Let $ABC$ be an acute scalene triangle. $D$ and $E$ are variable points in the half-lines $AB$ and $AC$ (with origin at $A$) such that the symmetric of $A$ over $DE$ lies on $BC$. Let $P$ be the intersection of the circles with diameter $AD$ and $AE$. Find the locus of $P$ when varying the line segment $DE$.
From the description $P$ is just the feet of altitute from $A$ to $DE$. Let $M$ and $N$ be the midpoints of $AB$ and $AC$ respectively and $P'$ any point in line $MN$. Notice that the reflection point from $A$ in $P'$ always lies in $BC$. Now let $D'$ and $E'$ be the intersections of the line through $P'$ perpendicular to $AP'$ with $AB$ and $AC$ respectively.
Finally see that as $P'$ varies in $MN$ so does the line $D'E'$ and therefore the position of $P'$ in $D'E'$ as the altitute as well and so we are done.