Problem
GEO-B2-M08-P021 Common Point of Three Circles
#21
★★★★★ Level 5 of 5
In a complete quadrilateral, circles \((ABE)\), \((ADF)\), \((BCF)\) have a common point \(M\ne A,B,F\). Prove that \(M\) lies on circle \((CDE)\).
This is the remaining circle in Miquel's theorem.
Since \(M\) lies on \((ABE)\) and \((ADF)\), the angle proof of Miquel's theorem implies that \(M\) must lie on \((CDE)\). Directly, \(\angle CME\) splits into a sum of angles equal to \(\angle CDE\). By the converse of the inscribed angle criterion, \(C,D,E,M\) are cyclic.
Level 5 because the missing circle must be recognised without being built first.