Problem
GEO-B3-M04-P014 Two Pedal Circles
B. New Original Problem. Prove that the pedal triangle of the circumcenter and the pedal triangle of the orthocenter of triangle \(ABC\) lie on one circle.
C. Hint 1. The pedal triangle of the circumcenter is the medial triangle.
D. Hint 2. The pedal triangle of the orthocenter is the triangle of altitude feet.
E. Full Solution.
By earlier facts, the projections of the circumcenter \(O\) onto the sides of \(ABC\) are the side midpoints, while the projections of the orthocenter \(H\) are the altitude feet.
The nine-point circle of triangle \(ABC\) passes through the side midpoints and the altitude feet. Therefore both pedal triangles have all their vertices on the same nine-point circle.
Good for connecting pedal geometry with classical triangle geometry.