Miquel's theorem
Let ABC be any triangle, and A´ , B´ and C´ be arbitrary points on lines BC, AC, and AB
respectively. Then the 3 circumcircles to triangles AB´C´, A´BC´, and A´B´C intersect in a unique point M, called Miquel's point.
Moreover, if the points A´, B´ and C´ are on the segments BC, AC, and AB , then the three angles MA´B, MB´C and MC´A are equal, and (of course also) the three supplementary angles MA´C, MB´A and MC´B.
Check what happens if the points A´, B´ and C´ are not on the segments BC, AC, and AB.
If the points A´, B´ and C´ are colinear, then the point M belongs to the circumcircle of the triangle
ABC. And vice-versa. To check this, the line B’C’ has been plotted (in red). Move the point A’ until it is on this line and check what happens.