Let $ABCD$ and $AB'C'D'$ be two squares with common vertex $A$. Let $E$ and $G$ be the midpoints of $B'D$ and $D'B$ respectively, and let $F$ and $H$ be the centers of the two squares. Then the quadrilateral $EFGH$ is a square as well

Triangle

Centroid

Centroid

Circumcenter

Incenter

Orthocenter

Euler Line

Gergonne Point

Ceva's Theorem

Ceva's Theorem

Isotomic Conjugate

Menelaus's Theorem

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Finsler-Hadwiger Theorem

Morley's Trisector Theorem

Viviani's Theorem

Van Aubel's Theorem

Thébault's I Problem

Fagnano's Problem

Droz-Farny Line Theorem

Let $ABCD$ and $AB'C'D'$ be two squares with common vertex $A$. Let $E$ and $G$ be the midpoints of $B'D$ and $D'B$ respectively, and let $F$ and $H$ be the centers of the two squares. Then the quadrilateral $EFGH$ is a square as well