Problem
GEO-B2-M08-P016 Proving a New Circle
#16
★★★★☆ Level 4 of 5
In a complete quadrilateral, \(M\) is the Miquel point. Prove that if point \(X\) lies on line \(l_3\) and \(\angle BXF=\angle BMF\), then \(B,F,M,X\) lie on one circle.
The equal angles should subtend the same segment \(BF\).
Angles \(\angle BXF\) and \(\angle BMF\) subtend segment \(BF\). By the converse of the inscribed angle criterion, points \(X\) and \(M\) lie on one circle with \(B\) and \(F\). Hence \(B,F,M,X\) are concyclic.
Shows how the Miquel point helps create new cyclic quadruples.