---
tag: 0YQQYET8YQ
kind: statement
role: theorem
name: Intersecting chords theorem
aliases: [chord-theorem]
msc: [51M04]
uses: [049DCJZ448, 0NPPGHQNXQ, 02E9VT1RVQ]
sources:
  - url: https://en.wikipedia.org/wiki/Intersecting_chords_theorem
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Intersecting_chords_theorem
    locator: Proof
same_as:
  wikidata: Q2266397
  wikipedia: Intersecting chords theorem
---

## Statement

Let $A$, $B$, $C$ and $D$ be [[concyclic points|049DCJZ448]] of the Euclidean plane $\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Suppose that the chords $AC$ and $BD$ meet at a point $S$ that is inside both segments.
Then
$$d(A, S)\, d(S, C) = d(B, S)\, d(S, D).$$

## Remarks

Both products are equal to $r^2 - d(S, O)^2$, where $O$ is the centre and $r$ the radius. This is minus the [[power|02E9VT1RVQ]] of $S$.
A proof: the triangles $ASD$ and $BSC$ are similar, by the inscribed angle theorem.
The theorem is Proposition 35 of Book III of Euclid's Elements.

## Explanations

### L1 human

Draw two chords of a circle that cross each other. Each chord is cut into two pieces. The product of the two pieces is the same for both chords.