Problem
GEO-B3-M05-P011 Common Nine-Point Circle
The altitudes of triangle \(ABC\) meet at \(H\). Prove that triangles \(ABC\), \(HBC\), \(AHC\), and \(ABH\) have the same nine-point circle.
C. Hint 1. Compare side midpoints and altitude feet.
D. Hint 2. For example, \(ABC\) and \(HBC\) already have three common points of their nine-point circles.
Compare triangles \(ABC\) and \(HBC\). The midpoint of \(BC\) is a side midpoint in both triangles. The midpoints of \(BH\) and \(CH\) lie on the nine-point circles of both triangles: for \(ABC\) they are midpoints from the orthocenter to vertices, while for \(HBC\) they are side midpoints.
Three common points determine a circle, so the nine-point circles of these two triangles coincide. The same comparison applies to \(ABC\) with \(AHC\) and \(ABH\). Therefore all four nine-point circles coincide.
This develops the idea that the orthocenter may be treated as a vertex.