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### Internal similitude center

In general, the internal similitude center of two circles and with centers given in Cartesian coordinates is given by(1)In trilinear coordinates, the internal center of similitude is given by , where(2)(3)(4)The incircle and circumcircle of a triangle have two similitude centers, namely the internal center of similitude Si and the external similitude center Se. The internal center of similitude of these two circles Si is the isogonal conjugate of the Gergonne point of . It is Kimberling center and has equivalent triangle center functions(5)(6)(7)The two points Si and Se share certain similar properties, but there seems to be no straightforward analogy between the two. For instance, Si is the homothetic center of the tangential, intangents, and extangents triangles of triangle taken pairwise, but the only comparable property of the external similitude center Se is more complicated: Se is the homothetic center of the tangential triangle..

### Fundamental theorem of directly similar figures

Let and denote two directly similar figures in the plane, where corresponds to under the given similarity. Let , and define . Then is also directly similar to .

### External similitude center

In general, the external similitude center of two circles and with centers given in Cartesian coordinates is given by(1)In trilinear coordinates, the external center of similitude is given by , where(2)(3)(4)The incircle and circumcircle of a triangle have two similitude centers, namely the internal similitude center Si and the external center of similitude Se. The external center of similitude of the circumcircle and incircle Se is the isogonal conjugate of the Nagel point of . It is Kimberling center and has equivalent triangle center functions(5)(6)(7)The two points Si and Se share certain similar properties, but there seems to be no straightforward analogy between the two. For instance, the internal similitude center Si is the homothetic center of the tangential, intangents, and extangents triangles of triangle taken pairwise, but the only comparable property of Se is more complicated: Se is the homothetic center of the tangential..

### Similitude center

If two similar figures lie in the plane but do not have parallel sides (i.e., they are similar but not homothetic), there exists a center of similitude, also called a self-homologous point, which occupies the same homologous position with respect to the two figures (Johnson 1929, p. 16).The similitude center of two triangles and can be constructed by extending each pair of corresponding sides of the triangles and locating their intersection, then drawing the circumcircle passing through two corresponding vertices of the triangles and the point of intersection of the pair of lines through corresponding sides that contain these points. Repeating for each of the three vertices gives three circles that intersect in a unique point, as illustrated above. This point is the similitude center (Johnson 1929).The locus of similitude centers of two nonconcentric circlesis another circle having the line joining the two homothetic centers as..

### Similarity

A transformation that preserves angles and changes all distances in the same ratio, called the ratio of magnification. A similarity can also be defined as a transformation that preserves ratios of distances.A similarity therefore transforms figures into similar figures. When written explicitly in terms of transformation matrices in three dimensions, similarities are commonly referred to as similarity transformations.Examples of similarities include the following. 1. Central dilation: a transformation of lines to parallel lines that is not merely a translation. 2. Geometric contraction: a transformationin which the scale is reduced. 3. Dilation: a transformation taking each line to a parallel line whose length is a fixed multiple of the length of the original line. 4. Expansion: a transformation in which the scale isincreased. 5. Isometry: a transformation that preserves distances.6. Reflection: a transformation in which all..

### Similar

Two figures are said to be similar when all corresponding angles are equal and all distances are increased (or decreased) in the same ratio, called the ratio of magnification (Coxeter and Greitzer 1967, p. 94). A transformation that takes figures to similar figures is called a similarity.Two figures are directly similar when all corresponding angles are equal and described in the same rotational sense. This relationship is written . (The symbol is also used to mean "is the same order of magnitude as" and "is asymptotic to.") Two figures are inversely similar when all corresponding angles are equal and described in the opposite rotational sense.

### Homothetic center

The meeting point of lines that connect corresponding points from homothetic figures. In the above figure, is the homothetic center of the homothetic figures and . For figures which are similar but do not have parallel sides, a similitude center exists (Johnson 1929, pp. 16-20).Given two nonconcentric circles, draw radii parallel and in the same direction. Then the line joining the extremities of the radii passes through a fixed point on the line of centers which divides that line externally in the ratio of radii. This point is called the external homothetic center, or external center of similitude (Johnson 1929, pp. 19-20 and 41).If radii are drawn parallel but instead in opposite directions, the extremities of the radii pass through a fixed point on the line of centers which divides that line internally in the ratio of radii (Johnson 1929, pp. 19-20 and 41). This point is called the internal homothetic center, or internal..

### Homothetic

Two figures are homothetic if they are related by an expansion or geometric contraction. This means that they lie in the same plane and corresponding sides are parallel; such figures have connectors of corresponding points which are concurrent at a point known as the homothetic center. The homothetic center divides each connector in the same ratio , known as the similitude ratio. For figures which are similar but do not have parallel sides, a similitude center exists.

### Spiral similarity

The combination of a central dilation and a rotation about the same center. However, the combination of a central dilation and a rotation whose centers are distinct is also a spiral symmetry. In fact, any two directly similar figures are related either by a translation or by a spiral symmetry (Coxeter and Greitzer 1967, p. 97).A spiral similarity tessellation of any ordinary tessellation can be constructed by placing a series of polygonal tiles of decreasing size on an equilateral spiral.

### Similarity point

External (or positive) and internal (or negative) similarity points of two circles with centers and and radii and are the points and on the lines such thator

### Geometric congruence

Two geometric figures are said to exhibit geometric congruence (or "be geometrically congruent") iff one can be transformed into the other by an isometry (Coxeter and Greitzer 1967, p. 80). This relationship is written . (Unfortunately, the symbol is also used to denote an isomorphism.)

### Homeomorphism

A homeomorphism, also called a continuous transformation, is an equivalence relation and one-to-one correspondence between points in two geometric figures or topological spaces that is continuous in both directions. A homeomorphism which also preserves distances is called an isometry. Affine transformations are another type of common geometric homeomorphism.The similarity in meaning and form of the words "homomorphism"and "homeomorphism" is unfortunate and a common source of confusion.

### Congruent

There are at least two meanings on the word congruent in mathematics. Two geometric figures are said to be congruent if one can be transformed into the other by an isometry (Coxeter and Greitzer 1967, p. 80). This relationship, called geometric congruence, is written . (Unfortunately, the symbol is also used to denote an isomorphism.)A number is said to be congruent to modulo if ( divides ).

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