Given a chord PQ with midpoint M, and two other chords AB and CD that intersect at M, then M is the midpoint of the intersections with PQ by AD and CB. Additionally (or alternatively), if AB and CD are considered the diagonals of cyclic quadrilateral CADB, then opposite sides of CADB intersect PQ at points equidistant from M.
Let's start with the traditional formulation of the Butterfly Theorem.
It begins with a chord, PQ, with midpoint M.
any two chords AB and CD are drawn that pass through M.
Now chords AD and CB are drawn, and the points where these chords intersect PQ are marked X and Y respectively.
According to the Butterfly Theorem, M is the midpoint of XY.
Another way to look at this theorem is to view those green chords, AB and CD, as diagonals of cyclic quadrilateral CADB. Under this interpretation, the statement of the Butterfly Theorem is:
If the diagonals of cyclic quadrilateral CADB intersect at the midpoint of arc PQ, then each pair of opposite sides of CADB intersect PQ at points equidistant from M.
In order to see the intersections formed by one pair of opposite sides, these sides must be extended.
According to the Butterfly Theorem, sides AD and BC of cyclic quadrilateral
CADB cut chord PQ at points X and Y, which are equidistant from the chord's
midpoint, M. Additionally, sides CA and DB cut the (extended) chord PQ at
points W and Z, which, too, are equidistant from M.
Mathworld: Butterfly Theorem proves the theorem.
Summary of geometrical theorems
Inscribed Angle Property -- (a.k.a. Central Angle Theorem) is a proof that all angles inscribed in a circle subtend an arc that is twice the arc subtended by the same central angle.
The Inscribed Angle Property: (also known as the Central Angle Theorem): The measure of an angle inscribed in a circle is half the measure of the arc it intercepts.
The Intersecting Chords Theorem says the products of the two segments of chords cut by their point of intersection are equal.
Ptolemy's Theorem -- given a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides.
Urquhart's Theorem: If lines OA and OB intersect at O, A' is on OA, B' is on OB, AB' and A'B intersect at O'; Then OA + AO' = OB + BO' iff OA' + A'O' = OB' + B'O'.
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