Problem
GEO-B3-M04-P019 Locus of Degenerate Pedal Triangles
#19
★★★★☆ Level 4 of 5
B. New Original Problem. For a fixed triangle \(ABC\), find the locus of points \(P\) whose pedal triangle has area \(0\).
Inspired by Prasolov Simson and pedal geometry method
C. Hint 1. Area \(0\) means collinearity of the three projections.
D. Hint 2. Use the direct and converse Simson theorems.
E. Full Solution.
The pedal triangle has area \(0\) if and only if its vertices, the three projections of \(P\), are collinear.
By the Wallace-Simson theorem, this happens if and only if \(P\) lies on the circumcircle of triangle \(ABC\). Therefore the desired locus is the circumcircle of \(ABC\).
Formally similar to level 2, but as a locus problem it requires a full iff argument.