X(68) = X(68) PRASOLOV POINT

Trilinears

\(cos A sec 2A : cos B sec 2B : cos C sec 2C\)

Barycentrics

\(tan 2A : tan 2B : tan 2C\)

\((b^2 + c^2 - a^2)/(a^4 + b^4 + c^4 - 2a^2b^2 - 2a^2c^2) : : Let A'B'C' be the 2nd Euler triangle. The lines AA', BB', CC' concur in :ref:`X(68) <X(68)>\), 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. Let Oa be the circle centered at the A-vertex of the orthic triangle and passing through A; define Ob and Oc cyclically. Then X(68) is the radical center of Oa, Ob, Oc. (Randy Hutson, November 2, 2017) The X(3)-Fuhrmann triangle is inversely similar to ABC, with similitude center X(3), and perspective to ABC at X(68). (Randy Hutson, November 3, 2017) 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(5)>`

Notes

Let A’B’C’ be the 2nd Euler triangle. 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.

Let Oa be the circle centered at the A-vertex of the orthic triangle and passing through A; define Ob and Oc cyclically. Then X(68) is the radical center of Oa, Ob, Oc. (Randy Hutson, November 2, 2017)

The X(3)-Fuhrmann triangle is inversely similar to ABC, with similitude center X(3), and perspective to ABC at X(68). (Randy Hutson, November 3, 2017)

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(68) = pedal antipodal perspector of X(4)

X(68) = pedal antipodal perspector of X(186)

X(68) = X(63)-isoconjugate of X(8745)

X(68) = crossdifference of every pair of points on line X(924)X(6753)

X(68) = trilinear product of vertices of X(3)-anti-altimedial triangle

X(68) = orthic-to-ABC barycentric image of X(52)

X(68) = cyclocevian conjugate of X(34287)