Problem
GEO-B2-M10-P017 An Angle from a Miquel Point
#17
★★★★☆ Level 4 of 5
In quadrilateral \(ABCD\), lines \(AB\) and \(CD\) meet at \(E\), while \(AD\) and \(BC\) meet at \(F\). Let \(M\) be the Miquel point of the four lines \(AB,BC,CD,DA\). Prove that \(\angle AMB=\angle DFC\).
Point \(M\) lies on circle \((ABF)\).
By Miquel's theorem, \(A,B,F,M\) lie on one circle. Therefore \(\angle AMB=\angle AFB\), since these angles subtend chord \(AB\). But \(FA\) is the same line as \(FD\), and \(FB\) is the same line as \(FC\). Hence \(\angle AFB=\angle DFC\). Therefore \(\angle AMB=\angle DFC\).
Here the Miquel point is used not to prove a common point, but to read an angle.