# Butterfly Theorem

The **butterfly theorem** is a well-known result from Euclidean geometry.

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## Statement

Let $M$ be the midpoint of chord $PQ$ of a circle.

Draw two other chords $AB$ and $CD$ through $M$.

If $AD$ and $BC$ intersect chord $PQ$ at $X$ and $Y$, respectively, $M$ is the midpoint of $XY$.

Looking at the diagram, you can probably tell how the butterfly theorem got its name!

## Proof

There are various proofs for the butterfly theorem. We're going to prove it by using the properties of similar triangles.

We're going to start by dropping perpendiculars $x_1$ and $x_2$ from $X$ to $AB$ and $CD$, respectively, and likewise $y_2$ and $y_1$ from $Y$ to $AB$ and $CD$, respectively.

Here's what the diagram looks like after we do that:

Let $MQ=PM=a$, $XM=x$, and $MY=y$.

Now we have a lot of similar triangles we can work with, and from these similar triangles we have a lot of ratios that are equal to one another:

$\begin{array}{c}&\dfrac{x}{y}=\dfrac{x_1}{y_2}, &\dfrac{x}{y}=\dfrac{x_2}{y_1}, &\dfrac{x_1}{y_1}=\dfrac{AX}{CY},& \dfrac{x_2}{y_2}=\dfrac{XD}{YB}.\end{array}$

So, we have $\dfrac{x^2}{y^2}=\dfrac{x_1}{y_2}\cdot \dfrac{x_2}{y_1}=\dfrac{x_1}{y_1} \cdot \dfrac{x_2}{y_2}=\dfrac{AX \cdot XD}{CY \cdot YB}$.

Now we're going to use the intersecting chords theorem. (Check the wiki out if you aren't familiar with it.)

From that theorem, we have, $\dfrac{AX\cdot XD}{CY \cdot YB}=\dfrac{PX \cdot XQ}{PY \cdot YQ}$.

So, we can write $\dfrac{x^2}{y^2}=\dfrac{PX \cdot XQ}{PY \cdot YQ}=\dfrac{(a-x)(a+x)}{(a+y)(a-y)}=\dfrac{a^2-x^2}{a^2-y^2}$.

In other words, $\dfrac{x^2}{y^2}=\dfrac{a^2-x^2}{a^2-y^2}$. But that can only happen if $x^2=y^2$.

This means $x=y$, which implies $M$ is the midpoint of $XY$. $_\square$

On the circumference of circle $\Gamma$, chord $AB$ with length $1100$ is drawn. Let $C$ be the midpoint of $AB$. Through $C$, 2 other chords $DE$ and $FG$ are also drawn, such that the points around the circle are $A, D, F, B, E, G$. The line segment $AB$ intersects $DG$ and $FE$ (internally) at $H$ and $I$, respectively.

If $AH=449$, what is $CI?$

## Practice Problems

- Let the incircle $($with center $I)$ of $\triangle ABC$ touch the side $BC$ at $X$, and let $A'$ be the midpoint of this side. Then prove that line $A'I$ (extended) bisects $AX$.
- Let $PT$ and $PB$ be two tangents to a circle, $AB$ the diameter through $B$, and $TH$ the perpendicular from $T$ to $AB$. Then prove that $AP$ bisects $TH$.

## See Also

**Cite as:**Butterfly Theorem.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/butterfly-theorem/