11.Let $ABC$ be a triangle,and let $D,E$ and $F$ be the point of tangency of the incircle of the triangle $ABC$ with the sides $BC,CA$ and $AB$ respectively.Let $X$ be in the interior of $ABC$ such that the incircle of $XBC$ touches $XB,XC$ and $BC$ in $Z,Y$ and $D$ respectively.Prove that $E,F,Z,Y$ are concyclic.
12.Let $ABC$ be a triangle, with P and Q arbitrary points on $CA, AB$ respectively. Let $PQ$ meet the circumcircle of $ABC$ at $X$ and $Y$. Prove that the midpoints of $BP, CQ, PQ$ and $XY$ are concyclic.
13.The incircle of triangle ABC touches BC at D and AB at F , intersects the line AD again at H and the line CF again at K. Prove that $\dfrac{FD \cdot HK}{DL \cdot FH}=3$
14.In a right angled-triangle $ABC$, $A\hat{C} B =90^{\circ} $ . Its incircle $O$ meets $BC,AC ,AB$ at $D,E,F$ respectively. $AD$ cuts $O$ at $P$. If $B\hat{P} C=90^{\circ}$, prove $AE+AP=PD$
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