Problem
GEO-B2-M03-P020 Two Tangents of a Radical Center
#20
★★★★☆ Level 4 of 5
Circles \(\omega_1\) and \(\omega_2\) are tangent at point \(A\), and \(\omega_2\) and \(\omega_3\) are tangent at point \(B\). The common tangents at \(A\) and \(B\) meet at point \(R\). Prove that \(R\) lies on the radical axis of circles \(\omega_1\) and \(\omega_3\).
The common tangent at a tangency point is the radical axis of the corresponding two circles.
The common tangent at \(A\) is the radical axis of \(\omega_1\) and \(\omega_2\), so \(R\) lies on this axis. The common tangent at \(B\) is the radical axis of \(\omega_2\) and \(\omega_3\), so \(R\) lies on it as well. By the radical center theorem, \(R\) lies on the radical axis of \(\omega_1\) and \(\omega_3\).
A nice problem: radical axes are not necessarily common chords.