The intercept theorem

The intercept theorem links parallelism to proportions: when several parallel lines cut two transversals, the segments they determine on them are in the same proportion. It is the foundation of the whole theory of similarity.

Theorem — the intercept theorem

A pencil of parallel lines cutting two transversals determines proportional segments on them. $r_1\parallel r_2\parallel r_3 \implies \frac{AB}{BC} = \frac{A'B'}{B'C'}$

The three parallel lines $r_1,r_2,r_3$ cut the two transversals in proportional segments: $AB:BC = A'B':B'C'$.

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Topics: Euclidean geometry
Concepts: Proportionality · Line · Parallel lines · Intercept theorem · Transversal
Skills: Synthetic geometry · Using formulae
People: Thales