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Homesimilarity in geometry
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similarity in geometry
Two figures $K$ and $K^{{\prime}}$ in a Euclidean plane or space are similar iff there exists a bijection $f$ from the set of points of $K$ onto the set of points of $K^{{\prime}}$ such that, for any $P,Q\in K$, the ratio
$\frac{P^{{\prime}}Q^{{\prime}}}{PQ}$ 
of the lengths of the line segments $P^{{\prime}}Q^{{\prime}}$ and $PQ$ is always the same number $k$, where $P^{{\prime}}=f(P)$ and $Q^{{\prime}}=f(Q)$.
The number $k$ is called the ratio of similarity or the line ratio of the figure $K^{{\prime}}$ with respect to the figure $K$ (N.B. the order in which the figures are mentioned!). The similarity of $K$ and $K^{{\prime}}$ is often denoted by
$K^{{\prime}}\sim K\;\;\;(\mbox{or}\;\;K\sim K^{{\prime}}).$ 
Examples

All cubes are similar.

All circles are similar.

All parabolas are similar.

All sectors of circle with equal central angle are similar.

All spheres are similar.

All equilateral triangles are similar.
Nonexamples

Not all rectangles are similar.

Not all rhombi are similar.

Not all rectangular prisms are similar.

Not all ellipses are similar.

Not all ellipsoids are similar.

Not all triangles are similar.
Properties

The corresponding angles (consisting of corresponding points) of two similar figures are equal.

The lengths of any corresponding arcs of two similar figures are proportional in the ratio $k$.

The areas of two similar regions are proportional in the ratio $k^{2}$ when $k$ is the line ratio of the regions.
Remarks

In any Euclidean space $E$, the relation of similarity (denoted $\sim$) on the set of figures in $E$ is an equivalence relation.
Mathematics Subject Classification
51F99 no label found51M05 no label found5100 no label found Forums
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