If any point P 1 is taken on the side A 2 A 3 of the triangle A 1 A 2 A 3 and P 1 P 2 is drawn parallel to A 2 A 1 to meet A 3 A 1 in P 2,P 2 P 3 parallel to A 3 A 2 to meet A 1 A 2 in P 3, P 3 Q 1 parallel to A 1 A 3 to meet A 2 A 3 in Q 1 and then Q 1 Q 2, Q 2 Q 3, Q 3 R 1 are drawn parallel to P 1 P 2, P 2 P 3, P 3 Q 1 in the same way, finishing with R 1 on A 2 A 3, it is easily shown that R 1 coincides with P 1, as in Fig. 1.