---
tag: 0M1GFS6PVA
kind: statement
role: theorem
name: Intercept theorem
aliases: [basic-proportionality-theorem, thales-theorem-on-proportions]
msc: [51M04]
uses: [0MP9BCF6DP, 0NPPGHQNXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Intercept_theorem
    locator: Formulation of the theorem
same_as:
  wikidata: Q3757563
  wikipedia: Intercept theorem
---

## Statement

Let $S$ be a point of the Euclidean plane, and let $\ell_1$ and $\ell_2$ be two distinct [[lines|0MP9BCF6DP]] through $S$. Let $A$ and $B$ be points of $\ell_1$, and let $C$ and $D$ be points of $\ell_2$, all different from $S$. Let $|XY|$ denote the [[distance|0NPPGHQNXQ]] between the points $X$ and $Y$.
Assume that the line $AC$ is parallel to the line $BD$. Then
$$\frac{|SA|}{|SB|} = \frac{|SC|}{|SD|} = \frac{|AC|}{|BD|}.$$

## Remarks

If also $A \neq B$ and $A$ and $B$ are on the same ray from $S$, then the theorem also gives $\frac{|SA|}{|AB|} = \frac{|SC|}{|CD|}$.
A converse holds for two rays from $S$: if $A, B$ are on one ray, $C, D$ on the other, and $\frac{|SA|}{|AB|} = \frac{|SC|}{|CD|}$, then $AC$ is parallel to $BD$.
The theorem says that the triangles $SAC$ and $SBD$ are [[similar|0YF6QXXWB5]]. The [[homothety|0ZG1PNEP35]] with the centre $S$ that sends $A$ to $B$ also sends $C$ to $D$.

## History

Tradition attributes the theorem to Thales of Miletus. He may have used it to find the heights of the pyramids of Egypt.

## Explanations

### L1 human

Draw two lines from one point, and cut them with two parallel lines. The parallel lines cut both lines in the same proportions. This is how you can find the height of a tree from the length of its shadow and the shadow of a stick.