## Butterfly Theorem

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.

Then,
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.

### Alternative formulation of the Butterfly Theorem

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.

### Internet references

Mathworld:
Butterfly Theorem proves the theorem.

### Related pages in this website:

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'.

The webmaster and author of this Math Help site is
Graeme McRae.