Intercept theorem

Statement

Let $S$ be a point of the Euclidean plane, and let $\ell_1$ and $\ell_2$ be two distinct lines 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 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. The homothety 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.