X(79) = ISOGONAL CONJUGATE OF X(35)¶
Trilinears
\(1/(1 + 2 cos A) : :\)
\(bc/(b2 + c2 - a2 + bc)\)
\((sin A/2)(sin 3B/2)(sin 3C/2) : :\)
\(sin(A/2) csc(3A/2) : :\)
Barycentrics
\(1/(b2 + c2 - a2 + bc) : :\)
\(1/(b c + 2 SA) : : :ref:`X(79) <X(79)>\) = (2r + 3R)*:ref:X(1) <X(1)> + 6r*:ref:X(2) <X(2)> - 6r*:ref:X(3) <X(3)> (Peter Moses, April 2, 2013) Let A’ be the reflection of X(1) in sideline BC, and define B’ and C’ cyclically. Then the lines AA’, BB’, CC’ concur in X(79). (Eric Danneels, Hyacinthos 7892, 9/13/03) A’B’C’ is also the reflection triangle of X(1). The lines AA’, BB’, CC’ concur in X(79). (Randy Hutson, July 20, 2016) Let P and Q be the intersections of line BC and circle {X(1),2r}. Let X = X(1). Let A’ be the circumcenter of triangle PQX, and define B’ and C’ cyclically. The lines AA’, BB’, CC’ concur in X(79). (Compare to X(592), where the circle is the 1st Lemoine circle) (Randy Hutson, July 20, 2016) Let A25B25C25 be Gemini triangle 25. Let A’ be the perspector of conic {A,B,C,B25,C25}}, and define B’ and C’ cyclically. The lines AA’, BB’, CC’ concur in X(79). (Randy Hutson, January 15, 2019) A construction for X(79) appears in Dasari Naga Vijay Krishna, On the Feuerbach Triangle. X(79) lies on these lines: 1,30 2,3647 8,758 9,46 12,484 21,36 33,1063 34,1061 35,226 57,90 65,80 104,946 314,320 388,1000 X(79) = reflection of X(191) in X(442) <X(442)>`
Notes
Let A’ be the reflection of X(1) in sideline BC, and define B’ and C’ cyclically. Then the lines AA’, BB’, CC’ concur in X(79). (Eric Danneels, Hyacinthos 7892, 9/13/03)
A’B’C’ is also the reflection triangle of X(1). The lines AA’, BB’, CC’ concur in X(79). (Randy Hutson, July 20, 2016)
Let P and Q be the intersections of line BC and circle {X(1),2r}. Let X = X(1). Let A’ be the circumcenter of triangle PQX, and define B’ and C’ cyclically. The lines AA’, BB’, CC’ concur in X(79). (Compare to X(592), where the circle is the 1st Lemoine circle) (Randy Hutson, July 20, 2016)
Let A25B25C25 be Gemini triangle 25. Let A’ be the perspector of conic {A,B,C,B25,C25}}, and define B’ and C’ cyclically. The lines AA’, BB’, CC’ concur in X(79). (Randy Hutson, January 15, 2019)
A construction for X(79) appears in Dasari Naga Vijay Krishna, On the Feuerbach Triangle.
X(79) lies on these lines: 1,30 2,3647 8,758 9,46 12,484 21,36 33,1063 34,1061 35,226 57,90 65,80 104,946 314,320 388,1000
X(79) = reflection of X(191) in X(442)
X(79) = isogonal conjugate of X(35)
X(79) = isotomic conjugate of X(319)
X(79) = cevapoint of X(481) and X(482)
X(79) = crosssum of X(55) and X(1030)
X(79) = anticomplement of X(3647)
X(79) = X(2914) of Fuhrmann triangle
X(79) = antigonal image of X(3065)
X(79) = trilinear pole of line X(650)X(4802)
X(79) = perspector of ABC and extraversion triangle of X(80)
X(79) = trilinear pole of line X(650)X(4802)
X(79) = trilinear product of vertices of reflection triangle of X(1)
X(79) = X(6152)-of-excentral-triangle