NL-NWO 2011 Problem 2

Let $ABC$ be a triangle. Points $P$ and $Q$ lie on side $BC$ and satisfy $|BP| =|PQ| = |QC| = \frac13 |BC|$ . Points $R$ and $S$ lie on side $CA$ and satisfy $|CR| =|RS| = |SA| = 1 3 |CA|$ . Finally, points $T$ and $U$ lie on side $AB$ and satisfy $|AT| = |TU| = |UB| =\frac13 |AB|$ . Points $P, Q,R, S, T$ and $U$ turn out to lie on a common circle. Prove that $ABC$ is an equilateral triangle.