Convex quadrangles

Question

there is a quadrangles ABCD with $|AB| + |BC| = |AD| + |DC|$. The beam $AB$ cuts the beam $DC$ in the point $X$. The beam $AD$ cuts the beam $BC$ in the point $Y$.

Now show that

\begin{equation*} |AX| + |XC| = |AY| + |YC|. \end{equation*}

Hope you can help me. I just realize that there are some circles that may be useful to show some equalities.

Thanks.

Answer 1

This theorem is known as Urquhart's theorem (or at least half of it). There is an elementary synthetic solution, which you can find at Cut The Knot.