Sharygin 2008, final round, grade 10, problem 3
Sharygin 2008, final round, grade 10, problem 3 is a plane geometry problem from the Sharygin Geometry Olympiad 2008, final round, grade 10, problem 3: three tangent circles and AB = CD.
Problem
$X$ and $Y$ are the common points of two circles $\omega_1$ and $\omega_2$. The third circle $\omega$ touches internally $\omega_1$ and $\omega_2$ in $P$ and $Q$ respectively. Segment $XY$ intersects $\omega$ in points $M$ and $N$. The rays $PM$ and $PN$ intersect $\omega_1$ in points $A$ and $D$; the rays $QM$ and $QN$ intersect $\omega_2$ in points $B$ and $C$ respectively. Prove that $AB = CD$.
Source Sharygin Geometry Olympiad 2008, final round, grade 10, problem 3
The diagram
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