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Reflections of O, G form SEVEN concurrent circles

Source: EGMO 2017 P6

April 9, 2017
geometrygeometric transformationreflectioncircumcircle

Problem Statement

Let ABCABC be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid GG and the circumcentre OO of ABCABC in its sides BC,CA,ABBC,CA,AB are denoted by G1,G2,G3G_1,G_2,G_3 and O1,O2,O3O_1,O_2,O_3, respectively. Show that the circumcircles of triangles G1G2CG_1G_2C, G1G3BG_1G_3B, G2G3AG_2G_3A, O1O2CO_1O_2C, O1O3BO_1O_3B, O2O3AO_2O_3A and ABCABC have a common point.
The centroid of a triangle is the intersection point of the three medians. A median is a line connecting a vertex of the triangle to the midpoint of the opposite side.
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