Problem
GEO-B2-M08-P022 An Angle on the Fourth Circle
#22
★★★★★ Level 5 of 5
In quadrilateral \(ABCD\), let \(E=AB\cap CD\), \(F=AD\cap BC\), and let \(M\) be the second intersection of circles \((ABF)\) and \((BCE)\). Prove that \(\angle DMC=\angle DFC\).
Consider the complete quadrilateral formed by the lines \(AB,BC,CD,DA\), and identify the fourth Miquel circle.
The lines \(AB,BC,CD,DA\) form a complete quadrilateral. Since \(M\) lies on circles \((ABF)\) and \((BCE)\), Miquel's theorem gives \(M\in (CDF)\). In circle \((CDFM)\), the angles \(\angle DMC\) and \(\angle DFC\) subtend the same chord \(DC\), so they are equal.
A good problem on passing from an ordinary quadrilateral to the complete quadrilateral of its side lines.