Question 170
Let $D$ be the midpoint of the side $AB$ of an acute triangle $ABC$. Let $A'$ and $B'$ be points on the segments $AC$ and $BC$ such that the triangles $ADA'$ and $DBB'$ are isosceles with common apex $D$. Show: if the line $CD$ is perpendicular to the line $A'B'$, then the triangle $ABC$ is isosceles.