[Math] Orthocenter of triangle $DEF$ is same as the circumcenter of triangle $ABC$

analytic geometrygeometry

$D,E,F$ are mid points of the sides of the triangle $ABC$,then prove that the orthocenter of triangle DEF is same as the circumcenter of triangle ABC.

I cannnot figure out what coordinates to suppose for A,B,C.I tried taking $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ but calculations go messy and clumsy.Can someone help me in proving this question?

Best Answer

Algebra isn't necessary. Instead, use geometrical definitions: in particular, the circumcentre $O$ of $ABC$ is at the intersection of the perpendicular bisectors of $AB$, $AC$, and $BC$. Do you see how to continue from there?

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