Problem
GEO-B2-M01-P007 The Angle Between Two Circles
#7
★★★☆☆ Level 3 of 5
Circles \(\omega_1\) and \(\omega_2\) intersect at \(A\) and \(B\). Point \(C\) lies on \(\omega_1\), and point \(D\) lies on \(\omega_2\). Prove that the angle between the circles at \(A\) equals \(\angle ACB-\angle ADB\) in oriented notation.
The angle between the circles is the angle between their tangents at \(A\).
Let \(t_1\) and \(t_2\) be the tangents to \(\omega_1\) and \(\omega_2\) at \(A\). By the tangent-chord theorem, \(\angle(t_1,AB)\equiv\angle ACB\), and \(\angle(t_2,AB)\equiv\angle ADB\). Thus \(\angle(t_1,t_2)\equiv\angle(t_1,AB)-\angle(t_2,AB)\equiv\angle ACB-\angle ADB\pmod{180^\circ}\).
This is the first problem on the angle between circles; later it becomes a standard tool.