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  1. MHB Apprentice

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    #1
    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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    #2
    Quote Originally Posted by pholee95 View Post
    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 ? 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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