Problem
GEO-B3-M04-P004 Pedal Triangle of the Circumcenter
#4
★☆☆☆☆ Level 1 of 5
B. New Original Problem. Let \(O\) be the circumcenter of triangle \(ABC\). Prove that the pedal triangle of \(O\) consists of the midpoints of the sides of \(ABC\).
Inspired by Prasolov Simson and pedal geometry method
C. Hint 1. The perpendicular from the center of a circle to a chord bisects the chord.
D. Hint 2. Apply this to the chords \(BC,CA,AB\).
E. Full Solution.
The side \(BC\) is a chord of the circumcircle. The perpendicular from the center \(O\) to the chord \(BC\) bisects it. Hence the projection of \(O\) onto \(BC\) is the midpoint of \(BC\). The same applies to \(CA\) and \(AB\).
Another entry point to the nine-point circle.