## X(68) (PRASOLOV POINT)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon , select the center name from the list and, then, click on the vertices A, B and C successively.

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This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           cos A sec 2A : cos B sec 2B : cos C sec 2C
Barycentrics    tan 2A : tan 2B : tan 2C

Let A'B'C' be the reflection of the orthic triangle of ABC in X(5). 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.

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)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.