Problem
GEO-B2-M08-P014 If Two Circles Already Meet
#14
★★★★☆ Level 4 of 5
In quadrilateral \(ABCD\), let \(E=AB\cap CD\), \(F=AD\cap BC\). Circles \((ABF)\) and \((BCE)\) meet at \(B\) and \(M\). Prove that \(M\in (CDF)\) and \(M\in (DAE)\).
This is Miquel's theorem for the four lines \(AB,BC,CD,DA\).
Lines \(AB,BC,CD,DA\) form a complete quadrilateral. Two of its circles, \((ABF)\) and \((BCE)\), meet again at \(M\). By Miquel's theorem, the same point lies on the other two circles of the complete quadrilateral: \((CDF)\) and \((DAE)\).
The problem teaches recognition of the theorem in an ordinary quadrilateral.