Question 33
Let $ABCD$ be a convex quadrilateral inscribed in a circle, and let diagonals $AC$ and $BD$ meet at $X$. The perpendiculars from $X$ meet the sides $AB$, $BC$, $CD$, $DA$ at $A'$, $B'$, $C'$, $D'$ respectively. Prove that
$$
|A'B'| + |C'D'| = |A'D'| + |B'C'|
$$
($|A'B'|$ is the length of line segment $A'B'$, etc.)
$$
|A'B'| + |C'D'| = |A'D'| + |B'C'|
$$
($|A'B'|$ is the length of line segment $A'B'$, etc.)