2018 Benelux #3

 Let $ABC$ be a triangle with orthocentre $H$, and let $D$, $E$, and $F$ denote the respective midpoints of line segments $AB$, $AC$, and $AH$. The reflections of $B$ and $C$ in $F$ are $P$ and $Q$, respectively.

(a) Show that lines $PE$ and $QD$ intersect on the circumcircle of triangle $ABC$.

(b) Prove that lines $PD$ and $QE$ intersect on line segment $AH$.

a)Obviously, $BHPA,CHQA,PQBC$ are all parallelograms, so $PQ=BC,PQ||BC$. Let $QD \cap PE =T$ then since $DE||BC||PQ,DE=1/2 PQ$ we deduce that $D,E$ are midpoints of $QT,PT$,which means that $TBQA,TCPA$ are parallelograms, so $BT=AQ=CH,BT||AQ||CH$ which means that $HCTB$ is also parallelogram, so $T$ is the reflection of $H$ in midpoint of $BC$ and lies on $(ABC)$. $\square$

b)If $G=QE \cap PD$ then $G$ is the centroid of $PQT$,but this means that $G$ is also the centroid of $QAC$ and since $AF$ is a median of this triangle, so $G \in AH$.$\square$


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