Problem
GEO-B2-M08-P023 Two Angle Chains
#23
★★★★★ Level 5 of 5
Let \(M\) be the Miquel point of a complete quadrilateral with notation \(A,B,C,D,E,F\). Prove the equalities \(\angle BME=\angle DMF\) and \(\angle CME=\angle CDE\).
Read the first equality from circles \((ABE)\) and \((ADF)\), and the second from circle \((CDE)\).
Since \(A,B,E,M\) are concyclic, \(\angle BME=\angle BAE\). Since \(A,D,F,M\) are concyclic, \(\angle DMF=\angle DAF\). The angles \(\angle BAE\) and \(\angle DAF\) are the same angle between the lines \(l_2\) and \(l_1\), so \(\angle BME=\angle DMF\). Also, \(C,D,E,M\) are concyclic; the angles \(\angle CME\) and \(\angle CDE\) subtend chord \(CE\), hence they are equal.
The problem trains extracting several angle equalities from one Miquel configuration.