Problem
GEO-B3-M04-P007 Degeneration of the Pedal Triangle
#7
★★☆☆☆ Level 2 of 5
B. New Original Problem. Prove that the pedal triangle of a point \(P\) with respect to triangle \(ABC\) has zero area if and only if \(P\) lies on the circumcircle of \(ABC\).
Inspired by Prasolov Simson and pedal geometry method
C. Hint 1. Zero area means collinearity of the three pedal points.
D. Hint 2. Apply the direct and converse Simson theorems.
E. Full Solution.
The pedal triangle has zero area if and only if its vertices \(A_1,B_1,C_1\) are collinear.
If \(P\) lies on the circumcircle, then by the Simson theorem \(A_1,B_1,C_1\) are collinear. Conversely, if they are collinear, then by the converse Simson theorem \(P\) lies on the circumcircle. The claim follows.
This formulation is useful for problems mentioning the area of the pedal triangle.