Similarity

1. Similarity

A Similarity is a transformation F of the plane onto itself defined by a tripple (O, ω, k), consisting from a point Ī, an
angle ω and a (positive) number k.Point O is the center of similarity and angle ω is the rotation angle of the
similarity. Number k is the ratio of the similarity. The standard reference here is Yaglom's book [Yaglom, vol. II] which
starts with an excellent review entitled "What is Geometry?".

For every point P different from O point F(P) = P' is defined by rotating P by the angle ω to point P₀ and then taking
P' on OP₀, such that OP'/OP₀ = k.Thus, by definition, the similarity is a composition of two transformations: the rotation
R_w and the homothety H_k.

It is easily shown that the order of factors in the composition is irrelevant.

2. Usual definition

A similarity transformation F is defined by giving two points {A,B} and prescribing their images
{A'=F(A),B'=F(B)}. The center of similarity is the intersection point O of the circles through the tripples
{A,A',I}, {B,B',I}, where I is the intersection point of the carrying lines of AB and A'B'. The ratio of similarity is
the ratio of the lengths r = |A'B'|/|AB| and the rotation-angle of the similarity is the angle ω of the carrying
lines.

When the carrying lines are parallel point I goes to infinity, the circles become straight lines through {A,A'},
{B,B'} correspondingly, and the similarity becomes a homothety with center at the intersection point of these
lines. If segments {AA', BB'} are parallel and equal then O is at infinity and the transformation mapping
AB to A'B' becomes a translation.

Remark A point X with given relative position with respect to AB (i.e. given k=XA/XB) maps by the
similarity to point X' with the same relative position with respect to A'B' (i.e. X'A'/X'B' is also k). This has
the consequence that all circles {X,X',I} pass through O. There is also another point of view for this, discussed
in Thales_General.html .

3. A prominent example

One of the most prominent examples in applying similarities, I think, is the following, in which a triangle circumscribes
another remaining all the time similar to itself. To create the example take an arbitrary triangle t₀ = DEF and draw on
its sides circles the points of which  (the outer arcs) view the sides under angles {φ, χ, ψ} with φ + χ + ψ = π.
Then take an arbitrary point A on the φ-arc and extend lines AE, AF until they cut again the corresponding circles at C, B.

The following facts follow directly from the definitions:
[1] Line BC passes from D and triangle t = ABC has angles {φ, χ, ψ}.
[2] The circles carrying A,B,C intersect at a common point O, characterized by the property to view the
      sides of DEF under the complementary angles {π-φ, π-χ, π-ψ}.
[3] The center of similarity of two such triangles ABC and A'B'C' is point O.
[4] O has the same relative position with respect to all triangles ABC. This means that drawing the vertical
      distances to the sides {OX, OY, OZ} the ratios between any pairs of them (OX : OY : OZ)  are the same for all
      triangles ABC.  This because for two triangles ABC , A'B'C' the corresponding  OX, OX' result by the same
      similarity mapping ABC to A'B'C', hence their angle is the similarity angle  and the ratio OX'/OX = r, where r is
      the similarity ratio.

[5] Inversely, points U having the same relative position describe a circle passing through O. This can be
      proved first for points U on the sides of ABC, by applying the above remark. Then it can be proved
      for arbitrary points by considering the intersection points of OU with the sides and using again the
      remark.
[6] The previous remark implies that lines L having fixed relative position with respect to ABC pass
      through a point O*, the same for all these L (but changing if the relative position of L changes). In
      fact, the fixed relative position of L is guaranteed by requiring that the projection O_L of O on
      L is a point with fixed relative position with respect to the varying ABC. Then O_L describes a circle
      through O and O* is the diametric of O on that circle. (The argument could be reversed, first proving
      this and then [5]).

Some further points on this subject are discussed in the files SimilarlyGliding.html , Similarly_Rotating.html ,
SimilarInscribed_0.html and SimilarInscribed.html and Pivot.html . The set of all similarities is not a group.
In order to become a group it has to be complemented with the set of translations. The reason is exposed in
Homotheties_Composition.html . Homotheties are similarities without rotation-part i.e. with w=0.

4. Pivot of rotation of a triangle into another triangle

O is called a pivot of rotation of a triangle similar to ABC about the triangle DEF. The trilinears of O (see
Trilinears.html ) relative to either of the two triangles (DEF and ABC) are uniquely determined by the angles of
the two triangles and the choice of which is opposite to which. Since we can pair any one of the angles {d,e,f}
of DEF with some of the angles {φ, χ, ψ} we come at six possible pivots. By allowing the vertices of
ABC to view the sides of DEF also under the complementary angles correspondingly {π-φ,π-χ,π-ψ} (see
image below) we come at twelve pivots relating two triangles with given angles. But this is another story (see
SixPivots.html ) handled usually from the (dual in some sense) point of view of turning DEF around its pivot in
ABC, so that it remains similar to itself all the time.

After this short digression on pivots, I come to my subject to state the results in the form of a theorem.

Theorem If a triangle ABC changes position but remains similar to itself and its side-lines pass
trhough three fixed points {D,E,F} then:
i) All similarities interchanging two such positions of ABC have the same center O.
ii) All points fixed relative to ABC describe circles through O.
iii) All lines L fixed relative to ABC pass through some point O_L, which is the diametric of O with
respect to the circle described by the foot of the perpendigular to L from O.

Bibliography

[Yaglom] Yaglom, I. M. Geometric Transformations (3 vols.) Washington DC, Math. Assoc. Ammer., 1962

Return to Gallery

http://users.math.uoc.gr/~pamfilos/

1. Similarity

2. Usual definition

3. A prominent example

4. Pivot of rotation of a triangle into another triangle

See Also

Bibliography