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Similarity geometry lies between Euclidean geometry (with group *I*(**R**^{n})) and Affine geometry (group *A*(**R**^{n})).

**Definition**

A **similarity transformation** or **similitude** is an affine map which preserves angles.

**Theorem**

*A similarity transformation can be written T _{a} λL with L ∈ O*(

**Proof**

Since *f* is an affine transformation it can be written as *T*_{a} *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

**Pythagoras's theorem**

*If a triangle ABC has a right angle at A then* *AB*^{2} + *AC*^{2} = *BC*^{2}

**Proof**

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 *AB*^{2} = *BP*.*BC* and *AC*^{2} = *BC*.*PC*

Thus *AB*^{2} + *AC*^{2} = (*BP* + *PC*).*BC* = *BC*^{2}

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