Question 97
Given is an acute triangle $ABC$. Let $C'$ be the center of the height at $AB$, and let $D$ and $E$ be different points at the distance $CC'$. Let $F$ and $G$ be the orthogonal projections of the point $D$ on the side $AC$ and $BC$, respectively. Prove that triangle $ABC$ is isosceles if quadrilateral $DGEF$ is a parallelogram.