Question 60
Let $ABC$ be an acute triangle and $D$ the foot of the altitude from $A$. Let $X$ and $Y$ be the two points on the circumcircle of $ABC$ with $AX = AY = AD$, with $B$ and $X$ on the same side of $(AD)$. Let $E$ be the intersection of $(BX)$ and $(CY)$. Let $F$ be the intersection other than $E$ of $(DE)$ and the circumcircle of $EXY$.
Prove that the circumcircles of $BCF$ and $EXY$ are tangent.
Prove that the circumcircles of $BCF$ and $EXY$ are tangent.