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ratio PA_1:PB_1:PC_1 wanted, starting with a 90-60-30 triangle, equilateral

Source: 1973 Swedish Mathematical Competition p3

March 26, 2021
ratioEquilateralgeometryright triangle

Problem Statement

ABCABC is a triangle with ∠A=90∘\angle A = 90^\circ, ∠B=60∘\angle B = 60^\circ. The points A1A_1, B1B_1, C1C_1 on BCBC, CACA, ABAB respectively are such that A1B1C1A_1B_1C_1 is equilateral and the perpendiculars (to BCBC at A1A_1, to CACA at B1B_1 and to ABAB at C1C_1) meet at a point PP inside the triangle. Find the ratios PA1:PB1:PC1PA_1:PB_1:PC_1.
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