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rotating triangle about it's centroid at angles 120^o and 240^o

Source: TOT 318 1991 Autumn A S5 - Tournament of Towns

June 9, 2024
geometryCentroidrotationgeometric transformationEquilateral

Problem Statement

Let MM be a centre of gravity (the intersection point of the medians) of a triangle ABCABC. Under rotation by 120120 degrees about the point MM, the point BB is taken to the point PP; under rotation by 240240 degrees about MM, the point CC is taken to the point QQ. Prove that either APQAPQ is an equilateral triangle, or the points A,P,QA, P, Q coincide. (Bykovsky, Khabarovsksk)
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