# Circles and an isosceles triangle

Let $ABCD, CDEF$ be two cyclic quadrilateral (with circumcircles $\Gamma_1, \Gamma_2$), such that $AE||BF$. Let point $P$ located on the circumscribed circle of $\triangle ADE$ such that, $P$ is an interior point of quadrilateral $ABFE$ and $PB=PF$. Prove that quadrilateral $BFPD$ is cyclic if $AE$ is tangent to circles $\Gamma_1, \Gamma_2$! I proved that line $CD$ bisects segment $AE$. Ok, I'm sure it is about the power of a point. I tried making equations, but it didn't help. Also tried angle hunting...

## locked by Jack D'AurizioFeb 1 '18 at 10:28

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• This question has been locked (and it will be unlocked April 2, 2018) since it belongs to an active competition, the Sharygin Geometry Olympiad. – Jack D'Aurizio Feb 1 '18 at 10:29