mathematics

Japanese theorem for cyclic polygons

Japanese theorem for cyclic polygons 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 Japanese theorem for cyclic polygons rather than just read about it. In short: In geometry, the Japanese theorem states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant. Conversely, if the sum of inradii is independent of the triangulation, then the polygon is cyclic.

Key takeaways

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

Reference excerpt

In geometry, the Japanese theorem states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant. Conversely, if the sum of inradii is independent of the triangulation, then the polygon is cyclic. The Japanese theorem follows from Carnot's theorem; it is a Sangaku problem.

Proof This theorem can be proven by first proving a special case: no matter how one triangulates a cyclic quadrilateral, the sum of inradii of triangles is constant. After proving the quadrilateral case, the general case of the cyclic polygon theorem is an immediate corollary. The quadrilateral rule can be applied to quadrilateral components of a general partition of a cyclic polygon, and repeated application of the rule, which "flips" one diagonal, will generate all the possible partitions from any given partition, with each "flip" preserving the sum of the inradii. The quadrilateral case follows from a simple extension of the Japanese theorem for cyclic quadrilaterals, which shows that a rectangle is formed by the two pairs of incenters corresponding to the two possible triangulations of the quadrilateral. The steps of this theorem require nothing beyond basic constructive Euclidean geometry. With the additional construction of a parallelogram having sides parallel to the diagonals, and tangent to the corners of the rectangle of incenters, the quadrilateral case of the cyclic polygon theorem can be proved in a few steps. The equality of the sums of the radii of the two pairs is equivalent to the condition that the constructed parallelogram be a rhombus, and this is easily shown in the construction. Another proof of the quadrilateral case is available due to Wilfred Reyes (2002). In the proof, both the Japanese theorem for cyclic quadrilaterals and the quadrilateral case of the cyclic polygon theorem are proven as a consequence of Thébault's problem III.

See also Carnot's theorem, which is used in a proof of the theorem above Equal incircles theorem Tangent lines to circles

Notes

References Claudi Alsina, Roger B. Nelsen: Icons of Mathematics: An Exploration of Twenty Key Images. MAA, 2011, ISBN 9780883853528, pp. 121-125 Wilfred Reyes: An Application of Thebault’s Theorem Archived 2018-10-24 at the Wayback Machine. Forum Geometricorum, Volume 2, 2002, pp. 183–185

External links Mangho Ahuja, Wataru Uegaki, Kayo Matsushita: In Search of the Japanese Theorem Japanese theorem at Mathworld Japanese Theorem interactive demonstration at the C.a.R. website Wataru Uegaki: "Japanese Theoremの起源と歴史" (On the Origin and History of the Japanese Theorem) http://hdl.handle.net/10076/4917

Illustrations

Worked examples

Example 1 — a first encounter with Japanese theorem for cyclic polygons

Start with the simplest possible case. Write down what Japanese theorem for cyclic polygons 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 Japanese theorem for cyclic polygons 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 Japanese theorem for cyclic polygons 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 Japanese theorem for cyclic polygons

In research: Japanese theorem for cyclic polygons 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 Japanese theorem for cyclic polygons 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: Japanese theorem for cyclic polygons is common in secondary-school and first-year university syllabi. It links to neighbouring topics Euclidean plane geometry, Japanese mathematics, Theorems about triangles and circles, so understanding it makes those chapters shorter.
In everyday life: Look for Japanese theorem for cyclic polygons 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 Japanese theorem for cyclic polygons in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Japanese theorem for cyclic polygons 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 Japanese theorem for cyclic polygons out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Japanese theorem for cyclic polygons in simple terms?

Why does Japanese theorem for cyclic polygons 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 Japanese theorem for cyclic polygons?

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 Japanese theorem for cyclic polygons.