Problem
GEO-B2-M01-P085 Angle Between Two Circles
Circles \(\omega_1\) and \(\omega_2\) meet at points \(A\) and \(B\). Point \(C\) is chosen on \(\omega_1\), and point \(D\) on \(\omega_2\), with \(C\) and \(D\) on the same side of line \(AB\). It is known that \(\angle ACB=83^\circ\), \(\angle ADB=50^\circ\). Find the acute angle between circles \(\omega_1\) and \(\omega_2\) at point \(A\).
C. Hint 1. The angle between the circles is the angle between their tangents at the common point.
D. Hint 2. Replace each angle between a tangent and \(AB\) by an inscribed angle subtending chord \(AB\).
E. Full solution. Draw tangents \(t_1\) and \(t_2\) at \(A\) to circles \(\omega_1\) and \(\omega_2\). The required angle is the acute angle between \(t_1\) and \(t_2\).
By the tangent-chord theorem, the angle between \(t_1\) and \(AB\) equals the inscribed angle \(\angle ACB=83^\circ\). Similarly, the angle between \(t_2\) and \(AB\) equals \(\angle ADB=50^\circ\).
Since \(C\) and \(D\) lie on the same side of \(AB\), the corresponding tangents are compared on the same side of line \(AB\). Hence the acute angle between the circles is the difference: \(83^\circ-50^\circ=33^\circ\).
Method comment. the angle between circles equals the angle between tangents, and each tangent can be replaced by an inscribed angle Thus the solution is not a brute-force chase of all angles in the diagram, but a deliberate choice of the right circle or tangent, after which the angles can be compared through the same chord or the same line.
If the auxiliary step is skipped, the problem looks almost arbitrary: the equal angles live in different parts of the diagram. That is why the hidden configuration is identified first, then the angle replacement is made, and only at the end the required conclusion follows.
F. Difficulty justification. This is Level 6: a medium regional-style problem. A direct angle chase quickly overloads the diagram, so one must see a hidden circle or replace an angle by a tangent argument; after that, the chain is short.
G. Check. This is not a one-step exercise: it requires 3 key ideas. First one must recognise the hidden geometric structure, then make an angle replacement or add a circle, and only after that complete the final conclusion.