Let ABC be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid G and the circumcentre O of ABC in its sides BC,CA,AB are denoted by G1,G2,G3 and O1,O2,O3, respectively. Show that the circumcircles of triangles G1G2C, G1G3B, G2G3A, O1O2C, O1O3B, O2O3A and ABC 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. geometrygeometric transformationreflectioncircumcircle