mathematics

Reuschle's theorem

Reuschle's theorem 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 Reuschle's theorem rather than just read about it. In short: In elementary geometry, Reuschle's theorem describes a property of the cevians of a triangle intersecting in a common point and is named after the German mathematician Karl Gustav Reuschle (1812–1875). It is also known as Terquem's theorem after the French mathematician Olry Terquem (1782–1862), who published it in 1842.

Key takeaways

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

Reference excerpt

In elementary geometry, Reuschle's theorem describes a property of the cevians of a triangle intersecting in a common point and is named after the German mathematician Karl Gustav Reuschle (1812–1875). It is also known as Terquem's theorem after the French mathematician Olry Terquem (1782–1862), who published it in 1842. In a triangle A B C {\displaystyle ABC} with its three cevians intersecting in a common point other than the vertices A {\displaystyle A} , B {\displaystyle B} or C {\displaystyle C} let P a {\displaystyle P_{a}} , P b {\displaystyle P_{b}} and P c {\displaystyle P_{c}} denote the intersections of the (extended) triangle sides and the cevians. The circle defined by the three points P a {\displaystyle P_{a}} , P b {\displaystyle P_{b}} and P c {\displaystyle P_{c}} intersects the (extended) triangle sides in the (additional) points P a ′ {\displaystyle P'_{a}} , P b ′ {\displaystyle P'_{b}} and P c ′ {\displaystyle P'_{c}} . Reuschle's theorem now states that the three new cevians A P a ′ {\displaystyle AP'_{a}} , B P b ′ {\displaystyle BP'_{b}} and C P c ′ {\displaystyle CP'_{c}} intersect in a common point as well.

References Friedrich Riecke (ed.): Mathematische Unterhaltungen. Volume I, Stuttgart 1867, (reprint Wiesbaden 1973), ISBN 3-500-26010-1, p. 125 (German) M. D. Fox, J. R. Goggins: "Cevian Axes and Related Curves." The Mathematical Gazette, volume 91, no. 520, 2007, pp. 3-4 (JSTOR).

External links

Terquem's theorem at cut-the-knot.org Weisstein, Eric W. "Cyclocevian Conjugate". MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Reuschle's theorem

Start with the simplest possible case. Write down what Reuschle's theorem 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 Reuschle's theorem 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 Reuschle's theorem 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 Reuschle's theorem

In research: Reuschle's theorem 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 Reuschle's theorem 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: Reuschle's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elementary geometry, Elementary geometry stubs, Theorems about triangles and circles, so understanding it makes those chapters shorter.
In everyday life: Look for Reuschle's theorem 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 Reuschle's theorem in 20 minutes

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

Frequently asked questions

What is Reuschle's theorem in simple terms?

Why does Reuschle's theorem 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 Reuschle's theorem?

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 Reuschle's theorem.