Similarity geometry lies between Euclidean geometry (with group I(Rn)) and Affine geometry (group A(Rn)).
A similarity transformation or similitude is an affine map which preserves angles.
A similarity transformation can be written Ta ∘ λL with L ∈ O(n) and λ > 0.
Since f is an affine transformation it can be written as Ta ∘ L with L linear. Since f preserves angles, so does L and such a map must stretch all vectors by the same amount and must be a non-zero scalar multiple of an orthogonal transformation.
Any two figures related by a similarity transformation are called similar.
For example, any two squares are similar; any two circles are similar; two triangles are similar if their corresponding angles are equal.
Many of the theorems of so-called Euclidean geometry are in fact theorems of Similarity geometry.
For example, the well-known theorem of Pythagoras can be proved by "similar triangle" methods
If a triangle ABC has a right angle at A then AB2 + AC2 = BC2
Drop a perpendicular to P as shown.
Then the triangles ABC, PBA and PAC are similar and so have their sides in proportion.
Hence AB/PB = BC/BA and AC/PC = BC/AC and so AB2 = BP.BC and AC2 = BC.PC
Thus AB2 + AC2 = (BP + PC).BC = BC2