Problem
GEO-B2-M03-P002 Tangent Circles
#2
★★☆☆☆ Level 2 of 5
Two circles are tangent at point \(A\). Prove that their common tangent at \(A\) is the radical axis of these circles.
For a point \(X\) on the common tangent, the power to each circle is \(XA^2\).
Let \(X\) lie on the common tangent at \(A\). With respect to the first circle, \(XA\) is a tangent, so the power equals \(XA^2\). With respect to the second circle, \(XA\) is also a tangent, so the power is again \(XA^2\). Hence the powers are equal for all points of this line.
An important extension: tangent circles also have a radical axis.