Problem
GEO-B2-M02-P010 Common Chord and Equal Powers
#10
★★★☆☆ Level 3 of 5
Two circles intersect at \(A\) and \(B\). Point \(P\) lies on line \(AB\) outside both circles. Prove that the powers of point \(P\) with respect to these circles are equal.
The same line \(PAB\) is a secant for both circles.
For the first circle, the power of point \(P\) equals \(PA\cdot PB\). For the second circle, the intersection points with line \(PAB\) are the same \(A\) and \(B\), so the power also equals \(PA\cdot PB\). Hence the powers are equal.
This is a gentle entry into the radical axis, without the formal definition.