centers play a central role is Bernard Gibert's Cubics in the Triangle Plane.

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

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

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 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).

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*G(a,b,c) + 0*H(a,b,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.

X(1) = INCENTER

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

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

The radii of the excircles are

If you have The Geometer's Sketchpad, 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,

where R denotes the radius of the circumcircle.

The incenter is the identity of the group of triangle centers under "trilinear multiplication" defined by

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)).

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): (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, 162
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), (78,1490), (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), (207,1490), (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) = orthocenter of X(4)X(8)X(5556)
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(2) = CENTROID

Barycentrics  1 : 1 : 1

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.

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(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): (3,381), (4,376), (210,354), (395,396), (668,3227), (670,3228)
X(2) = reflection of X(i) in X(j) for these (i,j): (1,551), (3,549), (4,381), (5,547), (6,597), (20,376), (69,599), (148,671), (376,3), (381,5), (549,140), (551,1125), (599,141), (671,115), (903,1086), (1121,1146)

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(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(3) = CIRCUMCENTER

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

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)

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

X(3) lies on the Thomson cubic, the Darboux cubic, the Napoleon cubic, the Neuberg cubic, 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) = {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), 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(3529,5072,3830), (3529,5076,5073), (3533,3543,5), (3533,3545,5067), (3533,3832,547), (3533,3850,1656), (3533,5059,3850), (3533,5067,2), (3534,5054,381), (3534,5076,3529), (3534,5079,3146), (3534,6644,5899), (3537,3538,4), (3543,3545,3845), (3543,3832,4), (3543,5056,3832), (3543,5067,3850), (3545,3832,3850), (3545,3845,381), (3545,5056,5), (3545,5059,3853), (3545,5067,5056), (3546,3547,2), (3546,3549,6640), (3546,6643,1368), (3547,3548,6639), (3547,6643,5), (3548,3549,2), (3549,6643,2072), (3557,3558,576), (3576,3579,1482), (3579,5482,1764), (3582,4330,4857), (3584,4325,5270), (3587,5709,40), (3592,3594,6), (3592,6200,6519), (3592,6396,6448), (3592,6410,6454), (3592,6419,3311), (3592,6420,6427), (3592,6425,371), (3592,6426,6420), (3592,6428,6417), (3592,6448,6418), (3592,6453,6447), (3592,6454,3312), (3594,6200,6447), (3594,6396,6522), (3594,6409,6453), (3594,6419,6428), (3594,6420,3312), (3594,6425,6419), (3594,6426,372), (3594,6427,6418), (3594,6447,6417), (3594,6453,3311), (3594,6454,6448), (3601,5122,5708), (3616,5603,5901), (3624,5259,4423), (3627,3628,3091), (3627,5072,3843), (3627,5076,3830), (3628,5072,5055), (3628,5076,3851), (3653,3656,551), (3683,5918,1709), (3736,5156,6), (3746,5563,1), (3784,3955,222), (3785,3926,69), (3785,6337,3933), (3796,3917,3167), (3830,3843,4), (3830,3851,3843), (3830,5055,381), (3830,5070,3851), (3830,5073,382), (3832,3850,381), (3832,5056,3545), (3832,5059,3543), (3832,5067,5), (3839,3855,3858), (3839,5066,381), (3839,5068,3855), (3839,5071,5066), (3843,3851,381), (3843,5055,3851), (3843,5070,5), (3843,5073,3830), (3845,3850,3832), (3845,3853,4), (3845,5059,382), (3850,3853,3845), (3851,5055,5), (3851,5070,5055), (3851,5073,4), (3851,5899,1598), (3853,5056,381), (3854,3856,381), (3855,3858,381), (3855,5068,5066), (3855,5071,5068), (3858,3861,3839), (3858,5066,3855), (3861,5066,3858), (3861,5068,381), (3869,4511,5730), (3911,4304,5722), (3916,4855,3940), (3916,5440,72), (3917,5562,1216), (3926,6337,6390), (3927,3940,72), (3933,6390,3926), (3951,3984,72), (4184,4210,2), (4184,4225,21), (4188,4189,2), (4188,4225,4191), (4188,5428,5054), (4210,6636,199), (4218,6636,859), (4251,4253,6), (4251,4262,4258), (4251,5030,4253), (4252,4255,6), (4253,4262,4251), (4253,5030,5022), (4254,5120,6), (4256,4257,6), (4256,4276,5132), (4258,5022,6), (4259,5135,6), (4260,5138,6), (4262,5030,6), (4263,5042,6), (4264,5105,6), (4265,5096,6), (4265,5124,3286), (4266,5053,6), (4268,4271,6), (4272,5115,6), (4273,5165,6), (4274,5114,6), (4275,5153,6), (4276,4278,58), (4277,5035,6), (4279,5145,6), (4283,5009,6), (4284,5037,6), (4287,5036,6), (4289,5043,6), (4290,5109,6), (4293,5218,495), (4297,5267,5450), (4297,5745,5787), (4313,5435,938), (4652,4855,72), (4652,5440,3927), (5004,5005,2), (5010,5204,3295), (5012,6101,195), (5013,5023,32), (5013,5171,1351), (5013,5206,1384), (5013,5210,3053), (5013,5585,5206), (5023,5210,5206), (5028,5033,1692), (5034,5052,6), (5044,5720,5780), (5054,5076,3628), (5054,5079,632), (5055,5070,1656), (5055,5073,3843), (5056,5059,4), (5056,5067,547), (5058,6567,371), (5059,5067,3845), (5062,6566,372), (5068,5071,5), (5070,5073,381), (5072,5076,546), (5072,5079,5), (5076,5079,381), (5092,5188,3398), (5131,5538,5535), (5204,5217,1), (5237,5238,6), (5237,5352,61), (5238,5351,62), (5277,5283,5275), (5351,5352,6), (5396,5398,6), (5406,5407,394), (5408,5409,394), (5418,6560,485), (5420,6561,486), (5433,6284,11), (5446,5462,51), (5446,5892,5462), (5463,5464,599), (5538,5885,1482), (5590,5870,6214), (5591,5871,6215), (5597,5598,354), (5603,6361,962), (5611,5615,1351), (5657,5744,5771), (5692,5693,5694), (5699,5700,5695), (5703,5758,5761), (5703,5759,5763), (5719,5763,5761), (5719,6147,3487), (5731,5744,5768), (5736,5757,5760), (5737,5786,5788), (5744,5768,5770), (5745,5787,5789), (5791,6245,5789), (5876,5944,156), (5889,5890,6102), (5980,5981,183), (6039,6040,5999), (6120,6123,358), (6121,6124,1135), (6122,6125,1137), (6199,6395,6), (6199,6408,3312), (6199,6417,3311), (6199,6418,6417), (6199,6445,6221), (6199,6446,6395), (6200,6221,6445), (6200,6396,6), (6200,6398,6199), (6200,6409,6455), (6200,6410,3312), (6200,6411,6451), (6200,6412,6398), (6200,6419,6453), (6200,6420,6425), (6200,6450,6417), (6200,6452,6395), (6200,6454,3592), (6200,6456,6418), (6200,6480,6433), (6200,6484,6486), (6200,6485,6431), (6200,6500,6472), (6200,6501,6474), (6221,6396,6395), (6221,6398,6), (6221,6408,6501), (6221,6412,6446), (6221,6428,3592), (6221,6447,6425), (6221,6448,6427), (6221,6449,1151), (6221,6450,3312), (6221,6451,6200), (6221,6452,6398), (6221,6455,6449), (6221,6456,372), (6221,6496,6455), (6221,6497,1152), (6221,6519,6453), (6221,6522,6428), (6395,6407,3311), (6395,6417,6418), (6395,6418,3312), (6395,6445,6199), (6395,6446,6398), (6396,6398,6446), (6396,6409,3311), (6396,6410,6456), (6396,6411,6221), (6396,6412,6452), (6396,6419,6426), (6396,6420,6454), (6396,6449,6418), (6396,6451,6199), (6396,6453,3594), (6396,6455,6417), (6396,6481,6434), (6396,6484,6432), (6396,6485,6487), (6396,6500,6475), (6396,6501,6473), (6398,6407,6500), (6398,6411,6445), (6398,6427,3594), (6398,6447,6428), (6398,6448,6426), (6398,6449,3311), (6398,6450,1152), (6398,6451,6221), (6398,6452,6396), (6398,6455,371), (6398,6456,6450), (6398,6496,1151), (6398,6497,6456), (6398,6519,6427), (6398,6522,6454), (6407,6408,6), (6407,6417,371), (6407,6418,6199), (6407,6445,1151), (6407,6446,6418), (6408,6417,6395), (6408,6418,372), (6408,6445,6417), (6408,6446,1152), (6409,6410,6), (6409,6412,372), (6409,6420,6519), (6409,6426,6425), (6409,6430,6437), (6409,6431,6484), (6409,6432,6480), (6409,6437,6486), (6409,6450,6199), (6409,6452,6418), (6409,6454,6447), (6409,6456,6417), (6409,6471,6468), (6409,6497,6395), (6410,6411,371), (6410,6419,6522), (6410,6425,6426), (6410,6429,6438), (6410,6431,6481), (6410,6432,6485), (6410,6438,6487), (6410,6449,6395), (6410,6451,6417), (6410,6453,6448), (6410,6455,6418), (6410,6470,6469), (6410,6496,6199), (6411,6412,6), (6411,6430,6484), (6411,6434,6480), (6411,6438,6433), (6411,6450,6407), (6411,6452,6199), (6411,6497,6417), (6412,6429,6485), (6412,6433,6481), (6412,6437,6434), (6412,6449,6408), (6412,6451,6395), (6412,6496,6418), (6417,6418,6), (6417,6446,372), (6418,6445,371), (6419,6420,6), (6419,6426,3312), (6419,6427,6417), (6419,6447,6199), (6419,6453,371), (6419,6454,3594), (6419,6522,6395), (6420,6425,3311), (6420,6428,6418), (6420,6448,6395), (6420,6453,3592), (6420,6454,372), (6420,6519,6199), (6421,6424,6), (6422,6423,6), (6425,6426,6), (6425,6427,6199), (6425,6453,6221), (6425,6454,6428), (6425,6522,6418), (6426,6428,6395), (6426,6453,6427), (6426,6454,6398), (6426,6519,6417), (6427,6428,6), (6427,6447,3592), (6427,6448,3312), (6427,6449,6425), (6427,6450,6448), (6427,6454,6395), (6427,6455,6519), (6427,6456,6454), (6427,6519,371), (6428,6447,3311), (6428,6448,3594), (6428,6449,6447), (6428,6450,6426), (6428,6453,6199), (6428,6455,6453), (6428,6456,6522), (6428,6522,372), (6429,6430,6), (6429,6431,371), (6429,6433,1151), (6429,6434,6432), (6429,6483,6428), (6429,6487,3312), (6430,6432,372), (6430,6433,6431), (6430,6434,1152), (6430,6482,6427), (6430,6486,3311), (6431,6432,6), (6431,6434,372), (6431,6438,6432), (6431,6482,6447), (6431,6486,6221), (6432,6433,371), (6432,6437,6431), (6432,6483,6448), (6432,6487,6398), (6433,6434,6), (6433,6437,6480), (6433,6486,6449), (6434,6438,6481), (6434,6487,6450), (6435,6436,6), (6437,6438,6), (6437,6480,6221), (6437,6485,3312), (6438,6481,6398), (6438,6484,3311), (6439,6440,6), (6441,6442,6), (6441,6479,3312), (6442,6478,3311), (6445,6446,6), (6445,6450,6501), (6446,6449,6500), (6447,6448,6), (6447,6449,6519), (6447,6450,3594), (6447,6452,6454), (6447,6454,6418), (6447,6456,6448), (6447,6519,6221), (6447,6522,6420), (6448,6449,3592), (6448,6450,6522), (6448,6451,6453), (6448,6453,6417), (6448,6455,6447), (6448,6519,6419), (6448,6522,6398), (6449,6450,6), (6449,6451,6455), (6449,6452,372), (6449,6455,6200), (6449,6456,3312), (6449,6496,6409), (6449,6497,6398), (6449,6522,6419), (6450,6451,371), (6450,6452,6456), (6450,6455,3311), (6450,6456,6396), (6450,6496,6221), (6450,6497,6410), (6450,6519,6420), (6451,6452,6), (6451,6455,6409), (6451,6456,3311), (6451,6497,3312), (6451,6522,6425), (6452,6455,3312), (6452,6456,6410), (6452,6496,3311), (6452,6519,6426), (6453,6454,6), (6453,6482,6480), (6453,6484,6482), (6453,6519,6407), (6454,6483,6481), (6454,6485,6483), (6454,6522,6408), (6455,6456,6), (6455,6496,6451), (6455,6497,372), (6456,6496,371), (6456,6497,6452), (6465,6466,6467), (6468,6469,6), (6468,6471,6470), (6469,6470,6471), (6470,6471,6), (6472,6473,6), (6472,6474,6221), (6473,6475,6398), (6474,6475,6), (6476,6477,6), (6478,6479,6), (6480,6481,6), (6480,6484,1151), (6480,6486,6484), (6480,6487,6432), (6481,6485,1152), (6481,6486,6431), (6481,6487,6485), (6482,6483,6), (6482,6487,6420), (6483,6486,6419), (6484,6485,6), (6484,6486,6433), (6485,6487,6434), (6486,6487,6), (6488,6489,6), (6490,6491,6), (6492,6493,6), (6494,6495,6), (6494,6499,6435), (6495,6498,6436), (6496,6497,6), (6496,6522,6519), (6497,6519,6522), (6498,6499,6), (6500,6501,6), (6519,6522,6), (6566,6567,1570), (6639,6640,2)

X(4) = ORTHOCENTER

Barycentrics  tan A : tan B : tan C

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)

If you have The Geometer's Sketchpad, 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.)

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 cubic, the Darboux cubic, the Napoleon cubic, the Neuberg cubic, and these lines:
1,33   2,3   6,53   7,273   8,72   9,10   11,56   12,55   13,61   14,62   15,17   16,18   32,98   35,498   36,499   37,1841   39,232   42,1860   46,90   48,1881   49,156   51,185   52,68   54,184   57,84   64,1853   65,158   67,338   69,76   74,107   78,908   79,1784   80,1825   83,182   93,562  94,143   96,231   99,114   100,119   101,118   102,124   103,116   109,117   110,113   111,1560   120,1292   121,1293   122,1294   123,1295   126,1296   127,1289   128,930   129,1303   130,1298   131,135   137,933   141,1350   145,149   147,148   150,152   155,254   162,270   165,1698   171,601   193,1351   195,399   204,1453   218,294   238,602   240,256   250,1553   252,1487   276,327   279,1565   282,3351   371,485   372,486   390,495   394,1217   477,1304   484,3483   487,489   488,490   496,999   512,879   523,1552   542,576   569,1179   572,1474   574,1506   575,598   579,1713   580,1714   590,1151   608,1518   615,1152   616,627   617,628   653,1156   774,1254   801,1092   842,935   937,1534   940,1396   941,1880   953,1309   1036,1065   1037,1067   1038,1076   1039,1096   1040,1074   1073,3350   1138,2132   1157,3482   1160,1162   1161,1163   1251,1832   1329,1376   1340,1348   1341,1349   1385,1538   1430,1468   1499,1550   1715,1730   1716,1721   1717,1718   1726,1782   3065,3464   3347,3472   3348,3355

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), (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) = eigencenter of cevian triangle of X(i) for I = 1, 88, 162
X(4) = eigencenter of anticevian triangle of X(i) for I = 1, 44, 513

X(4) = X(i) -Ceva conjugate of X(j) for these (i,j):
(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(5) = NINE-POINT CENTER

Barycentrics  a cos(B - C) : b cos(C - A) : c cos(A - B)
                        = h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = a²(b² + c²) - (b² - c²)²

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.

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

X(5) lies on the Napoleon cubic (also known as the Feuerbach cubic) and these lines:
1,11   2,3   6,68   8,1389   9,1729   10,517   13,18   14,17   15,2913   16,2912   32,230   33,1062   34,1060   39,114   40,1698   46,1836   49,54   51,52   53,216   55,498   56,499   57,1728   65,1737   69,1351   72,908   76,262   79,1749   83,98   85,1565   96,1166   113,125   116,118   117,124   122,133   127,132   128,137   129,130   131,136   141,211   142,971   156,184   182,206   183,315   195,3459   217,1625   225,1465   226,912   252,1157   264,1093   298,634   299,633   302,622   303,621   311,1225   316,1078   339,1235   371,590   372,615   386,1834   388,999   392,1512   491,637   492,638   515,1125   524,576   539,1493   542,575   570,1879   573,1213   578,1147   579,1901   582,1754   601,750   602,748   618,629   619,630   842,1287   920,1454   1073,1217   1090,1091   1155,1770   1173,1487   1181,1899   1214,1838   1498,1853   1848,1871   1861,1872   2120,2121   3460,3461   3462,3463   3468,3469   3470,3471

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) = 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)} (Randy Hutson, August 23, 2011)

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

Barycentrics  a² : b² : c²

style="font-weight: bold">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.

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

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, 1995. 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)

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) = insimilicenter of 1st and 2nd Kenmotu circles X(6) = harmonic center of circumcircle and Gallatly circle
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) = crossdifference of every pair of points on line X(30)X(511)
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(7) = GERGONNE POINT

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.

X(7) lies on 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(8) = NAGEL POINT

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 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)

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

X(8) lies on these lines:
1,2   3,100   4,72   5,1389   6,594   7,65   9,346   11,1320   19,1891   20,40   21,55   29,219   31,987   33,1039   34,1041   35,993   37,941   38,986   56,404   57,1219   58,996   76,668   79,758   80,149   81,1010   101,1311   140,1483   144,516   171,1468   175,1270   176,1271   177,556   178,236   181,959   190,528   192,256   193,894   194,730   197,1603   210,312   213,981   220,294   221,651   224,914   238,983   253,307   274,1002   277,1280   278,1257   291,330   314,1264   315,760   326,1442   344,480   348,664   392,1000   405,943   406,1061   442,495   443,942   474,999   475,1063   491,1267   595,1724   599,1086   631,1385   643,1098   726,1278   860,1068   908,946   961,1460   1015,1574   1016,1083   1034,1895   1036,1183   1124,1377   1211,1834   1281,1282   1317,1388   1335,1378   1500,1573   1672,1680   1673,1681   1674,1679   1675,1679   1857,1896

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) = reflection of X(i) in X(j) for these (i,j): (1,10), (4,355), (20,40), (100,1145), (145,1), (149,80), (192,984), (390,9), (944,2), (962,4), (1320,11), (1482,5), (1483,140)

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): (69,329), (72,2), (312,346), (314,312), (333,9)
X(8) = X(i) -cross conjugate of X(j) for these (i,j): (1,280), (9,2), (10,318), (11,522), (55,281), (72,78), (200,346), (210,9), (219,345), (497,7), (521,100)
X(8) = cevapoint 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)
X(8) = crosspoint of X(i) and X(j) for these (i,j): (75,312), (314,333)
X(8) = crosssum of X(i) and X(j) for these (i,j): (1,978), (31,604), (57,1423), (667,1015), (1042,1410), (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(9) = MITTENPUNKT

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

X(9) = (r + 2R)*X(1) - 6*X(2) -2r*X(3)

X(9) = perspector of ABC and the medial triangle of the extouch triangle of ABC. (Randy Hutson, 9/23/2011)

X(9): Let A' be the orthocorrespondent of the A-excenter, and define B', C' cyclically. The lines AA', BB', CC' concur in X(9). X(9): 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 similtide center X(9). (Randy Hutson, November 18, 2015)

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

X(9) lies on the Thomson cubic and these lines:
1,6   2,7   3,84   4,10   5,1729   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) = inverse-in-circumconic-centered-at-X(1) of X(6603)
X(9) = orthocenter of X(1)X(4)X(885)

X(10) = SPIEKER CENTER

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.

X(10) lies on 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) = 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) = 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(11) = FEUERBACH POINT

Barycentrics  a(1 - cos(B - C)) : b(1 - cos(C - A)) : c(1 - cos(A -B))
                         = h(a,b,c) : h(b,c,a) : h(c,a,b), where h(A,B,C) = (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.

If you have The Geometer's Sketchpad, 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 these lines:
1,5   2,55   3,499   4,56   7,658   8,1320   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(12) = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11)

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).

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 ABC and triangular hull of circumcircles of BCX(4), CAX(4), and ABX(4)
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(13) = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)

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(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.

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, resp. Let A' be the isogonal conjugate of Na', wrt NaNbNc, and define B', C' cyclically. The lines NaA', NbB', NcC' concur in X(13). (Randy Hutson, January 29, 2015)

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(14) = 2nd ISOGONIC CENTER

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)

If you have The Geometer's Sketchpad, 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(15) = 1st ISODYNAMIC POINT

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.)

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.

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(16) = 2nd ISODYNAMIC POINT

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.

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(17) = 1st NAPOLEON POINT

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.

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(18) = 2nd NAPOLEON POINT

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.

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(19) = CLAWSON POINT

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.

Further information is available from
Paul Yiu's Website.

Although John Clawson studied this point in 1925, it was studied earlier by Lemoine:

Emile Lemoine, "Quelques questions se rapportant à l'étude des antiparallèles des côtes d'un triangle", Bulletin de la S. M. F., tome 14 (1886), p. 107-128, specifically, on page 114. This article is available online at Numdam.

Let A'B'C' be the 4th Brocard triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(19). (Randy Hutson, November 18, 2015)

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

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.

X(20) = DE LONGCHAMPS POINT

Barycentrics  tan B + tan C - tan A : tan C + tan A - tan B : tan A + tan B - tan C
                        = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [-3a⁴ + 2a²(b² + c²) + (b² - c²)²]
X(20) = 2(r + 2R)*X(1) - (r +4R)*X(7) = 3X(2) - 4X(3) = X(8) - 2X(40)

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)

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

X(20) lies on the Darboux cubic and these lines:
1,7   2,3   8,40   10,165   33,1038   34,1040   35,1478   36,1479   55,388   56,497   57,938   58,387   64,69   68,74   72,144   78,329   97,1217   98,148   99,147   100,153   101,152   103,150   104,149   109,151   110,146   145,517   155,323   185,193   190,1265   243,1118   254,1300   346,1766   371,1587   372,1588   391,573   393,577   394,1032   485,1131   486,1132   487,638   488,637   616,633   617,635   621,627   622,628   936,1750   999,1058   1062,1870   1074,1838   1076,1785   1125,1699   1147,1614   1155,1788   1204,1899   1440,1804   1610,1633   2130,2131   3182,3347   3183,3348   3353,3354   3472,3473

X(20) is the {X(3),X(4)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(20), click Tables at the top of this page.

X(20) = reflection of X(i) in X(j) for these (i,j): (2,376), (3,550), (4,3), (5,548), (8,40), (69,1350), (145,944), (146,110), (147,99), (148,98), (149,104), (150,103), (151,109), (152,101), (153,100), (382,5), (962,1)

X(20) = isogonal conjugate of X(64)
X(20) = isotomic conjugate of X(253)
X(20) = cyclocevian conjugate of X(1032)
X(20) = inverse-in-circumcircle of X(2071)
X(20) = inverse-in-orthocentroidal-circle of X(3091)
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)
X(20) = crosssum of X(1) and X(1044)
X(20) = crossdifference of every pair of points on line X(647)X(657)
X(20) = X(i)-aleph conjugate of X(j) for these (i,j): (8,191), (9,1045), (188,1046), (333,2), (1043,20)
X(20) = X(i)-beth conjugate of X(j) for these (i,j): (664,20), (1043,280)
X(20) = X(4)-of-anticomplementary triangle
X(20) = X(52)-of-hexyl-triangle
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(21) = SCHIFFLER POINT

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).

If you have The Geometer's Sketchpad, 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) = 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)
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(86) and X(333)
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(22) = EXETER POINT

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.

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(23) = FAR-OUT POINT

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)

If you have The Geometer's Sketchpad, 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 harmonic 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(24) = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE

Barycentrics   tan A cos 2A : tan B cos 2B : tan C cos 2C
                        = tan A - sin 2A : tan B - sin 2B : tan C - sin 2C

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).

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

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).

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)

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(4)
X(25) = trilinear product of PU(18)
X(25) = barycentric product of PU(23)
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(26) = CIRCUMCENTER OF THE TANGENTIAL TRIANGLE

Barycentrics  h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = a²(b²cos 2B + c²cos 2C - a²cos 2A)

As a point on the Euler line, X(25) has Shinagawa coefficients (E + 4F, -3E - 4F).

If you have The Geometer's Sketchpad, 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

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).

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)

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).

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

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).

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

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)

As a point on the Euler line, X(30) has Shinagawa coefficients (1, -3).

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 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(31) = 2nd POWER POINT

Barycentrics  a³ : b³ : c³

X(31) = (r² + s²)*X(1) - 6rR*X(2) - 4r²*X(3)   (Peter Moses, April 2, 2013)

If you have The Geometer's Sketchpad, you can view X(31) (1), X(31) (2), X(31) (3).

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(32) = 3rd POWER POINT

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).

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(33) = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES

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).

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)

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).

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) = 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) = X(56)-Hirst inverse of X(1430)
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) = crossdifference of every pair of points on line X(521)X(652)

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

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).

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(36) = INVERSE-IN-CIRCUMCIRCLE OF INCENTER

If you have The Geometer's Sketchpad, 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-1), C(n-1) 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)

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) = 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(37) = CROSSPOINT OF INCENTER AND CENTROID

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)

If you have The Geometer's Sketchpad, 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) = 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(38) = CROSSPOINT OF X(1) AND X(75)

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(39) = BROCARD MIDPOINT

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) 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).

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 10: The Brocard Points.

X(39) lies 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) = 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(40) = BEVAN POINT

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

X(40) = X(1) - 2X(3) = 2R*X(4) - (r + 4R)*X(9)

If you have The Geometer's Sketchpad, 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 Randay 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).

X(40) lies on the Darboux cubic and these lines:
1,3   2,926   4,9   6,380   8,20   30,191   31,580   33,201   34,212   42,581   43,970   58,601   64,72   77,947   78,100   80,90   92,412   101,972   108,207   109,255   164,188   190,341   196,208   219,610   220,910   221,223   256,989   376,519   386,1064   387,579   390,938   392,474   511,1045   550,952   595,602   728,1018   936,960   958,1012   978,1050   2130,3354   2131,3472   3182,3346   3183,3347   3348,3353   3355,3473

X(40) is the {X(55),X(65)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(40), click Tables at the top of this page.

X(40) = midpoint of X(8) and X(20)
X(40) = reflection of X(i) in X(j) for these (i,j): (1,3), (4,10), (84,1158), (962,946), (1482,1385)
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(i) -Ceva conjugate of X(j) for these (i,j): (8,1), (20,1490), (63,9), (347,223)
X(40) = X(i) -cross conjugate of X(j) for these (i,j): (198,223), (221,1)
X(40) = crosspoint of X(i) and X(j) for these (i,j): (329,347)
X(40) = crosssum of X(56) and X(1413)
X(40) = crossdifference of every pair of points on line X(650)X(1459)
X(40) = X(i) -aleph conjugate of X(j) for these (i,j): (1,978), (2,57), (8,40), (188,1), (556,63)
X(40) = X(i) -beth conjugate of X(j) for these (i,j): (8,4), (40,221), (643,78), (644,728)
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(41) = X(6)-CEVA CONJUGATE OF X(31)

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(42)  CROSSPOINT OF INCENTER AND SYMMEDIAN POINT

If you have The Geometer's Sketchpad, you can view X(42).

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(43) = X(6)-CEVA CONJUGATE OF X(1)

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)

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(45) = X(9)-BETH CONJUGATE OF X(1)

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)

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(47) = X(110)-BETH CONJUGATE OF X(34)

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)

Barycentrics  a sin 2A : b sin 2B : c sin 2C

X(48) = (r² + 4rR² + 4R² + s²)*X(1) - 6R(2R + r)*X(2) - 2(r² + 2rR - s²)*X(3)    (Peter Moses, April 2, 2013)

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

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)

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)

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(51) = CENTROID OF ORTHIC TRIANGLE

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)

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) = 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(52) = ORTHOCENTER OF ORTHIC TRIANGLE

Barycentrics  tan A (sec 2B + sec 2C) : tan B (sec 2C + sec 2A) : tan C (sec 2A + sec 2B)

X(52) = (r² + 2rR + s²)*X(1) - 6rR*X(2) - (r² - 4rR - 2R² + s²)*X(3)    (Peter Moses, April 2, 2013)

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(53) = SYMMEDIAN POINT OF ORTHIC TRIANGLE

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

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(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(55) = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)

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)

X(55) lies on these lines:
1,3   2,11   4,12   5,498   6,31   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   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

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(56) = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)

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)

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'(47), where PU'(47) are the 1st & 2nd Bicentrics of X(56) (the isogonal conjugates of PU(47))
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(57)  ISOGONAL CONJUGATE OF X(9)

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) = 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) = 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)

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.

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(59)  ISOGONAL CONJUGATE OF X(11)

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)

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)

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(62) = ISOGONAL CONJUGATE OF X(18)

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(63) = ISOGONAL CONJUGATE OF X(19)

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) lies on these lines:
1,21   2,7   3,72   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) = 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) = 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(64) = ISOGONAL CONJUGATE OF X(20)

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(65) = ORTHOCENTER OF THE INTOUCH TRIANGLE

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

X(65) is the perspector of ABC and the extangents triangle.

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) = 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)
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(66) = ISOGONAL CONJUGATE OF X(22)

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(67) = ISOGONAL CONJUGATE OF X(23)

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(68)  PRASOLOV POINT

Let A'B'C' be the reflection of the orthic triangle of ABC in X(5). 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.

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(69) = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE

Barycentrics  cot A : cot B : cot C
                        = b² + c² - a² : c² + a² - b² : a² + b² - c²

X(69) = 2(r² + 2rR + s²)*X(1) + 3(r² - s²)*X(2) - 4r²*X(3)    (Peter Moses, April 2, 2013)

X(69) lies on 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.

X(69) is the perspector of the orthic-of-medial triangle and the reference triangle. (César Lozada, November 29, 2010)
X(69) is the perspector of ABC and the pedal triangle of X(20). (Randy Hutson, August 23, 2011)
X(69) is the perspector of ABC and (reflection in X(2) of the pedal triangle of X(2)). (Randy Hutson, August 23, 2011)

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): (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(70) = ISOGONAL CONJUGATE OF X(26)

X(70) lies on this line: 74,1288

X(70) = isogonal conjugate of X(26)

X(71) = ISOGONAL CONJUGATE OF X(27)

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)

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)

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 extouch triangle and the triangle formed by the lines through the external pairs of extouch points. (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(73)  CROSSPOINT OF INCENTER AND CIRCUMCENTER

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(74)  ISOGONAL CONJUGATE OF EULER INFINITY POINT

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.

X(74) = circumcircle-antipode of X(110)
X(74) is the point of intersection, other than A, B, and C of the circumcircle and Jerabek hyperbola; also, X(74) is the antipode of X(4) on the Jerabek hyperbola. X(74) lies on the Darboux septic curve.

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).

X(74) lies on the Neuberg cubic 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) = 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): The tangents at A, B, C to the Neuberg cubic K001 concur in X(74)
. 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) = perspector of ABC and the (degenerate) side-triangle of the (equilateral) circumcevian triangles of X(15) and X(16)

X(75)  ISOTOMIC CONJUGATE OF INCENTER

Barycentrics  1/a : 1/b : 1/c
Barycentrics  csc A : csc B : csc C

This is the center X(37) of the anticomplementary triangle.

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   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(76) = 3rd BROCARD POINT

Barycentrics  1/a² : 1/b² : 1/c²

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(77) = ISOGONAL CONJUGATE OF X(33)

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)

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).

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)

Barycentrics  h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = 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)

X(79) lies on these lines:
1,30   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(80)  REFLECTION OF INCENTER IN FEUERBACH POINT

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/(b² + c² - a² - bc)

X(80) = (2r + R)*X(1) - 6r*X(2) + 2r*X(3)    (Peter Moses, April 2, 2013)

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(81)  CEVAPOINT OF INCENTER AND SYMMEDIAN POINT

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)

If you have The Geometer's Sketchpad, 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(82) = ISOGONAL CONJUGATE OF X(38)

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

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)

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(84) = ISOGONAL CONJUGATE OF X(40)

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. Also, X(84) is X(68)-of-the-hexyl-triangle.

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(85)  ISOTOMIC CONJUGATE OF X(9)

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

X(86) = 2(r² + 2rR + s²)*X(1) + 3(r² + s²)*X(2) - 4r²*X(3)    (Peter Moses, April 2, 2013)

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(87)   X(2)-CROSS CONJUGATE OF X(1)

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)

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) = 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(89) = ISOGONAL CONJUGATE OF X(45)

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)

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(91) = ISOGONAL CONJUGATE OF X(47)

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

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).

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(93) = ISOGONAL CONJUGATE OF X(49)

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)

X(94) lies on 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(95)  CEVAPOINT OF CENTROID AND CIRCUMCENTER

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)

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(97) = ISOGONAL CONJUGATE OF X(53)

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)

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.

X(98) = TARRY POINT

If you have The Geometer's Sketchpad, you can view X(98).

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)

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)

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   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) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,X(3),X(25),X(32)}} (isogonal conjugate of line X(4)X(69))
X(98) = 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(99) = STEINER POINT

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

B>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)

X(99) = circumcircle-antipode of X(98)
X(99) = the point of intersection, other than A, B, and C, of the circumcircle and Steiner ellipse. X(99) = Ψ(X(6), X(2))

Let L_A be the reflection of the line X(3)X(6) in line BC, and define L_B and L_C cyclically. Let A' = L_B∩L_C, B' = L_C∩L_A, C' = L_A∩L_B. The lines AA', BB', CC' concur in X(99). (Randy Hutson, 9/23/2011)

X(99) is the only point on the circumcircle whose isotomic conjugate lies on the line at infinity. (Randy Hutson, 9/23/2011)

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 information on the Steiner circum-ellipse, 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

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(100)  ANTICOMPLEMENT OF FEUERBACH POINT

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)

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 (viz., the Jerabek hyperbola of the excentral triangle)
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(101) = Ψ(INCENTER, SYMMEDIAN POINT)

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

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)

X(101) = circumcircle-antipode of X(103)
X(101) = Ψ(X(1), X(6))
X(101) = X(114)-of-the-hexyl-triangle

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)

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(102) = Λ(INCENTER, ORTHOCENTER)

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = (sin A)/[sin B (sec A - sec B) + sin C (sec A - sec C)]

X(102) = circumcircle-antipode of X(109)
X(102) = Λ(X(1), X(4))

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(103) = ANTIPODE OF X(101)

Barycentrics  f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²/[(a - b) cot C + (a - c) cot B]

X(103) = circumcircle-antipode of X(101)
X(103) = Ψ(X(101), X(3))
X(103) = X(115)-of-the-hexyl-triangle

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(104) = ANTIPODE OF X(100)

X(104) = 2R*X(1) - 3R*X(2) + (2R - 2r)*X(3)    (Peter Moses, April 2, 2013)

X(104) = circumcircle-antipode of X(100)
X(104) is the 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

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)

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(105) = Λ(INCENTER, SYMMEDIAN POINT)

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(106) = Λ(INCENTER, CENTROID)

X(106) = Λ(X(1), X(2))
X(106) = Ψ(X(101), X(6))
X(106) = X(122)-of-the-hexyl-triangle

X(106) lies on these lines:
1,88   2,121   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(107) = Ψ(SYMMEDIAN POINT, ORTHOCENTER)

Barycentrics  1/[(b² - c²)(b² + c² - a²)²] : 1/[(c² - a²)(c² + a² - b²)²] : 1/[(a² - b²)(a² + b² - c²)²]

X(107) = Ψ(X(6), X(4))

X(107) lies on these lines:
2,122   4,74   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

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(108) = Ψ(CIRCUMCENTER, INCENTER)

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   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(109) = Ψ(INCENTER, CIRCUMCENTER)

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

X(109) = circumcircle-antipode of X(102)
X(109) = Ψ(X(1), X(3))
X(109) = trilinear product X(1381)*X(1382)

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)

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) = trilinear product of X(1381) and X(1382)

X(110) = FOCUS OF KIEPERT PARABOLA

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.

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) = Ψ(X(6), X(3))
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), (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(111) = PARRY POINT

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)

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(112) = Ψ(ORTHOCENTER, SYMMEDIAN POINT)

Barycentrics  a²/(sin 2B - sin 2C) : b²/(sin 2C - sin 2A) : c²/(sin 2A - sin 2B)

X(112) = Ψ(X(4), X(6))

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)

X(112) lies on these lines:
2,127   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   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) = 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)

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)

X(113) = JERABEK ANTIPODE

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where
                        g(A,B,C) = (sin C)/(cos C - 2 cos A cos B) + (sin B)/(cos B - 2 cos A cos C)

                        = h(a,b,c) : h(b,c,a) : h(c,a,b),
                        where h(a,b,c) = b²/(b²S_B - 2S_AS_C) + c²/(c²S_C - 2S_AS_B),
                        S_A = (b² + c² - a²)/2, and S_B and S_C are defined cyclically (Peter J. C. Moses, 3/2003)

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 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(114) = KIEPERT ANTIPODE

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)

X(114) lies 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) = 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(115) = CENTER OF KIEPERT HYPERBOLA

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)

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, and 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(116) = MIDPOINT OF X(4) AND X(103)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b) where
                        g(a,b,c) = (b - c)²(b² + bc + c² - ab - ac)

X(116) lies on the nine-point circle
X(116) = X(101)-of-medial triangle.

X(116) lies on 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(117) = MIDPOINT OF X(4) AND X(109)

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(118) = MIDPOINT OF X(4) AND X(101)

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

X(118) lies on the nine-point circle
X(118) = X(103)-of-medial triangle.

X(118) lies on 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(119) = FEUERBACH ANTIPODE

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B) where
                        g(A,B,C) = (-1 + cos B + cos C)[sin 2B + sin 2C + 2(-1 + cos A)(sin B + sin C)]

X(119) = nine-point-circle-antipode of X(11)
X(119) = X(104)-of-medial triangle.

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(120) = X(105)-OF-MEDIAL-TRIANGLE

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = [2abc - (b + c)(a² + (b - c)²)](b² + c² - ab -ac)

X(120) lies on the nine-point circle
X(120) = X(105)-of-medial triangle.

X(120) lies on 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(121) = X(106)-OF-MEDIAL-TRIANGLE

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

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = a(b² - c²)²(cos A - cos B cos C) cot²A

X(122) lies on the nine-point circle
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) lies on 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(123) = X(108)-OF-MEDIAL-TRIANGLE

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B) where
                        g(A,B,C) = (sec B - sec C)[(sec A)(sin²B - sin²C) + sin C tan C - sin B tan B]

X(123) lies on the nine-point circle
X(123) = X(108)-of-medial triangle.

X(123) lies on these lines: 2,108   3,119   10,117   113,960

X(123) = complementary conjugate of X(521)
X(123) = X(4)-Ceva conjugate of X(521)

X(124) = X(109)-OF-MEDIAL-TRIANGLE

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(125) = CENTER OF JERABEK HYPERBOLA

Barycentrics  (sin 2A)[sin(B - C)]² : (sin 2B)[sin(C - A)]² : (sin 2C)[sin(A - B)]²

X(125) lies on the nine-point circle
X(125) = X(110)-of-medial triangle
X(125) = X(100)-of-orthic triangle, if ABC is acute

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)

X(125) 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 line X(110)X(112)
X(125) = X(115)-Hirst inverse of X(868)
X(125) = X(2)-line conjugate of X(110)

X(126) = X(111)-OF-MEDIAL-TRIANGLE

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(127) = X(112)-OF-MEDIAL-TRIANGLE

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where
                        g(a,b,c) = (sin 2B - sin 2C)[(b² - c²)sin 2A - b²sin 2B + c²sin 2C]

X(127) lies on the nine-point circle
X(127) = X(112)-of-medial triangle
X(127) = center of the rectangular hyperbola that passes through A, B, C, and X(22)

X(127) lies on these lines: 2,112   3,114   5,132   113,141   115,338   133,381   125,140

X(127) = reflection of X(132) in X(5)
X(127) = complement of X(112)
X(127) = anticomplementary conjugate of X(525)
X(127) = X(4)-Ceva conjugate of X(525)

X(128) = X(74)-OF-ORTHIC-TRIANGLE

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

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

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)

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) = INTERSECTION OF LINES X(2)X(107) AND X(4)X(32)

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
X(132) = X(105)-of-orthic triangle if ABC is acute

X(132) lies on 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(133) = INTERSECTION OF LINES X(4)X(74) AND X(5)X(122)

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

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)

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)

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

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

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

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

X(140) = MIDPOINT OF X(3) AND X(5)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = b cos(C - A) + c cos(B - A)

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)

X(140) lies on these lines:
2,3   10,214   11,35   12,36   15,18   16,17   39,230   54,252   55,496   56,495   61,395   62,396   95,340   125,128   141,182   143,511   195,323   298,628   299,627   302,633   303,634   343,569   371,615   372,590   524,575   576,597   601,748   602,750   618,630   619,629

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), (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(141) = COMPLEMENT OF SYMMEDIAN POINT

Barycentrics  b² + c² : c² + a² : a² + b²

X(141) = X(6)-of-medial triangle

X(141) lies 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(142) = COMPLEMENT OF X(9)

X(142) = X(9)-of- medial triangle
X(142) = centroid of the set {X(1), X(4), X(7), X(40)}

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)

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   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(143) = NINE-POINT CENTER OF ORTHIC TRIANGLE

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)

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)

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(145) = ANTICOMPLEMENT OF NAGEL POINT

X(145) = X(8)-of-anticomplementary-triangle

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)

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(146) = REFLECTION OF X(20) IN X(110)

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

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(148) = STEINER POINT OF ANTICOMPLEMENTARY TRIANGLE

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(149) = REFLECTION OF X(20) IN X(104)

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)

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)

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)

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)

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)

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin² A)(tan B + tan C - tan A)

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

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(155)  EIGENCENTER OF ORTHIC TRIANGLE

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) = eigencenter of cevian triangle of X(4)
X(155) = eigencenter of anticevian triangle of X(3)
X(155) = X(4)-Ceva conjugate of X(3)
X(155) = crosssum of X(136) and X(523)

X(156) = X(5)-OF-TANGENTIAL-TRIANGLE

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

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

X(157) = X(6)-of-tangential triangle

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

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

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

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

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)

Barycentrics  a/(sin 2B - sin 2C) : b/(sin 2C - sin 2A) : c/(sin 2A - sin 2B)

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(163) = TRILINEAR PRODUCT X(6)*X(110)

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(164)  INCENTER OF EXCENTRAL TRIANGLE

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

Barycentrics  a[tan(B/2) + tan(C/2) - tan(A/2)] : b[tan(C/2) + tan(A/2) - tan(B/2)] : c[tan(A/2) + tan(B/2) - tan(C/2)]

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)

If DEF is the pedal triangle of X(165), then |AE| + |AF| = |BF| + |BD| = |CD| + |CE|. (Seiichi Kirikami, October 8, 2010.)

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) = 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) = X(i) -beth conjugate of X(j) for these (i,j): (100,165), (643,200)

X(166)  GERGONNE POINT OF EXCENTRAL TRIANGLE

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

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

X(168) lies on these lines: 1,173   9,164   165,166

X(168) = X(188)-aleph conjugate of X(363)

X(169) = X(85)-CEVA CONJUGATE OF X(1)

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)

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)

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(172) = TRILINEAR PRODUCT X(6)*X(171)

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(173) = CONGRUENT ISOSCELIZERS POINT

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 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)
(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

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(175) = ISOPERIMETRIC POINT

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.

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). (Randy Hutson, 9/23/2011)

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(176) = EQUAL DETOUR POINT

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)

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(177) = 1st MID-ARC POINT

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(178) = 2nd MID-ARC POINT

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

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

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        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)

See also

Clark Kimberling, Shiko Iwata, and Hidetosi Fukagawa, Problem 1091 and Solution, Crux Mathematicorum 13 (1987) 128-129; 217-218. [proposed 1985].

X(181) lies on these lines:
{1,970}, {6,197}, {8,959}, {10,12}, {11,2051}, {25,2175}, {31,51}, {33,3022}, {42,228}, {43,57}, {44,375}, {55,573}, {56,386}, {58,1324}, {81,5061}, {171,511}, {182,5329}, {200,3779}, {213,2333}, {373,748}, {389,3072}, {518,3687}, {553,1463}, {575,5363}, {612,3688}, {750,3917}, {756,2171}, {942,5530}, {994,1361}, {1124,1685}, {1254,1425}, {1317,3032}, {1335,1686}, {1356,5213}, {1358,3034}, {1364,5348}, {1376,4259}, {1395,1843}, {1672,1683}, {1673,1684}, {1674,1693}, {1675,1694}, {1695,1697}, {2007,2019}, {2008,2020}, {2330,5285}, {2534,2538}, {2535,2539}, {3027,3029}, {3028,3031}, {3056,5269}, {3340,4517}, {3781,5268}, {3792,3819}, {4276,5172}

X(181) = isogonal conjugate of X(261)
X(181) = X(i) -Ceva conjugate of X(j) for these (i,j): (12,2197), (59,4559), (65,2171), (2171,1500)
X(181) = X(i) -cross conjugate of X(j) for these (i,j): (872,1500), (2643,512)
X(181) = crosspoint of X(i) and X(j) for these (i,j): (42,1824), (59,4559), (65,1400), (1354,2171)
X(181) = crosssum of X(i) and X(j) for these (i,j): (2,2975), (11,4560), (21,333), (81,4225), (86,1444), (1098,2185)
X(181) = crossdifference of X(3904) and X(3910)
X(181) = X(i) -beth conjugate of X(j) for these (i,j): (42,181), (660,181), (756,756)

X(181) = trilinear product of X(i) and X(j) for these {I,J}:
{1,181}, {6,2171}, {7,872}, {10,1402}, {12,31}, {19,2197}, {25,201}, {33,1425}, {34,3690}, {37,1400}, {42,65}, {55,1254}, {56,756}, {57,1500}, {59,2643}, {71,1880}, {73,1824}, {109,4705}, {115,2149}, {210,1042}, {213,226}, {225,228}, {227,2357}, {349,2205}, {512,4551}, {594,604}, {608,3949}, {651,4079}, {661,4559}, {762,1412}, {798,4552}, {1020,3709}, {1089,1397}, {1110,1365}, {1214,2333}, {1334,1427}, {1395,3695}, {1409,1826}, {1415,4024}, {1426,2318}, {1441,1918}, {3063,4605}, {3124,4564}, {4017,4557}

X(181) = barycentric product of X(i) and X(j) for these {I,J}:
{{1,2171}, {2,181}, {4,2197}, {6,12}, {7,1500}, {9,1254}, {10,1400}, {19,201}, {34,3949}, {37,65}, {42,226}, {56,594}, {57,756}, {59,115}, {71,225}, {72,1880}, {73,1826}, {85,872}, {109,4024}, {210,1427}, {213,1441}, {227,1903}, {278,3690}, {281,1425}, {307,2333}, {321,1402}, {349,1918}, {512,4552}, {523,4559}, {604,1089}, {608,3695}, {651,4705}, {661,4551}, {663,4605}, {664,4079}, {762,1014}, {1018,4017}, {1020,4041}, {1042,2321}, {1091,2150}, {1109,2149}, {1214,1824}, {1252,1365}, {1262,4092}, {1334,3668}, {1404,4013}, {1411,4053}, {1415,4036}, {1426,3694}, {2222,2610}, {2643,4564}, {3124,4998}, {3709,4566}

X(181) = X(i) -isoconjugate of X(j) for these (i,j):
(1,261), (2,2185), (7,1098), (8,757), (9,1509), (21,86), (27,1812), (28,332), (29,1444), (55,873), (58,314), (60,75), (69,270), (76,2150), (81,333), (99,3737), (200,552), (249,4858), (274,284), (283,286), (304,2189), (310,2194), (312,593), (348,2326), (514,4612), (645,1019), (649,4631), (650,4610), (662,4560), (663,4623), (693,4636), (763,2321), (849,3596), (1014,1043), (1021,4573), (1434,2287), (2170,4590), (4391,4556)

X(182) = MIDPOINT OF BROCARD DIAMETER

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(183) = TRILINEAR PRODUCT X(75)X(182)

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

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 the polar conjugates of PU(38)
X(184) = X(75)-isoconjugate of X(4)

X(185) = NAGEL POINT OF THE ORTHIC TRIANGLE

Alexei Myakishev has noted that X(185) is the Nagel point of the orthic triangle only is ABC is an acute triangle.

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(186) = INVERSE-IN-CIRCUMCIRCLE OF X(4)

Barycentrics  (sin A)(4 cos A - sec A) : (sin B)(4 cos B - sec B) : (sin C)(4 cos C - sec C)

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

Barycentrics  a²(2a² - b² - c²) : b²(2b² - c² - a²) : c²(2c² - a² - b²)

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.

If you have The Geometer's Sketchpad, you can view X(1316), which includes X(187).

X(187) lies on 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   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(188) = 2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE

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)

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(189) = CYCLOCEVIAN CONJUGATE OF X(8)

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

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)

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) = 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) = 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)

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.

X(191) = X(10)-CEVA CONJUGATE OF X(1)

X(191) = isogonal-conjugate-with-respect-to-excentral-triangle of X(3) (Randy Hutson, 9/23/2011)

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

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)

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)

X(193) lies on these lines:
2,6   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

X(193) = reflection of X(i) in X(j) for these (i,j): (3,1353), (4,1351), (69,6), (1352,576)
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(194) = X(6)-CEVA CONJUGATE OF X(2)

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 radical center of the Neuberg circles.

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) = X(6)-Ceva conjugate of X(2)
X(194) = X(3)-Hirst inverse of X(385)

X(195) = X(5)-CEVA CONJUGATE OF X(3)

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)

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(196) = X(7)-CEVA CONJUGATE OF X(4)

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)

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(198) = X(9)-CEVA CONJUGATE OF X(6)

X(198) lies on these lines:
3,9   6,41   19,25   45,1030   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)

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)

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

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(201) = X(10)-CEVA CONJUGATE OF X(12)

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       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       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)

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)

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)

This is also X(66) of the medial triangle.

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(207) = X(1)-CEVA CONJUGATE OF X(34)

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)

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)

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)

X(210) lies on these lines:
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(211) = X(4)-CEVA CONJUGATE OF X(39)

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)

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)

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) = 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(214) = X(2)-CEVA CONJUGATE OF X(44)

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(215) = X(1)-CEVA CONJUGATE OF X(50)

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)

X(216) is the perspector of triangle ABCand the tangential triangle of the Johnson circumconic. (Randy Hutson, 9/23/2011)

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

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(217) = X(6)-CEVA CONJUGATE OF X(51)

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)

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(219) = X(8)-CEVA CONJUGATE OF X(55)

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

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   101,102   144,347   200,282   206,692   255,268   278,329   332,345   346,644   572,947   577,906   604,672

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(220) = X(9)-CEVA CONJUGATE OF X(55)

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

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(221) = X(1)-CEVA CONJUGATE OF X(56)

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(222) = X(7)-CEVA CONJUGATE OF X(56)

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)

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) = 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)

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) = 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)

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)

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)

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

X(226) = X(63)-of-medial-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)

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(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)

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)

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)

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)

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(231) = X(2)-CEVA CONJUGATE OF X(128)

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)

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) annd 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(233) = X(2)-CEVA CONJUGATE OF X(140)

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)

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)

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

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.

X(237) = X(3)-LINE CONJUGATE OF X(2)

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(238) = X(1)-LINE CONJUGATE OF X(37)

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

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

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(241) = X(1)-LINE CONJUGATE OF X(55)

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

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(242) = X(4)-LINE CONJUGATE OF X(71)

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(243) = X(4)-LINE CONJUGATE OF X(73)

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

X(244) lies on 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(245) = X(1)-LINE CONJUGATE OF X(110)

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)

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)

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)

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.

X(249) = ISOGONAL CONJUGATE OF X(115)

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)

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)

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)

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(252) = ISOGONAL CONJUGATE OF X(143)

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)

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)

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)

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

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(257) = ISOGONAL CONJUGATE OF X(172)

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(258) = CONGRUENT INCIRCLES ISOSCELIZER POINT

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)

Barycentrics  sin A cos A/2 : sin B cos B/2 : sin C cos C/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(260) = ISOGONAL CONJUGATE OF X(177)

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)

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)

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) = isogonal conjugate of X(182)
X(262) = isotomic conjugate of X(183)

X(263) = ISOGONAL CONJUGATE OF X(183)

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

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)

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) = 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)

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)

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) = symgonal of X(5)
X(265) = Johnson-circumconic antipode of X(110)
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(266) = ISOGONAL CONJUGATE OF X(188)

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)

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)

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)

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)

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)

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)

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)

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)

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

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(275) = CEVAPOINT OF ORTHOCENTER AND SYMMEDIAN POINT

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)

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(277) = ISOGONAL CONJUGATE OF X(218)

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)

Barycentrics  tan A tan A/2 : tan B tan B/2 : tan C tan C/2

        nbsp;= 1 - sec A : 1 - sec B : 1 - sec C

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(279) = ISOGONAL CONJUGATE OF X(220)

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)

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(281) = X(37)-CROSS CONJUGATE OF X(9)

Barycentrics  tan A cot A/2 : tan B cot B/2 : tan C cot C/2
        = 1 + sec A : 1 + sec B : 1 + sec C

X(281) lies on these lines:
1,282   2,92   4,9   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(282) = X(6)-CROSS CONJUGATE OF X(9)

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)

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)

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

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)

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)

X(287) lies on 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(288) = CEVAPOINT OF X(6) AND X(54)

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)

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)

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

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(291) = 2nd SHARYGIN POINT

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

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(292) = X(1)-HIRST INVERSE OF X(291)

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) = X(1)-Hirst inverse of X(291)

X(293) = ISOGONAL CONJUGATE OF X(240)

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(294) = ISOGONAL CONJUGATE OF X(241)

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(295) = ISOGONAL CONJUGATE OF X(242)

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)

X(296) = isogonal conjugate of X(243)

X(297) = X(2)-HIRST INVERSE OF X(4)

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 X9i) 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) = 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.

X(298)  ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER

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

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

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

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

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

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)

Barycentrics  (cos A)csc²A : (cos B)csc²B : (cos C)csc²C

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)