Problem
GEO-B2-M08-P018 Two Pairs of Segments
#18
★★★★☆ Level 4 of 5
Let \(M\) be the Miquel point of a complete quadrilateral. Prove that segments \(BE\) and \(DF\) are seen from \(M\) under equal angles: \(\angle BME=\angle DMF\).
Compare both angles with the angle between \(l_2\) and \(l_1\).
Since \(A,B,E,M\) are cyclic, \(\angle BME=\angle BAE\). Since \(A,D,F,M\) are cyclic, \(\angle DMF=\angle DAF\). Both right-hand angles are the angle between lines \(l_2\) and \(l_1\), so \(\angle BME=\angle DMF\).
This forms a bridge to spiral similarity.