Problem
GEO-B3-M04-P003 Pedal Triangle of the Orthocenter
#3
★☆☆☆☆ Level 1 of 5
B. New Original Problem. In an acute triangle \(ABC\), let \(H\) be the orthocenter. Prove that the pedal triangle of \(H\) consists of the feet of the altitudes of \(ABC\).
Inspired by Prasolov Simson and pedal geometry method
C. Hint 1. The projection of \(H\) onto \(BC\) lies on the altitude from \(A\).
D. Hint 2. Argue similarly for the other two sides.
E. Full Solution.
Since \(H\) lies on the altitude from \(A\), the line \(AH\) is perpendicular to \(BC\). Therefore the foot of the perpendicular from \(H\) to \(BC\) coincides with the foot of the altitude from \(A\). Similarly, the projections of \(H\) onto \(CA\) and \(AB\) coincide with the feet of the altitudes from \(B\) and \(C\).
Preparation for the connection between the pedal circle and the nine-point circle.