Problem

Source: 2015 Taiwan TST Round 1 Quiz2 P2

Tags: geometry



Given any triangle $ABC.$ Let $O_1$ be it's circumcircle, $O_2$ be it's nine point circle, $O_3$ is a circle with orthocenter of $ABC$, $H$, and centroid $G$, be it's diameter. Prove that: $O_1,O_2,O_3$ share axis. (i.e. chose any two of them, their axis will be the same one, if $ABC$ is an obtuse triangle, the three circle share two points.)