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collinear symmetrics wrt midpoints , equilateral (2015 Kyiv City MO Round2 11.2)

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September 4, 2020
geometryEquilateralcollinearSymmetric

Problem Statement

The line passing through the center of the equilateral triangle ABC ABC intersects the lines AB AB , BC BC and CA CA at the points C1 {{C} _ {1}} , A1 {{A} _ {1}} and B1 {{B} _ {1}} , respectively. Let A2 {{A} _ {2}} be a point that is symmetric A1 {{A} _ {1}} with respect to the midpoint of BC BC ; the points B2 {{B} _ {2}} and C2 {{C} _ {2}} are defined similarly. Prove that the points A2 {{A} _ {2}} , B2 {{B} _ {2}} and C2 {{C} _ {2}} lie on the same line tangent to the inscribed circle of the triangle ABC ABC .
(Serdyuk Nazar)
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