Problem
GEO-B2-M08-P020 Circle from Two Angles
#20
★★★★☆ Level 4 of 5
In a complete quadrilateral, point \(M\) is chosen so that \(A,B,E,M\) and \(A,D,F,M\) are cyclic. Prove without citing Miquel's theorem that \(B,C,F,M\) are cyclic.
Repeat the angle proof: \(\angle BMF=\angle BMA+\angle AMF\).
From \(A,B,E,M\) cyclic, \(\angle BMA=\angle BEA\). From \(A,D,F,M\) cyclic, \(\angle AMF=\angle ADF\). The sum of these angles equals the angle between \(l_3\) and \(l_4\), namely \(\angle BCF\). Therefore \(\angle BMF=\angle BCF\), hence \(B,C,F,M\) are cyclic.
A strong check of understanding, not just citation.