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.

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Euclidean distance concept
Line concept
Intercept theorem statement

Tag	`0M1GFS6PVA`
URI	`m2://0M1GFS6PVA`
Aliases	`basic-proportionality-theorem`, `thales-theorem-on-proportions`
Kind	statement, theorem
Writing	W1 stated
MSC	51M04
Status	This copy of the site has no build data. Run `m2 status 0M1GFS6PVA`.
Uses	Line, Euclidean distance
Same as	Wikidata Q3757563 Wikipedia Intercept theorem

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