Problem
GEO-B2-M10-P015 Converse Power of a Point
#15
★★★★☆ Level 4 of 5
Two lines meet at \(P\). Points \(A,B\) lie on one ray from \(P\), and points \(C,D\) lie on another ray, with \(PA
Draw the circle through \(A,C,D\) and look where it meets ray \(PB\) for the second time.
Let the circle through \(A,C,D\) meet ray \(PB\) for the second time at \(X\). By power of point \(P\), \(PA\cdot PX=PC\cdot PD\). By the condition, \(PC\cdot PD=PA\cdot PB\). Since \(PA\ne 0\), \(PX=PB\). On one ray such a point is unique, so \(X=B\). Hence \(A,C,B,D\) are concyclic.
This is an important converse technique: a product of segments can prove a circle.