Euclidean Geometry Revisited - Main Theorem

TheoremFundamental Theorem of Similarity

Two triangles are similar if and only if one of the following conditions holds:

AA (Angle-Angle): Two angles of one triangle are congruent to two angles of the other
SAS (Side-Angle-Side): Two sides are proportional and the included angles are congruent
SSS (Side-Side-Side): All three pairs of corresponding sides are proportional

Furthermore, if triangles $\triangle ABC \sim \triangle DEF$ with similarity ratio $k$ , then:

\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)} = k^2

This theorem is fundamental to Euclidean geometry and distinguishes it from non-Euclidean geometries where similar triangles of different sizes may not exist. The proof relies on the parallel postulate.

The similarity theorem has profound implications. It allows us to deduce properties of inaccessible objects by studying similar models. Ancient Greek mathematicians used similar triangles to measure the height of pyramids and the distance to ships at sea. Modern applications include computer graphics, where objects are scaled while preserving shape.

DefinitionHomothety

A homothety (or similarity transformation) with center $O$ and ratio $k \neq 0$ is the transformation:

h_{O,k}(P) = O + k \cdot \overrightarrow{OP}

Homotheties preserve angles and ratios of lengths. The composition of a homothety with an isometry yields the most general similarity transformation.

The relationship between similarity and area reveals a deep principle: when linear dimensions scale by a factor $k$ , areas scale by $k^2$ , and volumes by $k^3$ . This scaling law has applications throughout physics and engineering, from the structural strength of buildings to the metabolism of animals.

ExampleThales' Theorem

If a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally. Specifically, if $DE \parallel BC$ in triangle $\triangle ABC$ with $D$ on $AB$ and $E$ on $AC$ , then:

\frac{AD}{DB} = \frac{AE}{EC}

This result, attributed to Thales of Miletus (c. 624-546 BCE), is a direct consequence of similar triangles.

The theory of similar triangles extends to three dimensions with similar tetrahedra and more general polyhedra. The key property—that similarity preserves angles and ratios—remains valid. In higher dimensions, the relationship between similarity ratio and volume generalizes: if two $n$ -dimensional objects are similar with ratio $k$ , their $n$ -dimensional volumes have ratio $k^n$ .

Remark

The existence of similar triangles of arbitrary size is equivalent to the parallel postulate. In hyperbolic geometry, all triangles with the same angles are congruent—there is an absolute scale. In spherical geometry, large similar triangles distort differently than small ones. This makes similarity theory particularly distinctive to Euclidean geometry.

Similarity transformations form a group under composition. This group contains the Euclidean group $E(n)$ as a subgroup (the similarities with ratio $k=1$ ) and includes all scalings. The full group of similarities, denoted $\text{Sim}(n)$ , can be written as:

\text{Sim}(n) = \mathbb{R}^n \rtimes (\mathbb{R}^+ \times O(n))

where $\mathbb{R}^+$ represents positive scalings. This structure underlies the uniformity and scaling properties of Euclidean space.