Problem
GEO-B2-M08-P007 The Third Miquel Circle
#7
★★★☆☆ Level 3 of 5
In complete quadrilateral \(A,B,C,D,E,F\), point \(M\) lies on circles \((ABE)\) and \((ADF)\). Prove that \(B,C,F,M\) lie on one circle.
Prove the equality \(\angle BMF=\angle BCF\).
We have \(\angle BMF=\angle BMA+\angle AMF\). From \(A,B,E,M\) cyclic, \(\angle BMA=\angle BEA\), the angle between \(l_3\) and \(l_1\). From \(A,D,F,M\) cyclic, \(\angle AMF=\angle ADF\), the angle between \(l_1\) and \(l_4\). The sum is the angle between \(l_3\) and \(l_4\), namely \(\angle BCF\). Therefore \(B,C,F,M\) are cyclic.
The main working angle in Miquel's theorem.