Problem
GEO-B2-M04-P019 The Angle Between Images
Points \(B,A,D\) are collinear, and points \(C,A,E\) are collinear. Circles \((ABC)\) and \((ADE)\) meet at points \(A\) and \(M\). Prove that the angle between lines \(MB\) and \(MC\) equals the angle between lines \(MD\) and \(ME\).
Translate the statement into \(\angle BMC=\angle DME\).
The angle between \(MB\) and \(MC\) is \(\angle BMC\), and the angle between \(MD\) and \(ME\) is \(\angle DME\). Since \(A,B,C,M\) lie on one circle, \(\angle BMC=\angle BAC\). Since \(A,D,E,M\) lie on one circle, \(\angle DME=\angle DAE\). Lines \(BAD\) and \(CAE\) determine the same angle at \(A\), so \(\angle BAC=\angle DAE\). Therefore the required angles are equal.
This is the same configuration with a different wording: students must recognise the structure.