Triangle conic

Key takeaways

Triangle 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 Triangle conic to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Triangle conic from memory before moving on to harder problems.

Reference excerpt

In Euclidean geometry, a triangle conic is a conic in the plane of the reference triangle and associated with it in some way. For example, the circumcircle and the incircle of the reference triangle are triangle conics. Other examples are the Steiner ellipse, which is an ellipse passing through the vertices and having its centre at the centroid of the reference triangle; the Kiepert hyperbola which is a conic passing through the vertices, the centroid and the orthocentre of the reference triangle; and the Artzt parabolas, which are parabolas touching two sidelines of the reference triangle at vertices of the triangle. The terminology of triangle conic is widely used in the literature without a formal definition; that is, without precisely formulating the relations a conic should have with the reference triangle so as to qualify it to be called a triangle conic (see ). However, Greek mathematician Paris Pamfilos defines a triangle conic as a "conic circumscribing a triangle △ABC (that is, passing through its vertices) or inscribed in a triangle (that is, tangent to its side-lines)". The terminology triangle circle (respectively, ellipse, hyperbola, parabola) is used to denote a circle (respectively, ellipse, hyperbola, parabola) associated with the reference triangle is some way. Even though several triangle conics have been studied individually, there is no comprehensive encyclopedia or catalogue of triangle conics similar to Clark Kimberling's Encyclopedia of Triangle Centres or Bernard Gibert's Catalogue of Triangle Cubics.

Equations of triangle conics in trilinear coordinates The equation of a general triangle conic in trilinear coordinates x : y : z has the form

r x 2 + s y 2 + t z 2 + 2 u y z + 2 v z x + 2 w x y = 0. {\displaystyle rx^{2}+sy^{2}+tz^{2}+2uyz+2vzx+2wxy=0.}

The equations of triangle circumconics and inconics have respectively the forms

u y z + v z x + w x y = 0 l 2 x 2 + m 2 y 2 + n 2 z 2 − 2 m n y z − 2 n l z x − 2 l m x y = 0 {\displaystyle {\begin{aligned}&uyz+vzx+wxy=0\\[2pt]&l^{2}x^{2}+m^{2}y^{2}+n^{2}z^{2}-2mnyz-2nlzx-2lmxy=0\end{aligned}}}

Perspector and dual conics The perspector of a circumconic or inconic is the perspector of the reference triangle and its polar triangle with respect to the conic.

A circumconic is the locus of trilinear poles of lines through its perspector. Conversely, the perspector of a circumconic lies on the trilinear polar of any point on the conic other than the triangle vertices. The perspector of an inconic is its Brianchon point. A circumconic and an inconic are said to be dual if, using barycentric coordinates, coordinates of any point on the circumconic yield coefficients of an equation of a tangent to the inconic.

Pairs of dual conics include the Steiner ellipse and inellipse, and the Kiepert hyperbola and parabola. Perspectors of dual conics are isotomic conjugates. The dual circumconic of an inconic is the isotomic conjugate of the trilinear polar of its perspector. Note: Paris Pamfilos describes a different notion of dual conics by the property of sharing the same perspector. This notion also includes the Steiner ellipse and inellipse. Not all conics associated with a triangle are circumconics or inconics; for instance, the Artzt parabolas each only touch two vertices.

Special triangle conics In the following, a few typical special triangle conics are discussed. In the descriptions, the standard notations are used: the reference triangle is always denoted by △ABC. The angles at the vertices A, B, C are denoted by A, B, C and the lengths of the sides opposite to the vertices A, B, C are respectively a, b, c. The equations of the conics are given in the trilinear coordinates x : y : z. The conics are selected as illustrative of the several different ways in which a conic could be associated with a triangle.

Triangle circles

Triangle ellipses

Triangle hyperbolas

Note: The pedal circle of any point on a rectangular circumhyperbola passes through the hyperbola's center. Since all such hyperbolas pass through the orthocenter, their centers all lie on the nine-point circle.

Triangle parabolas

Families of triangle conics

Hofstadter ellipses

An Hofstadter ellipse is a member of a one-parameter family of ellipses in the plane of △ABC defined by the following equation in trilinear coordinates:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Triangle conic

Start with the simplest possible case. Write down what Triangle 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 Triangle 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 Triangle 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 Triangle conic

In research: Triangle 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 Triangle 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: Triangle conic is common in secondary-school and first-year university syllabi. It links to neighbouring topics Triangle geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Triangle 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.

How to study Triangle conic in 20 minutes

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

Frequently asked questions

What is Triangle conic in simple terms?

In Euclidean geometry, a triangle conic is a conic in the plane of the reference triangle and associated with it in some way. For example, the circumcircle and the incircle of the reference triangle are triangle conics.

Why does Triangle 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 Triangle 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 Triangle conic.