Miquel's theorem

Miquel's theorem is a theorem in geometry, named after Auguste Miquel, about the intersection pattern of circles defined from six points on a triangle. 

From any triangle, and any three points on the three sides of the triangle, one may define three circles that each pass through a vertex of the triangle and the two points on its adjacent sides; Miquel's theorem states that these three circles meet in a single point, called the Miquel point.

Formally, let $ABC$ be a triangle, and let $A´,B´$ and $C´$ be three points on sides $BC,AC$, and $AB$ of the triangle, respectively. Draw three circles circumscribing the three triangles $AB´C´,A´BC´$, and $A´B´C$. Then Miquel's theorem states that these three circles intersect in a single point $M$, the Miquel point. In addition, the three angles $MA´B,MB´C$ and $MC´A$ (green in the diagram) are all equal to each other, as are the three angles $MA´C,MB´A$ and $MC´B$.

Read more »