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AC_1^2 + BA_1^2 + CB_1^2 = AB_1^2 + BC_1^2 + CA_1^2

Source: 1967 Polish MO finals p2

August 27, 2024
geometry

Problem Statement

Prove that if points A1,B1,C1 A_1, B_1, C_1 lying on the sides BC,CA,AB BC, CA, AB of a triangle ABC ABC are the orthogonal projections of a point P P of the triangle onto these sides, then AC12+BA12+CB12=AB12+BC12+CA12. AC_1^2 + BA_1^2 + CB_1^2 = AB_1^2 + BC_1^2 + CA_1^2.
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