Homothetic center

In geometry, a homothetic center (also called a center of similarity or a center of similitude) is a point from which at least two geometrically similar figures can be seen as a dilation/contraction of one another. If the center is "external", the two figures are directly similar to one another; their angles have the same rotational sense. If the center is "internal", the two figures are scaled mirror images of one another; their angles have the opposite sense.

General polygons

If two geometric figures possess a homothetic center, they are similar to one another; in other words, they must have the same angles at corresponding points and differ only in their relative scaling. The homothetic center and the two figures need not lie in the same plane; they can be related by a projection from the homothetic center.

Homothetic centers may be external or internal. If the center is internal, the two geometric figures are scaled mirror images of one another; in technical language, they have opposite chirality. A clockwise angle in one figure would correspond to a counterclockwise angle in the other. Conversely, if the center is external, the two figures are directly similar to one another; their angles have the same sense.

Circles

Circles are geometrically similar to one another and mirror symmetric. Hence, any pair of circles has both types of homothetic centers, internal and external; their two homothetic centers lie on the line joining the centers of the two given circles, which is called the "line of centers" (Figure 3).

For a given pair of circles, the internal and external homothetic centers may be found as follows. Two radii are drawn in the two circles such that they make the same angle with the line joining their centers (the angle α in Figure 3). A line drawn through corresponding endpoints of those radii (e.g., the points A₁ and A₂ in Figure 3) intersects the line of centers at the "external" homothetic center. Conversely, a line drawn through one endpoint and the diametric opposite endpoint of its counterpart (e.g., the points A₁ and B₂ in Figure 3) intersects the line of centers at the "internal" homothetic center.

As a limiting case of this construction, a line tangent to both circles, where the circles fall on opposite sides, passes through the internal homothetic center, as illustrated by the line joining the points B₁ and A₂ in Figure 3. Conversely, a line tangent to both given circle, where the circles fall on the same side, passes through the external homothetic center. See also Tangent lines to circles.

Homologous and antihomologous points

In general, a ray emanating from a homothetic center will intersect each of its circles in two places. Of these four points, two are said to be "homologous" if radii drawn to them make the same angle with the line connecting the centers, e.g., the points A₁ and A₂ in Figure 3. Two other pairs of points are said to "antihomologous", e.g., points P and Q in Figure 4; such points make complementary angles with the line of centers.

Pairs of antihomologous points lie on a circle

When two rays from the same homothetic center intersect the circles, the two pairs of homologous points lie on a circle. Consider the ray EPQ in Figure 4; since the triangles EPC and EQD are similar, then the ratios must be equal

:$frac\{d\_\{EP\{d\_\{EC\; =\; frac\{d\_\{ED\{d\_\{EQ$

which may be written

:$d\_\{EP\}\; cdot\; d\_\{EQ\}\; =\; d\_\{EC\}\; cdot\; d\_\{ED\}\; =\; constant$

The same argument can be applied to another ray EST, giving the general law

:$d\_\{EP\}\; cdot\; d\_\{EQ\}\; =\; d\_\{ES\}\; cdot\; d\_\{ET\}$

Since the two rays are arbitrary, this equality of products implies that the points P, Q, S and T all lie on the same circle, by the secant power theorem.

Relation with the radical axis

Two circles have a radical axis, which is the line of points from which tangents to both circles have equal length. More generally, every point on the radical axis has the property that its powers relative to the circles are equal. The radical axis is always perpendicular to the line of centers, and if two circles intersect, their radical axis is the line joining their points of intersection. For three circles, three radical axes can be defined, one for each pair of circles ("C"₁/"C"₂, "C"₁/"C"₃, and "C"₂/"C"₃); remarkably, these three radical axes intersect at a single point, the radical center. Tangents drawn from the radical to the three circles would all have equal length.

Any two pairs of antihomologous points can be used to find a point on the radical axis. Consider the two rays emanating from the external homothetic center E in Figure 4. These rays intersect the two given circles (green and blue in Figure 4) in two pairs of antihomologous points, P and Q for the first ray, and S and T for the second ray. These four points lie on a single circle, that intersects both given circles. By definition, the line "PS" is the radical axis of the new circle with the green given circle, whereas the line "QT" is the radical axis of the new circle with the blue given circle. These two lines intersect at the point G, which is the radical center of the new circle and the two given circles. Therefore, the point G also lies on the radical axis of the two given circles.

As a limiting case of this construction, when the two rays become very close, tangent lines at two antihomologous points on the circles intersect on the radical axis. This property is exploited in Joseph Diaz Gergonne's general solution to Apollonius' problem.

ee also

* Similarity (geometry)
* Homothetic transformation
* Radical axis, radical center

Bibliography

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