Problem
GEO-B3-M04-P001 Circle with Diameter \(PC\)
#1
★☆☆☆☆ Level 1 of 5
B. New Original Problem. From a point \(P\), perpendiculars are dropped to the lines \(BC\) and \(CA\), with feet \(A_1\) and \(B_1\). Prove that \(P,A_1,C,B_1\) lie on one circle.
Inspired by Prasolov Simson and pedal geometry method
C. Hint 1. Find two right angles.
D. Hint 2. The circle with diameter \(PC\) contains all points from which \(PC\) is seen under a right angle.
E. Full Solution.
Since \(A_1\in BC\) and \(PA_1\perp BC\), we have \(\angle PA_1C=90^\circ\). Similarly, \(B_1\in CA\) and \(PB_1\perp CA\), so \(\angle PB_1C=90^\circ\).
Thus \(A_1\) and \(B_1\) lie on the circle with diameter \(PC\). Therefore \(P,A_1,C,B_1\) are cyclic.
A basic building block for the Simson line.