Simson's Law

Introduction

Point P is on a circumcircle of the triangle ABC. PQ and AB, PR and BC, PS and CA cross perpendicularly. What do you observe about three point Q,R and S.

Applet

Proof

Since points C, S, P and R are on the same circle, angle PCS = angle SRP. Since points A, B, P and C are on the same circle, angle PBQ = angle PCS. Since points P, R, B and Q are on the same circle, angle PBQ + angle PRQ = 180 deg. . Therefore, angle SRP + angle PRQ = 180 deg. .