Question 125
Let $A, B, C, D$ be four points on a circle (occurring in clockwise order), with $A B \lt A D$ and $B C \gt C D$. Let the bisector of angle $B A D$ meet the circle at $X$ and the bisector of angle $B C D$ meet the circle at $Y$. Consider the hexagon formed by these six points on the circle. If four of the six sides of the hexagon have equal length, prove that $B D$ must be a diameter of the circle.
