Problem
GEO-B2-M08-P010 Miquel of the Side Lines
#10
★★★☆☆ Level 3 of 5
In quadrilateral \(ABCD\), lines \(AB\) and \(CD\) meet at \(E\), and \(AD\) and \(BC\) meet at \(F\). Prove that circles \((ABF)\), \((BCE)\), \((CDF)\), \((DAE)\) have one common point.
Consider the complete quadrilateral of the four lines \(AB,BC,CD,DA\).
The four lines \(AB,BC,CD,DA\) form a complete quadrilateral. Its four circles are exactly \((ABF)\), \((BCE)\), \((CDF)\), and \((DAE)\). By Miquel's theorem, they pass through one point.
A direct application of the theorem in familiar notation.