X(65) = ORTHOCENTER OF THE INTOUCH TRIANGLE¶

Trilinears

\(cos B + cos C : cos C + cos A : cos A + cos B\)

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

\(sin(A/2) cos(B/2 - C/2) : sin(B/2) cos(C/2 - A/2) : sin(C/2) cos(A/2 - B/2)\)

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

Barycentrics

\(a(b + c)/(b + c - a) : b(c + a)/(c + a - b) : c(a + b)/(a + b - c) Let A' be the intersections of the tangents to the Yiu conic at the points where they meet the A-excircle. Define B' and C' similarly. The lines AA', BB', CC' concur in :ref:`X(65) <X(65)>\). (Randy Hutson, July 20, 2016) Let Ab, Ac be the points where the A-excircle touches lines CA and AB resp., and define Bc, Ba, Ca, Cb cyclically. Let Ta be the intersection of the tangents to the Yiu conic (defined at X(478)) at Bc and Ca, and define Tb, Tc cyclically. Let Ta’ be the intersection of the tangents to the Yiu conic at Ba and Cb, and define Tb’, Tc’ cyclically. Let Sa = TbTc∩Tb’Tc’, Sb = TcTa∩Tc’Ta’, Sc = TaTb∩Ta’Tb’. The lines ASa, BSb, CSc concur in X(65). (See also X(1903).) (Randy Hutson, July 20, 2016) Let A’ be the homothetic center of the orthic triangles of the intouch and A-extouch triangles, and define B’ and C’ cyclically. The triangle A’B’C’ is perspective to the extouch triangle at X(65). (Randy Hutson, July 20, 2016) Let A’B’C’ be the orthic triangle. Let B’C’A″ be the triangle similar to ABC such that segment A’A″ crosses the line B’C’. Define B″ and C″ cyclically. Equivalently, A″ is the reflection of A in B’C’, and cyclically for B″ and C″. Let Ia be the incenter of B’C’A″, and define Ib and Ic cyclically. The circumcenter of triangle IaIbIc is X(65). Let A* be the intersection of lines A″Ia and B’C’, and define B* and C* cyclically. The lines A’A*, B’B*, C’C* concur in X(65). (Randy Hutson, July 20, 2016) Let Ab, Ac be the points where the A-excircle touches lines CA and AB resp., and define Bc, Ba, Ca, Cb cyclically. Let IaIbIc be the intouch triangle. Let Oa be the circle through Ab, Ac, Ib, Ic, and define Ob, Oc cyclically. X(65) is the radical center of Oa, Ob, Oc. (Randy Hutson, July 20, 2016) Let A’B’C’ be the intouch triangle of the extangents triangle, if ABC is acute. Then A’B’C’ is perspective to the intouch triangle and 4th and 5th extouch triangles at X(65). (Randy Hutson, December 2 2017) Let OA be the circle centered at the A-vertex of the Wasat triangle and passing through A; define OB and OC cyclically. X(65) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020) Let A’ be the isogonal conjugate of A wrt the A-extouch triangle. Define B’ and C’ cyclically. The lines AA’, BB’, CC’ concur in X(65). (Randy Hutson, August 30, 2020) In the plane of a triangle ABC, let`

Notes

Let A’ be the intersections of the tangents to the Yiu conic at the points where they meet the A-excircle. Define B’ and C’ similarly. The lines AA’, BB’, CC’ concur in X(65). (Randy Hutson, July 20, 2016)

Let Ab, Ac be the points where the A-excircle touches lines CA and AB resp., and define Bc, Ba, Ca, Cb cyclically. Let Ta be the intersection of the tangents to the Yiu conic (defined at X(478)) at Bc and Ca, and define Tb, Tc cyclically. Let Ta’ be the intersection of the tangents to the Yiu conic at Ba and Cb, and define Tb’, Tc’ cyclically. Let Sa = TbTc∩Tb’Tc’, Sb = TcTa∩Tc’Ta’, Sc = TaTb∩Ta’Tb’. The lines ASa, BSb, CSc concur in X(65). (See also X(1903).) (Randy Hutson, July 20, 2016)

Let A’ be the homothetic center of the orthic triangles of the intouch and A-extouch triangles, and define B’ and C’ cyclically. The triangle A’B’C’ is perspective to the extouch triangle at X(65). (Randy Hutson, July 20, 2016)

Let A’B’C’ be the orthic triangle. Let B’C’A″ be the triangle similar to ABC such that segment A’A″ crosses the line B’C’. Define B″ and C″ cyclically. Equivalently, A″ is the reflection of A in B’C’, and cyclically for B″ and C″. Let Ia be the incenter of B’C’A″, and define Ib and Ic cyclically. The circumcenter of triangle IaIbIc is X(65). Let A* be the intersection of lines A″Ia and B’C’, and define B* and C* cyclically. The lines A’A*, B’B*, C’C* concur in X(65). (Randy Hutson, July 20, 2016)

Let Ab, Ac be the points where the A-excircle touches lines CA and AB resp., and define Bc, Ba, Ca, Cb cyclically. Let IaIbIc be the intouch triangle. Let Oa be the circle through Ab, Ac, Ib, Ic, and define Ob, Oc cyclically. X(65) is the radical center of Oa, Ob, Oc. (Randy Hutson, July 20, 2016)

Let A’B’C’ be the intouch triangle of the extangents triangle, if ABC is acute. Then A’B’C’ is perspective to the intouch triangle and 4th and 5th extouch triangles at X(65). (Randy Hutson, December 2 2017)

Let OA be the circle centered at the A-vertex of the Wasat triangle and passing through A; define OB and OC cyclically. X(65) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let A’ be the isogonal conjugate of A wrt the A-extouch triangle. Define B’ and C’ cyclically. The lines AA’, BB’, CC’ concur in X(65). (Randy Hutson, August 30, 2020)

In the plane of a triangle ABC, let Ba = reflection of A in the external angular bisector of angle B, and define Cb and Ac cyclically; Ca = reflection of A in the external angular bisector of angle C, and define Ab and Bc cyclically; Pa = AcBc∩AbCb, and define Pb and Pc cyclically; Ka = AbBa∩AcCa, and define Kb and Kc cyclically. Then PaPbPc and ABC, and also KaKbKc and ABC, are perspective, and the perspector is X(65). (Dasari Naga Vijay Krishna, June 19, 2021)

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),:ref:X(40) <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) = complement of X(3869)

X(65) = anticomplement of X(960)

X(65) = circumcircle-inverse of X(5172)

X(65) = incircle-inverse of X(1319)

X(65) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,73), (4,225), (7,226), (10,227), (109,513), (226,37)

X(65) = X(42)-cross conjugate of X(37)

X(65) = crosspoint of X(i) and X(j) for these (i,j): (1,4), (7,57)

X(65) = crosssum of X(i) and X(j) for these (i,j): (1,3), (9,55), (56,1394), (1805,1806)

X(65) = crossdifference of every pair of points on line X(521)X(650)

X(65) = X(1284)-Hirst inverse of X(1400)

X(65) = X(i)-beth conjugate of X(j) for these (i,j): (1,65), (8,72), (10,10), (65,1042), (80,65), (100,65), (101,213), (291,65), (668,65), (1018, 65)

X(65) = bicentric sum of PU(15)

X(65) = PU(15)-harmonic conjugate of X(650)

X(65) = trilinear product of PU(81)

X(65) = trilinear pole of line X(647)X(661)

X(65) = perspector of intouch triangle and inverse(n(hexyl triangle))

X(65) = orthologic center of inverse(n(hexyl triangle)) to hexyl triangle; the reciprocal orthologic center is X(84)

X(65) = perspector of ABC and the extangents triangle

X(65) = X(1986)-of-Fuhrmann-triangle

X(65) = X(40) of Mandart-incircle triangle

X(65) = homothetic center of intangents triangle and reflection of extangents triangle in X(40)

X(65) = homothetic center of extangents triangle and reflection of intangents triangle in X(1)

X(65) = reflection of X(3057) in X(1)

X(65) = {X(1),:ref:X(3) <X(3)>}-harmonic conjugate of X(2646)

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

X(65) = {P,Q}-harmonic conjugate of X(1463), where P and Q are the intersections of the incircle and line X(7)X(8)

X(65) = pairwise perspector of: intouch triangle, 4th extouch triangle, 5th extouch triangle

X(65) = perspector of [reflection of incentral triangle in X(1)] and tangential triangle, wrt incentral triangle, of circumconic of incentral triangle centered at X(1) (bicevian conic of X(1) and X(57))

X(65) = inverse-in-{incircle, circumcircle}-inverter of X(2078)

X(65) = pedal-isogonal conjugate of X(1)

X(65) = X(5) of reflection triangle of X(1)

X(65) = radical trace of circumcircle and circumcircle of reflection triangle of X(1)

X(65) = X(188)-of-orthic-triangle if ABC is acute

X(65) = perspector of ABC and cross-triangle of ABC and 4th extouch triangle

X(65) = perspector of ABC and cross-triangle of ABC and 5th extouch triangle

X(65) = polar conjugate of X(31623)

X(65) = pole wrt polar circle of trilinear polar of X(31623) (line X(521)X(1948))

X(65) = perspector of ABC and anti-tangential midarc triangle

X(65) = homothetic center of extangents triangle and anti-tangential midarc triangle

X(65) = excentral-to-intouch similarity image of X(1)