Circumcenter of Triangle is Orthocenter of Medial
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Theorem
Let $\triangle ABC$ be a triangle.
Let $\triangle DEF$ be the medial triangle of $\triangle ABC$.
Let $K$ be the circumcenter of $\triangle ABC$.
Then $K$ is the orthocenter of $\triangle DEF$.
Proof
Let $FG$, $DH$ and $EJ$ be the perpendicular bisectors of the sides of $AC$, $AB$ and $BC$ respectively.
From Circumscribing Circle about Triangle, the point $K$ where they intersect is the circumcenter of $\triangle ABC$.
From Perpendicular Bisector of Triangle is Altitude of Medial Triangle, $FG$, $DH$ and $EJ$ are the altitudes of $\triangle DEF$.
The result follows by definition of orthocenter.
$\blacksquare$