Problem
GEO-B2-M03-P014 Three Points of Equal Powers
#14
★★★★☆ Level 4 of 5
For two fixed circles, points \(X,Y,Z\) have equal powers with respect to these circles. Prove that \(X,Y,Z\) lie on one line.
All such points lie on the same radical axis.
The set of points having equal powers with respect to two given circles is, by definition, their radical axis. The radical axis is a line. Therefore all points \(X,Y,Z\) lie on this line.
A short but important idea: collinearity is often proved by the same reason for all points.