Question 2
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 $Q$.
Prove that $AQ=D_2P$.
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 $Q$.
Prove that $AQ=D_2P$.