IMO 1971 LL GDR19

In a triangle P1P2P3 let PiQi be the altitude from Pi for

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IMO 1971 LL GDR19

Origin: GDR

Problem

In a triangle P1P2P3 let PiQi be the altitude from Pi for i = 1, 2, 3 (Qi being the foot of the altitude). The circle with diameter PiQi meets the two corresponding sides at two points different from Pi. Denote the length of the segment whose endpoints are these two points by li. Prove that l1 = l2 = l3.