MHB Proving Miquel's Theorem: Need Help!

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To prove Miquel's Theorem, one should start by identifying point M as the intersection of two circumcircles from the triangles AEF, BDF, and CDE. By drawing lines MD, ME, and MF, the properties of cyclic quadrilaterals can be utilized to demonstrate that M also lies on the third circumcircle. The theorem asserts that for any triangle ABC with points D, E, and F on sides BC, AC, and AB, the circumcircles intersect at point M. A reference to the Wikipedia page on Miquel's Theorem is suggested for additional insights. This approach effectively establishes the necessary proof for the theorem.
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I don't know how to start proving this theorem, so can someone please help? I need to prove that the circumcircles all intersect at a point M. Thank you!

Miquel's Theorem: If triangleABC is any triangle, and points D, E, F are chosen in the interiors of the sides BC, AC, and AB, respectively, then the circumcircles for triangleAEF, triangleBDF, and triangleCDE intersect in a point M.

I have attached here the figure of theorem.
 

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pholee95 said:
I don't know how to start proving this theorem, so can someone please help? I need to prove that the circumcircles all intersect at a point M. Thank you!

Miquel's Theorem: If triangleABC is any triangle, and points D, E, F are chosen in the interiors of the sides BC, AC, and AB, respectively, then the circumcircles for triangleAEF, triangleBDF, and triangleCDE intersect in a point M.

I have attached here the figure of theorem.
Have you looked at https://en.wikipedia.org/wiki/Miquel's_theorem? The trick seems to be to take $M$ to be the point where two of the three circles meet, draw the lines $MD$, $ME$ and $MF$, then use properties of cyclic quadrilaterals to show that $M$ also lies on the third circle.
 
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