Twin circles

Key takeaways

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

Reference excerpt

In geometry, the twin circles are two special circles associated with an arbelos. An arbelos is determined by three collinear points A, B, and C, and is the curvilinear triangular region between the three semicircles that have AB, BC, and AC as their diameters. If the arbelos is partitioned into two smaller regions by a line segment through the middle point of A, B, and C, perpendicular to line ABC, then each of the two twin circles lies within one of these two regions, tangent to its two semicircular sides and to the splitting segment. These circles first appeared in the Book of Lemmas, which showed (Proposition V) that the two circles are congruent. Thābit ibn Qurra, who translated this book into Arabic, attributed it to Greek mathematician Archimedes. Based on this claim the twin circles, and several other circles in the Arbelos congruent to them, have also been called Archimedes's circles. However, this attribution has been questioned by later scholarship.

Construction

Specifically, let A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} be the three corners of the arbelos, with B {\displaystyle B} between A {\displaystyle A} and C {\displaystyle C} . Let D {\displaystyle D} be the point where the larger semicircle intercepts the line perpendicular to the A C {\displaystyle AC} through the point B {\displaystyle B} . The segment B D {\displaystyle BD} divides the arbelos in two parts. The twin circles are the two circles inscribed in these parts, each tangent to one of the two smaller semicircles, to the segment B D {\displaystyle BD} , and to the largest semicircle. Each of the two circles is uniquely determined by its three tangencies. Constructing it is a special case of the Problem of Apollonius. Alternative approaches to constructing two circles congruent to the twin circles have also been found. These circles have also been called Archimedean circles. They include the Bankoff circle, Schoch circles, and Woo circles.

Properties Let a and b be the diameters of two inner semicircles, so that the outer semicircle has diameter a + b. The diameter of each twin circle is then

d = a b a + b . {\displaystyle d={\frac {ab}{a+b}}.}

Alternatively, if the outer semicircle has unit diameter, and the inner circles have diameters s {\displaystyle s} and 1 − s {\displaystyle 1-s} , the diameter of each twin circle is

d = s ( 1 − s ) . {\displaystyle d=s(1-s).\,}

The smallest circle that encloses both twin circles has the same area as the arbelos.

Illustrations

Worked examples

Example 1 — a first encounter with Twin circles

Start with the simplest possible case. Write down what Twin circles 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 Twin circles 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 Twin circles 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 Twin circles

In research: Twin circles 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 Twin circles 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: Twin circles is common in secondary-school and first-year university syllabi. It links to neighbouring topics Ancient Greek mathematics, Arbelos, Archimedes, so understanding it makes those chapters shorter.
In everyday life: Look for Twin circles 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.

How to study Twin circles in 20 minutes

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

Frequently asked questions

What is Twin circles in simple terms?

In geometry, the twin circles are two special circles associated with an arbelos. An arbelos is determined by three collinear points A, B, and C, and is the curvilinear triangular region between the three semicircles that have AB, BC, and AC as their diameters.

Why does Twin circles 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 Twin circles?

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 Twin circles.