Problem
GEO-B3-M04-P018 Locus of Midpoints \(PH\)
B. New Original Problem. A point \(P\) moves on the circumcircle of \(ABC\), and \(H\) is the orthocenter. Prove that the midpoint of \(PH\) moves on the nine-point circle and lies on the Simson line of \(P\).
C. Hint 1. The nine-point circle is the image of the circumcircle under homothety centered at \(H\) with ratio \(\frac12\).
D. Hint 2. The second fact is the theorem about the midpoint of \(PH\) on the Simson line.
E. Full Solution.
The homothety with center \(H\) and ratio \(\frac12\) sends \(P\) to the midpoint \(M\) of \(PH\). Since \(P\) moves on the circumcircle, the image of this circle under the homothety is the nine-point circle. Hence \(M\) moves on the nine-point circle.
By the Simson line midpoint property, for each point \(P\), this same midpoint \(M\) lies on the Simson line of \(P\). Thus both statements hold simultaneously.
This connects the motion of the Simson line with the nine-point circle.