Butterfly theorem

Dependencies: Inscribed angles on same arc are equal; angle in a semicircle is right (inscribed angles on the same arc are equal), Opposite angles of a cyclic quadrilateral are supplementary (Opposite angles of a cyclic quadrilateral are supplementary), AA similarity (AA similarity), Intersecting chords: PA·PB = PC·PD (the "power" identity at an interior point), Perpendicular from a point to a line exists and is unique (perpendicular from a point + foot).

Statement

Let $\odot O$ be a circle, $\overline{PQ}$ a chord, and $M$ the midpoint of $\overline{PQ}$. Through $M$, draw two arbitrary chords $\overline{AB}$, $\overline{CD}$ (with endpoints $A, B, C, D$ on $\odot O$). Let the two diagonals

meet $\overline{PQ}$ at $X$ and $Y$ respectively. Then $M$ also bisects $\overline{XY}$:

Commonly known as the butterfly theorem — together with $PQ$, the four chords $AB$, $CD$, $AD$, $BC$ form a figure resembling a butterfly with outstretched wings, and "the midpoint $M$ bisects the central rib $\overline{XY}$" is its core symmetry.