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Nine-point conic

Nine-point conic 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 Nine-point conic rather than just read about it. In short: In geometry, the nine-point conic of a complete quadrangle is a conic that passes through the three diagonal points and the six midpoints of sides of the complete quadrangle. The nine-point conic was described by Maxime Bôcher in 1892.

Nine-point conic — main illustration
Nine-point conic — illustration

Key takeaways

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

Reference excerpt

In geometry, the nine-point conic of a complete quadrangle is a conic that passes through the three diagonal points and the six midpoints of sides of the complete quadrangle. The nine-point conic was described by Maxime Bôcher in 1892. The better-known nine-point circle is an instance of Bôcher's conic. The nine-point hyperbola is another instance. Bôcher used the four points of the complete quadrangle as three vertices of a triangle with one independent point:

Given a triangle △ABC and a point P in its plane, a conic can be drawn through the following nine points: the midpoints of the sides of △ABC, the midpoints of the lines joining P to the vertices, and the points where these last named lines cut the sides of the triangle. The conic is an ellipse if P lies in the interior of △ABC or in one of the regions of the plane separated from the interior by two sides of the triangle, otherwise the conic is a hyperbola. Bôcher notes that when P is the orthocenter, one obtains the nine-point circle, and when P is on the circumcircle of △ABC, then the conic is an equilateral hyperbola. In 1912 Maud Minthorn showed that the nine-point conic is the locus of the center of a conic through four given points. The nine-point conic with respect to a line l is the conic through the six harmonic conjugates of the intersection of the sides of the complete quadrangle with l.

References

Fanny Gates (1894) Some Considerations on the Nine-point Conic and its Reciprocal, Annals of Mathematics 8(6):185–8, link from Jstor. Eric W. Weisstein Nine-point conic from MathWorld. Michael DeVilliers (2006) The nine-point conic: a rediscovery and proof by computer from International Journal of Mathematical Education in Science and Technology, a Taylor & Francis publication. Christopher Bradley The Nine-point Conic and a Pair of Parallel Lines Archived 2016-03-04 at the Wayback Machine from University of Bath.

Further reading W. G. Fraser (1906) "On relations of certain conics to a triangle", Proceedings of the Edinburgh Mathematical Society 25:38–41. Thomas F. Hogate (1894) On the Cone of Second Order which is Analogous to the Nine-point Conic, Annals of Mathematics 7:73–6. P. Pinkerton (1905) "On a nine-point conic, etc.", Proceedings of the Edinburgh Mathematical Society 24:31–3.

External links Nine-point conic and Euler line generalization at Dynamic Geometry Sketches

Illustrations

Nine-point conic: .mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}  Four constituent points of the quadrangle (A, B, C, P)   Six constituent lines of the quadrangle   Nine-point conic (a nine-point hyperbola, since P is across side AC) If P were inside triangle △ABC, the nine-point conic would be an ellipse.
.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}  Four constituent points of the quadrangle (A, B, C, P)   Six constituent lines of the quadrangle   Nine-point conic (a nine-point hyperbola, since P is across side AC) If P were inside triangle △ABC, the nine-point conic would be an ellipse.

Worked examples

Example 1 — a first encounter with Nine-point conic

Start with the simplest possible case. Write down what Nine-point conic 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 Nine-point conic 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 Nine-point conic 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 Nine-point conic

In research
Nine-point conic 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 Nine-point conic 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
Nine-point conic is common in secondary-school and first-year university syllabi. It links to neighbouring topics Euclidean plane geometry, Projective geometry, so understanding it makes those chapters shorter.
In everyday life
Look for Nine-point conic 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 Nine-point conic in 20 minutes

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

Frequently asked questions

What is Nine-point conic in simple terms?

In geometry, the nine-point conic of a complete quadrangle is a conic that passes through the three diagonal points and the six midpoints of sides of the complete quadrangle. The nine-point conic was described by Maxime Bôcher in 1892.

Why does Nine-point conic 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 Nine-point conic?

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 Nine-point conic.

Tags

  • Euclidean plane geometry
  • Projective geometry

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