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ratio of circumradii, symmetric points wrt sides and midpoints related

Source: 2016 Oral Moscow Geometry Olympiad grades 10-11 p6

July 29, 2019
ratiogeometrycircumradiuscircumcirclesymmetry

Problem Statement

Given an acute triangle ABCABC. Let A′A' be a point symmetric to AA with respect to BC,OABC, O_A is the center of the circle passing through AA and the midpoints of the segments A′BA'B and A′C.OBA'C. O_B and OCO_C points are defined similarly. Find the ratio of the radii of the circles circumscribed around the triangles ABCABC and OAOBOCO_AO_BO_C.
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