Sharygin 2010, correspondence round, problem 23
Sharygin 2010, correspondence round, problem 23 is a plane geometry problem from the Sharygin Geometry Olympiad 2010, correspondence round, problem 23 (grades 10–11): A cyclic hexagon whose main diagonals concur.
Problem
A cyclic hexagon $ABCDEF$ is such that $AB \cdot CF = 2\,BC \cdot FA$, $CD \cdot EB = 2\,DE \cdot BC$, and $EF \cdot AD = 2\,FA \cdot DE$. Prove that the lines $AD$, $BE$ and $CF$ concur.
Source Sharygin Geometry Olympiad 2010, correspondence round, problem 23 (grades 10–11)
The diagram
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