Sim transitive — Similarity is transitive · Principia

Dependencies: AA similarity (provides the working definition of "similar": corresponding angles equal and corresponding sides in proportion).

Statement

Let $\triangle A$ , $\triangle B$ , $\triangle C$ be three triangles. If

\triangle A \sim \triangle B \quad \text{(with similarity ratio } k_1\text{)}, \qquad \triangle B \sim \triangle C \quad \text{(with similarity ratio } k_2\text{)},

then

\triangle A \sim \triangle C \quad \text{(with similarity ratio } k_1 k_2\text{)}.

In other words, the similarity relation is transitive: two layers of similarity can be "stacked" into one, with similarity ratios multiplying.

$Transitivity of similarity: \triangle A \sim \triangle B (ratio k_1), \triangle B \sim \triangle C (ratio k_2) ⇒ \triangle A \sim \triangle C (ratio k_1 k_2)$

Transitivity of similarity

Statement

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