Problem
GEO-B2-M03-P018 Construct the Radical Axis From Two Points
#18
★★★★☆ Level 4 of 5
Two disjoint circles have centres \(O_1,O_2\). Points \(P\) and \(Q\) are such that from each of them the tangent lengths to the two circles are equal. Prove that \(PQ\) is the radical axis of these circles and \(PQ\perp O_1O_2\).
First prove that \(P\) and \(Q\) lie on the radical axis.
Equality of tangent lengths from point \(P\) means equality of powers of \(P\) with respect to the two circles, so \(P\) lies on the radical axis. Similarly, \(Q\) lies on the same axis. Therefore line \(PQ\) is the radical axis. The radical axis of two nonconcentric circles is perpendicular to the line of centres, hence \(PQ\perp O_1O_2\).
The problem shows how to find the radical axis even when there is no common chord.