X(56) = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)

Trilinears

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

\(1 - cos A : 1 - cos B : 1 - cos C\)

\(sin2(A/2) : sin2(B/2) : sin2(C/2)\)

\(a(tan A/2) : :\)

\(Ra - r : Rb - r : Rc - r, where Ra, Rb, Rc are the exradii\)

\(a*Ra : b*Rb : c*Rc, where Ra, Rb, Rc are the exradii\)

\(a cos A - (c + a) cos B - (a + b) cos C : :\)

Barycentrics

\(a2/(b + c - a) : b2/(c + a - b) : c2/(a + b - c)\)

\(area(A'BC) : : , where A'B'C' = 2nd circumperp triangle\)

Notes

X(56) is the perspector of the tangential triangle and the reflection of the intangents triangle in X(1).

Let A’B’C’ be the Fuhrmann triangle. Let La be the line through A’ parallel to BC, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. The lines AA″, BB″, CC″ concur in X(56). Also, let AaBaCa be the poristic triangle (i.e., a triangle with common circumcircle and incircle as ABC) such that BaCa is parallel to BC. Define AbBbCb and AcBcCc cyclically. The lines AAa, BBb, CCc concur in X(56). (Randy Hutson, November 18, 2015)

Let A’B’C’ be the intouch triangle. Let A″ be the barycentric product of the circumcircle intercepts of line B’C’, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(56). (Randy Hutson, June 27, 2018)

See Dasari Naga Vijay Krishna, On a Conic Through Twelve Notable Points, Int. J. Adv. Math. and Mech. 7(2) (2019) 1-15.

If you have Geometer’s Sketchpad, X(56). If you have GeoGebra, you can view X(56).

In the plane of a triangle ABC, let A’B’C’ = circumcevian triangle of X(1); Ta = line tangent to circumcircle at A’, and define Tb and Tc cyclically; Va = Tb∩Tc, and define Vb and Vc cyclically. The triangle VaVbVc is perspective to ABC, and the perspector is X(56). (Dasari Naga Vijay Krishna, June 19, 2021)

In the plane of a triangle ABC, let (Oa) be the circle tangent internally to the incircle and tangent internally to the circumcircle at A. Define (Ob) and (Oc) cyclically. The radical center of (Oa), (Ob), (Oc) is X(56). (Ivan Pavlov, February 24, 2022)

X(56) lies on these lines: 1,3 2,12 4,11 5,499 6,41 7,21 8,404 10,474 19,207 20,497 22,977 25,34 28,278 30,496 31,154 32,1015 33,963 38,201 58,222 61,202 62,203 63,960 72,997 77,1036 78,480 81,959 85,870 87,238 100,145 101,218 105,279 106,109 140,495 181,386 182,611 197,227 212,939 219,579 220,672 223,937 226,405 255,602 266,289 269,738 330,385 376,1058 411,938 511,613 551,553 607,911 631,1056 667,764 946,1012 978,979 1025,1083 1070,1074 1072,1076 1345,2464

X(56) is the {X(1),:ref:X(3) <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) = complement of X(3436)

X(56) = anticomplement of X(1329)

X(56) = circumcircle-inverse of X(1319)

X(56) = antigonal conjugate of X(17101)

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)

X(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),:ref:X(40) <X(40)>}-harmonic conjugate of X(3057)

X(56) = {X(1),:ref:X(57) <X(57)>}-harmonic conjugate of X(65)

X(56) = trilinear square of X(266)

X(56) = trilinear square root of X(1106)

X(56) = X(92)-isoconjugate of X(219)

X(56) = vertex conjugate of PU(93)

X(56) = inverse-in-{circumcircle, incircle}-inverter of X(3660)

X(56) = perspector of ABC and extraversion triangle of X(55)

X(56) = perspector of ABC and unary cofactor triangle of Gemini triangle 15

X(56) = homothetic center of intangents triangle and reflection of tangential triangle in X(3)

X(56) = homothetic center of tangential triangle and reflection of intangents triangle in X(1)

X(56) = Brianchon point (perspector) of inellipse that is isogonal conjugate of isotomic conjugate of incircle

X(56) = pole wrt polar circle of trilinear polar of X(7017) (line X(2804)X(4397))

X(56) = X(48)-isoconjugate (polar conjugate) of X(7017)

X(56) = barycentric product of PU(46)

X(56) = bicentric sum of PU(60)

X(56) = PU(60)-harmonic conjugate of X(650)

X(56) = trilinear product of PU(92)

X(56) = perspector of ABC and cross-triangle of ABC and Apus triangle

X(56) = perspector of ABC and cross-triangle of ABC and Hutson intouch triangle

X(56) = homothetic center of ABC and cross-triangle of ABC and 1st Johnson-Yff triangle

X(56) = homothetic center of midarc triangle and 1st-circumperp-of-1st-circumperp triangle (which is also 2nd-circumperp-of-2nd-circumperp triangle)

X(56) = homothetic center of 2nd midarc triangle and 2nd-circumperp-of-1st-circumperp triangle (which is also 1st-circumperp-of-2nd-circumperp triangle)

X(56) = Cundy-Parry Phi transform of X(517)

X(56) = Cundy-Parry Psi transform of X(104)

X(56) = {X(3513),:ref:X(3514) <X(3514)>}-harmonic conjugate of X(55)

X(56) = X(4)-of-2nd-Johnson-Yff-triangle

X(56) = homothetic center of tangential triangle and anti-tangential midarc triangle

X(56) = Ursa-major-to-Ursa-minor similarity image of X(4)

X(56) = barycentric product of (nonreal) circumcircle intercepts of the Gergonne line