Sharygin 2005, final round, grade 10, problem 6
Sharygin 2005, final round, grade 10, problem 6 is a plane geometry problem from the Sharygin Geometry Olympiad 2005, final round, grade 10, problem 6: the circle on diameter XH and three concurrent lines.
Problem
Let $H$ be the orthocenter in the triangle $ABC$ and $X$ be an arbitrary point. The circle with diameter $XH$ has second intersections with lines $AH$, $BH$, $CH$ at points $A_1$, $B_1$, $C_1$, while with lines $AX$, $BX$, $CX$ at points $A_2$, $B_2$, $C_2$. Prove that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ concur.
Source Sharygin Geometry Olympiad 2005, final round, grade 10, problem 6 · proposed by A. A. Zaslavsky
The diagram
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