Butterfly theorem

A circle-geometry theorem stating that if two chords through a fixed point meet a given chord at four points, the midpoint of the given chord bisects the segment joining the midpoints of the other two chords.

Intersecting chords theorem: For two chords intersecting inside a circle, the products of the lengths of their respective segments are equal. The theorem’s configurations use intersecting chords, whose segment products supply key algebraic relations.
Chord (geometry): A line segment whose endpoints lie on a circle. The theorem is built from three chords arranged through one interior point.
William Wallace: A Scottish mathematician who worked on geometry and taught at the University of Edinburgh. The theorem is often called Wallace’s theorem, though its history is not firmly settled.
Miquel's theorem: A theorem asserting concyclicity of certain intersection points formed by lines through the vertices of a triangle. Both results use linked circle incidences to force a non-obvious geometric relation.
Butterfly theorem generalizations: Extensions of the butterfly theorem that alter the curve, configuration, or midpoint condition. Generalizations test which parts of the circle configuration are essential to the bisection result.
Power of a point: A signed quantity, constant for a point and circle, equal to the product of the lengths of a secant’s segments. This invariant expresses the equal products underlying the butterfly configuration.
Midpoint: The point on a segment that divides it into two equal lengths. Both the given chord and the segment between chord midpoints are bisected.
The Ladies' Diary: An English annual periodical that published mathematical problems and solutions from the eighteenth century onward. A problem in its 1803 volume is linked to an early appearance of the theorem.
Simson line: The line through the feet of the perpendiculars from a point to a triangle’s three sidelines. It is another classical circle-geometry result often approached through projection and cyclicity.
Butterfly theorem converse: A proposed reverse statement asking whether a midpoint-bisection condition forces the original circle configuration. The converse probes whether the theorem’s conclusion characterizes its hypotheses.

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