IMO 1983 SL 4
On the sides of the triangle ABC, three similar isosceles tri-
IMO 1983 SL 4
Origin: BEL
Problem
On the sides of the triangle ABC, three similar isosceles tri- angles ABP (AP = PB), AQC (AQ = QC), and BRC (BR = RC) are constructed. The first two are constructed externally to the triangle ABC, but the third is placed in the same half-plane determined by the line BC as the triangle ABC. Prove that APRQ is a parallelogram.
Solution
The rotational homothety centered at C that sends B to R also sends A to Q; hence the triangles ABC and QRC are similar. For the same reason,
\triangleABC and \trianglePBR are similar. Moreover, BR = CR; hence \triangleCRQ ∼= \triangleRBP. Thus PR = QC = AQ and QR = PB = PA, so APQR is a parallelogram.