point sharing

[Tahmid Hasan]

$D,E,F$ are the midpoints of $BC,AC,AB$ of a triangle respectively.prove that h circumcircles of $ \Delta AEF ,\Delta BDF ,\Delta CDE $ share a point.locate the point(it co-incides with a well-defined point.)

[Moon]

Actually this problem can be generalized further. For any three points $D,E,F$ those three circles will have a common point. (The point is known as the Miquel Point.)

Here in this special case the circumcenter of $\triangle ABC$ is the Miquel point.

[Tahmid Hasan]

isn't it called pivot's point?

[Moon]

Yeah, some people mentioned in the math camp that it is also called Pivot point. However, afaik this point is called Miquel point.