RU-ALLRUSSIAN 1964 Problem 15

A circle is inscribed in ABCD. AB is parallel to CD, and BC = AD. The diagonals AC, BD meet at E. The circles inscribed in ABE, BCE, CDE, DAE have radius $\mathrm{r}_1$ , $\mathrm{r}_2$ , $\mathrm{r}_3$ , $\mathrm{r}_4$ respectively. Prove that $1 / \mathrm{r}_1 + 1 / \mathrm{r}_3 = 1 / \mathrm{r}_2 + 1 / \mathrm{r}_4$ .