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trisections of sides, equilaterals outside triang3l

Source: Polish MO second round 1966 p5

August 28, 2024
geometryEquilateralequilaterals

Problem Statement

Each of the sides BC,CA,AB BC, CA, AB of the triangle ABC ABC was divided into three equal parts and on the middle sections of these sides as bases, equilateral triangles were built outside the triangle ABC ABC , the third vertices of which were marked with the letters A′,B′,C′ A', B' , C' respectively. In addition, points A′′,B′′,C′′ A'', B'', C'' were determined, symmetrical to A′,B′,C′ A', B', C' respectively with respect to the lines BC,CA,AB BC, CA, AB . Prove that the triangles A′B′C′ A'B'C' and A′′B′′C′′ A''B''C'' are equilateral and have the same center of gravity as the triangle ABC ABC .
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