Problem
GEO-B3-M04-P010 Simson Line and the Midpoint of \(PH\)
B. New Original Problem. Let \(H\) be the orthocenter of triangle \(ABC\), and let \(P\) lie on its circumcircle. Prove that the Simson line of \(P\) passes through the midpoint of \(PH\).
C. Hint 1. Consider the homothety with center \(P\) and ratio \(\frac12\).
D. Hint 2. It connects the altitudes from \(H\) with the projections from \(P\).
E. Full Solution.
Let \(M\) be the midpoint of \(PH\), and let \(B_1,C_1\) be the projections of \(P\) onto \(CA\) and \(AB\). Since \(BH\perp AC\), and \(PB_1\perp AC\), we have \(PB_1\parallel BH\). Similarly, \(PC_1\parallel CH\).
Under the homothety with center \(P\) and ratio \(\frac12\), the point \(H\) maps to \(M\), while the directions \(BH\) and \(CH\) correspond to the directions through \(B_1\) and \(C_1\). From the parallelism of corresponding segments, we get \(M\in B_1C_1\). But \(B_1C_1\) is the Simson line, so it passes through \(M\).
This is one of the main links between the Simson line and the nine-point circle.