Miquel's Theorem

Using the pointer, drag the vertices of triangle ABC. What do you notice about the angle measurements at the vertices and the angle measurements around the point E?
Now drag the point A'. When you drag point A', what happens to the angle measurements around B' and C'? Drag the points B' and C'. What happens?

Miquel's Theorem states that if A', B', and C' are marked on the sides of a triangle ABC, then the three circumcircles (also called Miquel's Circles) around triangles AB'C', BA'B', and CA'C' are concurrent at a single point E.

Angles AC'E, BA'E, and CB'E are congruent as are angles AB'E, BC'E, and CA'E.

Lastly, if E lies on the inside of triangle ABC and line segments are constructed from A'E, B'E and C'E, three quadrilaterals are formed AB'EC', BC'EA', and CA'EB'. As a result, angle BAC and angle C'EB' sum to 180, angle ABC and angle C'EA' sum to 180, and angle ACB and angle A'EB' sum to 180.

Michael Smith, Created with GeoGebra