Question 59
In an acute triangle $ABC$, let $O$ be the circumcenter and let $D$ be the foot of the altitude from $A$.
Let $M,N,P,Q$ be the midpoints of $AB,AC,BD,CD$, respectively. Prove that one of the intersection points of the circumcircles of triangles $AMN$ and $POQ$ lies on the altitude $AD$.
Let $M,N,P,Q$ be the midpoints of $AB,AC,BD,CD$, respectively. Prove that one of the intersection points of the circumcircles of triangles $AMN$ and $POQ$ lies on the altitude $AD$.