Problem
GEO-B3-M04-P002 The Pedal Triangle
B. New Original Problem. Let \(A_1,B_1,C_1\) be the feet of perpendiculars from \(P\) to the lines \(BC,CA,AB\). Prove that each side of the pedal triangle \(A_1B_1C_1\) is a chord of one of the circles with diameters \(PA,PB,PC\).
C. Hint 1. Consider pairs of projections lying on two sides issuing from one vertex.
D. Hint 2. For example, \(A_1B_1\) lies on the circle with diameter \(PC\).
E. Full Solution.
By the previous fact, \(A_1\) and \(B_1\) lie on the circle with diameter \(PC\), so \(A_1B_1\) is a chord of this circle. Similarly, \(B_1\) and \(C_1\) lie on the circle with diameter \(PA\), and \(C_1\) and \(A_1\) lie on the circle with diameter \(PB\).
Thus all three sides of the pedal triangle are naturally connected with the circles on \(PA,PB,PC\) as diameters.
Good for reinforcing the symmetry of notation \(A_1,B_1,C_1\).