Question 159
Given an arbitrary triangle $\triangle ABC$, let $O_1$ be its circumcircle and $O_2$ its nine-point circle; let $O_3$ be the circle with diameter $HG$, where $H$ is the orthocenter and $G$ is the centroid of $\triangle ABC$. Prove that $O_1, O_2, O_3$ are coaxial. (That is, there is a line such that the powers of all points on it with respect to the three circles are equal. The power of a point with respect to a circle is the difference between the square of the distance from the point to the center and the square of the radius.)
Note: The nine-point circle of a triangle is the circle through the nine points consisting of the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments joining the orthocenter to each of the three vertices.
Note: The nine-point circle of a triangle is the circle through the nine points consisting of the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments joining the orthocenter to each of the three vertices.