Circumcenter of Triangle is Orthocenter of Medial

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

 * CircumcenterMedialOrthocenter.png

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.