Sharygin 2006, correspondence round, problem 11
Sharygin 2006, correspondence round, problem 11 is a plane geometry problem from the Sharygin Geometry Olympiad 2006, correspondence round, problem 11 (grades 9–10): reflections of the vertices in the opposite sides.
Problem
Let point $O$ be the center for the circumcircle of the triangle $ABC$; let $A'$, $B'$, $C'$ be the points symmetrical to $A$, $B$, $C$ about respective opposite sides; let $A_1$, $B_1$, $C_1$ be the points of intersection between the lines $OA'$ and $BC$, $OB'$ and $AC$, $OC'$ and $AB$. Prove that the lines $AA_1$, $BB_1$, $CC_1$ intersect at the same point.
Source Sharygin Geometry Olympiad 2006, correspondence round, problem 11 (grades 9–10)
The diagram
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