Question 6
Let $ABC$ be a triangle all of whose angles are acute, with $AB \neq AC$. Let $\Gamma$ be the circumcircle of $ABC$, and $D$ the midpoint of the arc $\widehat{BC}$ not containing $A$. Let $E$ and $F$ be points on the segments $[AB]$ and $[AC]$ respectively, such that $AE = AF$. Let $P$ be the intersection point, other than $A$, of $\Gamma$ and the circumcircle of $AEF$.
Let $G$ and $H$ be the intersection points, other than $P$, of $\Gamma$ with the lines $(PE)$ and $(PF)$ respectively. Finally, let $J$ be the intersection point of the lines $(AB)$ and $(DG)$, and let $K$ be the intersection point of the lines $(AC)$ and $(DH)$.
Prove that the midpoint of the segment $[BC]$ lies on the line $(JK)$.
Let $G$ and $H$ be the intersection points, other than $P$, of $\Gamma$ with the lines $(PE)$ and $(PF)$ respectively. Finally, let $J$ be the intersection point of the lines $(AB)$ and $(DG)$, and let $K$ be the intersection point of the lines $(AC)$ and $(DH)$.
Prove that the midpoint of the segment $[BC]$ lies on the line $(JK)$.