# Juggling Points On An Incircle

Geometry Level 4

Let $$ABC$$ be a triangle and let $$\omega$$ be its incircle.

Denote by $$D_1$$ and $$E_1$$ the points where $$\omega$$ is tangent to sides $$BC$$ and $$AC$$, respectively. Denote by $$D_2$$ and $$E_2$$ the points on sides $$BC$$ and $$AC$$, respectively, such that $$CD_2=BD_1$$ and $$CE_2=AE_1$$, and denote by $$P$$ the point of intersection of segments $$AD_2$$ and $$BE_2$$.

Circle $$\omega$$ intersects segment $$AD_2$$ at two points, the closer of which to the vertex $$A$$ is denoted by $$Q$$. Given that $$D_2P = 5$$, find $$AQ.$$

×