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triangles P_1A_2A_3, A_1P_2A_3, A_1A_2P_3 cover the triangle A_1A_2A_3

Source: Czech and Slovak Olympiad 1988, National Round, Problem 6

September 13, 2024
geometrygeometric inequalitycovering

Problem Statement

Inside the triangle A1A2A3A_1A_2A_3 with sides a1a_1, a2a_2, a3a_3, three points are given, which we label P1P_1, P2P_2, P3P_3 so that the product of their distances from the corresponding sides a1a_1, a2a_2, a3a_3 is as large as possible. Prove that the triangles P1A2A3P_1A_2A_3, A1P2A3A_1P_2A_3, A1A2P3A_1A_2P_3 cover the triangle.
[hide=original wording]V trojúhelníku A1A2A3 se stranami a1, a2, a3 jsou dány tři body, které označíme Pi, P2, P3 tak, aby součin jejich vzdáleností od odpovídajících stran a1, a2, a3 byl co největší. Dokažte, že trojúhelníky P1A2A3, A1P2A3, A1A2P3 pokrývají trojúhelník.
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