Problem
GEO-B2-M01-P019 Orthogonal Circles
Circles \(\omega_1\) and \(\omega_2\) intersect at \(A\) and \(B\). Their tangents at \(A\) are perpendicular. Point \(C\) lies on \(\omega_1\), and point \(D\) lies on \(\omega_2\). Prove that \(\angle ACB-\angle ADB\equiv90^\circ\pmod{180^\circ}\).
The angle between the circles equals the angle between their tangents at \(A\).
Let \(t_1,t_2\) be the tangents to the circles at \(A\). By the tangent-chord theorem, \(\angle(t_1,AB)\equiv\angle ACB\), and \(\angle(t_2,AB)\equiv\angle ADB\). Therefore the angle between the circles equals \(\angle ACB-\angle ADB\) in oriented notation. By the condition, \(t_1\perp t_2\), so this angle is \(90^\circ\) modulo \(180^\circ\). Hence \(\angle ACB-\angle ADB\equiv90^\circ\pmod{180^\circ}\).
This is a correct form of a problem on the angle between two intersecting circles.