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Homothetic center

Homothetic center is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Homothetic center rather than just read about it. In short: 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 or 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.

Homothetic center — main illustration
Homothetic center — illustration

Key takeaways

  • Homothetic center belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Homothetic center to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Homothetic center from memory before moving on to harder problems.

Reference excerpt

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 or 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, a pair of circles has both types of homothetic centers, internal and external, unless the centers are equal or the radii are equal; these exceptional cases are treated after general position. These two homothetic centers lie on the line joining the centers of the two given circles, which is called the line of centers (Figure 3). Circles with radius zero can also be included (see exceptional cases), and negative radius can also be used, switching external and internal.

Computing homothetic centers

For a given pair of circles, the internal and external homothetic centers may be found in various ways. In analytic geometry, the internal homothetic center is the weighted average of the centers of the circles, weighted by the opposite circle's radius – distance from center of circle to inner center is proportional to that radius, so weighting is proportional to the opposite radius. Denoting the centers of the circles C1, C2 by (x1, y1), (x2, y2) and their radii by r1, r2 and denoting the center by (x0, y0), this is:

( x 0 , y 0 ) = r 2 r 1 + r 2 ( x 1 , y 1 ) + r 1 r 1 + r 2 ( x 2 , y 2 ) . {\displaystyle (x_{0},y_{0})={\frac {r_{2}}{r_{1}+r_{2}}}(x_{1},y_{1})+{\frac {r_{1}}{r_{1}+r_{2}}}(x_{2},y_{2}).}

The external center can be computed by the same equation, but considering one of the radii as negative; either one yields the same equation, which is:

( x e , y e ) = − r 2 r 1 − r 2 ( x 1 , y 1 ) + r 1 r 1 − r 2 ( x 2 , y 2 ) . {\displaystyle (x_{e},y_{e})={\frac {-r_{2}}{r_{1}-r_{2}}}(x_{1},y_{1})+{\frac {r_{1}}{r_{1}-r_{2}}}(x_{2},y_{2}).}

… excerpt ends here. Continue reading the full article.

Illustrations

Homothetic center: Figure 1: The point O is an external homothetic center for the two triangles. The size of each figure is proportional to its distance from the homothetic center.
Figure 1: The point O is an external homothetic center for the two triangles. The size of each figure is proportional to its distance from the homothetic center.
Homothetic center: Figure 2: Two geometric figures related by an external homothetic center S. The angles at corresponding points are the same and have the same sense; for example, the angles ∠ABC, ∠A'B'C' are both clockwise and equal in magnitude.
Figure 2: Two geometric figures related by an external homothetic center S. The angles at corresponding points are the same and have the same sense; for example, the angles ∠ABC, ∠A'B'C' are both clockwise and equal in magnitude.
Homothetic center: The external (above) and internal (below) homothetic centers of the two circles (red) are shown as black points.
The external (above) and internal (below) homothetic centers of the two circles (red) are shown as black points.
Homothetic center: Figure 3: Two circles have both types of homothetic centers, internal (I) and external (E). The radii of the circles (r1, r2) are proportional to the distance (d) from each homothetic center. The points A1, A2 are homologous, as are the points B1, B2.
Figure 3: Two circles have both types of homothetic centers, internal (I) and external (E). The radii of the circles (r1, r2) are proportional to the distance (d) from each homothetic center. The points A1, A2 are homologous, as are the points B1, B2.
Homothetic center: Figure 4: Lines through corresponding antihomologous points intersect on the radical axis of the two given circles (green and blue). The points Q, P' are antihomologous, as are S, R'. These four points lie on a circle that intersects the two given circles; the lines through the intersection points of the new circle with the two given circles must intersect at the radical center G of the three circles, which lies on the radical axis of the two given circles.
Figure 4: Lines through corresponding antihomologous points intersect on the radical axis of the two given circles (green and blue). The points Q, P' are antihomologous, as are S, R'. These four points lie on a circle that intersects the two given circles; the lines through the intersection points of the new circle with the two given circles must intersect at the radical center G of the three circles, which lies on the radical axis of the two given circles.

Worked examples

Example 1 — a first encounter with Homothetic center

Start with the simplest possible case. Write down what Homothetic center claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Homothetic center before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Homothetic center ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Homothetic center

In research
Homothetic center appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Homothetic center in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Homothetic center is common in secondary-school and first-year university syllabi. It links to neighbouring topics Circles, Euclidean geometry, Geometric centers, so understanding it makes those chapters shorter.
In everyday life
Look for Homothetic center outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Homothetic center in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Homothetic center means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Homothetic center out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Homothetic center in simple terms?

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 or contraction of one another. If the center is external, the two figures are directly similar to one another; t…

Why does Homothetic center matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Homothetic center?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Homothetic center.

Tags

  • Circles
  • Euclidean geometry
  • Geometric centers

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