Question 67
Let $K$ be the circumscribed circle of a triangle $ABC$. Let $D$ be the midpoint of the arc $AB$ that does not contain the point $C$, and let $E$ the midpoint of the arc $AC$ that does not contain the point $B$. Let $F$ and $G$ denote the points of tangency of the inscribed circle of the triangle $ABC$ with the sides $AB$ and $AC$, respectively. Let $X$ be the intersection point of the lines $EG$ and $DF$. Suppose that $DEX$ is an isosceles triangle with the top angle at $X$. Prove that the triangle $ABC$ is also isosceles with the top angle at $A$.