Sharygin 2009, correspondence round, problem 12
Sharygin 2009, correspondence round, problem 12 is a plane geometry problem from the Sharygin Geometry Olympiad 2009, correspondence round, problem 12 (grades 9–10): reflections in the bisector CL.
Problem
Let $CL$ be a bisector of triangle $ABC$. Points $A_1$ and $B_1$ are the reflections of $A$ and $B$ in $CL$; points $A_2$ and $B_2$ are the reflections of $A$ and $B$ in $L$. Let $O_1$ and $O_2$ be the circumcenters of triangles $AB_1B_2$ and $BA_1A_2$ respectively. Prove that angles $O_1CA$ and $O_2CB$ are equal.
Source Sharygin Geometry Olympiad 2009, correspondence round, problem 12 (grades 9–10)
The diagram
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