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One-seventh area triangle

One-seventh area triangle 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 One-seventh area triangle rather than just read about it. In short: In plane geometry, a triangle ABC contains a triangle having one-seventh of the area of ABC, which is formed as follows: the sides of this triangle lie on cevians p, q, r where p connects A to a point on BC that is one-third the distance from B to C, q connects B to a point on CA that is one-third the distance from C to A, r connects C to a point on AB that is one-third the distance from A to B. The proof of the exi…

One-seventh area triangle — main illustration
One-seventh area triangle — illustration

Key takeaways

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

Reference excerpt

In plane geometry, a triangle ABC contains a triangle having one-seventh of the area of ABC, which is formed as follows: the sides of this triangle lie on cevians p, q, r where

p connects A to a point on BC that is one-third the distance from B to C, q connects B to a point on CA that is one-third the distance from C to A, r connects C to a point on AB that is one-third the distance from A to B. The proof of the existence of the one-seventh area triangle follows from the construction of six parallel lines:

two parallel to p, one through C, the other through q.r two parallel to q, one through A, the other through r.p two parallel to r, one through B, the other through p.q. The suggestion of Hugo Steinhaus is that the (central) triangle with sides p,q,r be reflected in its sides and vertices. These six extra triangles partially cover ABC, and leave six overhanging extra triangles lying outside ABC. Focusing on the parallelism of the full construction (offered by Martin Gardner through James Randi’s on-line magazine), the pair-wise congruences of overhanging and missing pieces of ABC is evident. As seen in the graphical solution, six plus the original equals the whole triangle ABC.

An early exhibit of this geometrical construction and area computation was given by Robert Potts in 1859 in his Euclidean geometry textbook. According to Cook and Wood (2004), this triangle puzzled Richard Feynman in a dinner conversation; they go on to give four different proofs. A more general result is known as Routh's theorem. Also see Marion Walter’s theorem.

References

H. S. M. Coxeter (1969) Introduction to Geometry, page 211, John Wiley & Sons.

Illustrations

One-seventh area triangle: The area of the pink triangle is one-seventh of the area of the large triangle ABC.
The area of the pink triangle is one-seventh of the area of the large triangle ABC.
One-seventh area triangle: Congruence of edge lengths allows rotation of the selected triangles to form three equal-area parallelograms, which bisect into six triangles of equal size to the original interior triangle.
Congruence of edge lengths allows rotation of the selected triangles to form three equal-area parallelograms, which bisect into six triangles of equal size to the original interior triangle.

Worked examples

Example 1 — a first encounter with One-seventh area triangle

Start with the simplest possible case. Write down what One-seventh area triangle 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 One-seventh area triangle 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 One-seventh area triangle 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 One-seventh area triangle

In research
One-seventh area triangle 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 One-seventh area triangle 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
One-seventh area triangle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Affine geometry, Area, Objects defined for a triangle, so understanding it makes those chapters shorter.
In everyday life
Look for One-seventh area triangle 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 One-seventh area triangle in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what One-seventh area triangle 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 One-seventh area triangle out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is One-seventh area triangle in simple terms?

In plane geometry, a triangle ABC contains a triangle having one-seventh of the area of ABC, which is formed as follows: the sides of this triangle lie on cevians p, q, r where p connects A to a point on BC that is one-third the distance from B to C, q connects B to a point on CA that is one-third…

Why does One-seventh area triangle 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 One-seventh area triangle?

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 One-seventh area triangle.

Tags

  • Affine geometry
  • Area
  • Objects defined for a triangle

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