---
tag: E9TX1NDAHB
kind: statement
role: theorem
name: Butterfly theorem
msc: [51M04]
uses: [060W5CBZP6, 0S706JBZBG]
about: [060W5CBZP6]
sources:
  - url: https://en.wikipedia.org/wiki/Butterfly_theorem
same_as:
  wikidata: Q377047
  wikipedia: Butterfly theorem
---

## Statement

Let $PQ$ be a chord of a [[circle|060W5CBZP6]] in the Euclidean plane and let $M$ be its [[midpoint|0S706JBZBG]].
Let $AB$ and $CD$ be two further chords through $M$.
Suppose that the segment $AD$ meets $PQ$ at $X$ and the segment $BC$ meets $PQ$ at $Y$.
Then $M$ is the midpoint of $XY$.

## Remarks

One proof drops perpendiculars from $X$ and $Y$ to the chords and uses similar triangles together with the power of the point $M$, $MA \cdot MB = MC \cdot MD = MP \cdot MQ$.
A projective proof uses the involution that the pencil of conics through $A, B, C, D$ induces on the line $PQ$.
Example: let $CD$ be the mirror image of $AB$ in the diameter through $M$ perpendicular to $PQ$, with $C$ the image of $A$.
Then the figure is symmetric and $MX = MY$ is clear.
The theorem stays true when the circle is replaced by any conic.

## Explanations

### L1 human

Take a chord of a circle and its midpoint.
Through the midpoint draw two more chords, and join their ends crosswise to make a shape like the wings of a butterfly.
The wings cut the first chord at two points that are exactly equally far from the midpoint.