Question

The Butterfly Theorem. Suppose M is the midpoint of a chord P Q of a circle and AB and CD are two other chords that pass through M. Let AD and BC intersect P Q at X and Y , respectively. Then M is also the midpoint of XY .

1. Prove the Butterfly Theorem. [10]

Hint: You know a lot about angles in a circle, and about triangles, and cross ratios, and all sorts of things . . .

Answer 1

Given : A chord PQ of a circle has a mid point 'M'. There are two other chords AB and CD passing through the midpoint 'M'. Chord AD intersects PQ at X and chord CB intersects PQ at Y.

Prove That : 'M' is the midpoint of XY. i.e. XM = MY

Construction : Draw , , ,

Proof :

BY Similarity of Right Angled Triangles (A-A test) the Pair of Similar Triangles and its Corresponding Ratios are as follows:

One pair of right angles and .......Vertically opp angles

.. (1)..

One pair of right angles and ...... Vertically opp angles

.. (2).

One pair of Right angles and ..........(Angles intercept arc BD)

.. (3)..

One pair of Right angles and ..........(Angles intercept arc AC)

.. (4)...

Multiplying Sts 1 and 2

From Sts 3 and 4

.......... (5)

BY INTERSECTING CHORDS THEOREM

............. (6)

From Sts 5 and 6

............... (P - X - M - Y - Q)

'M' is the midpoint of chord PQ

PM = MQ

BY Theorem on Equal Ratios

................... (By deriving Square roots)

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