Problem
GEO-B3-M05-P010 Center of the Nine-Point Circle
Prove that the center of the nine-point circle of triangle \(ABC\) is the midpoint of \(OH\), where \(O\) is the circumcenter and \(H\) is the orthocenter.
C. Hint 1. Consider the midpoints of \(AH,BH,CH\).
D. Hint 2. They are images of \(A,B,C\) under a homothety centered at \(H\).
The homothety centered at \(H\) with ratio \(\frac12\) sends \(A,B,C\) to the midpoints of \(AH,BH,CH\). Therefore the circle through these three midpoints is the image of the circumcircle of \(ABC\).
The image of the center \(O\) under this homothety is the midpoint of \(OH\). Hence this point is the center of the circle through the midpoints of \(AH,BH,CH\). Since this circle is the nine-point circle, its center is the midpoint of \(OH\).
A short but important problem for later Euler-line connections.