X(80) = X(80) REFLECTION OF INCENTER IN FEUERBACH POINT

Trilinears

\(1/(1 - 2 cos A) : 1/(1 - 2 cos B) : 1/(1 - 2 cos C)\)

\(bc/(b2 + c2 - a2 - bc) : :\)

\(cos(A/2) sec(3A/2) : :\)

Barycentrics

\(1/(b2 + c2 - a2 - bc) : :\)

\(1/(bc - 2 SA) : :\)

Notes

Let A’ be the reflection in BC of the A-vertex of the excentral triangle, and define B’ and C’ cyclically. The circumcircles of A’BC, B’CA, and C’AB concur in X(80). Also, the lines AA’, BB’, CC’ concur in X(80). (Randy Hutson, December 10, 2016)

Let A’B’C’ be the Fuhrmann triangle. Let La be the line through A parallel to B’C’, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. The lines A’A″, B’B″, C’C″ concur in X(80). (Randy Hutson, December 10, 2016)

Let A’B’C’ be the orthic triangle. Let La be the antiorthic axis of AB’C’, and define Lb and Lc cyclically. Let A″ = Lb∩Lc. B″ = Lc∩La, C″ = La∩Lb. The triangle A″B″C″ is inversely similar to ABC, with similitude center X(9). The incenter of triangle A″B″C″ is X(80). Also, the lines AA″, BB″, CC″ concur in X(80).(Randy Hutson, December 10, 2016)

Let A’B’C’ be the excentral triangle. Let A″ be the isogonal conjugate, wrt A’BC, of A. Define B″, C″ cyclically. (A″ is also the reflection of A’ in BC, and cyclically for B″ and C″). The lines AA″, BB″, CC″ concur in X(80). (Randy Hutson, January 29, 2018)

Let A’B’C’ be the excentral triangle. Let Oa be the A’-Johnson circle of triangle A’BC, and define Ob and Oc cyclically. X(80) is the radical center of Oa, Ob, Oc. (Randy Hutson, June 27, 2018)

X(80) lies on these lines: 1,5 2,214 7,150 8,149 9,528 10,21 30,484 33,1061 34,1063 36,104 40,90 46,84 65,79 313,314 497,1000 499,944 516,655 519,908 943,950

X(80) = midpoint of X(8) and X(149)

X(80) = reflection of X(i) in X(j) for these (i,j): (1,11), (100,10), (1317,1387)

X(80) = isogonal conjugate of X(36)

X(80) = isotomic conjugate of X(320)

X(80) = circumcircle-inverse of X(10260)

X(80) = incircle-inverse of X(1387)

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

X(80) = complement of X(6224)

X(80) = anticomplement of X(214)

X(80) = cevapoint of X(10) and X(519)

X(80) = X(i)-cross conjugate of X(j) for these (i,j): (44,2), (517,1)

X(80) = X(8)-beth conjugate of X(100)

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

X(80) = syngonal conjugate of X(10)

X(80) = X(186)-of-Fuhrmann triangle

X(80) = orthology center of ABC and Fuhrmann triangle

X(80) = reflection of any vertex of ABC in the corresponding side of the Fuhrmann triangle

X(80) = perspector of ABC and reflection of Fuhrmann triangle in X(11)

X(80) = trilinear pole of line X(37)X(650)

X(80) = inverse-in-circumconic-centered-at-X(1)-of-X(1807)

X(80) = perspector of ABC and extraversion triangle of X(79)

X(80) = X(1986)-of-excentral triangle

X(80) = perspector of ABC and mid-triangle of 1st and 2nd extouch triangles

X(80) = inner-Garcia-to-outer-Garcia similarity image of X(1)

X(80) = X(100)-of-outer-Garcia-triangle

X(80) = Conway-circle-inverse of X(35638)