Modern triangle geometry

Emile Lemoine (1840–1912)

In mathematics, modern triangle geometry, or new triangle geometry, is the body of knowledge relating to the properties of a triangle discovered and developed roughly since the beginning of the last quarter of the nineteenth century. Triangles and their properties were the subject of investigation since at least the time of Euclid. In fact, Euclid's Elements contains description of the four special points – centroid, incenter, circumcenter and orthocenter - associated with a triangle. Even though Pascal and Ceva in the seventeenth century, Euler in the eighteenth century and Feuerbach in the nineteenth century and many other mathematicians had made important discoveries regarding the properties of the triangle, it was the publication in 1873 of a paper by Emile Lemoine (1840–1912) with the title "On a remarkable point of the triangle" that was considered to have, according to Nathan Altschiller-Court, "laid the foundations...of the modern geometry of the triangle as a whole." The American Mathematical Monthly, in which much of Lemoine's work is published, declared that "To none of these [geometers] more than Émile-Michel-Hyacinthe Lemoine is due the honor of starting this movement of modern triangle geometry". The publication of this paper caused a remarkable upsurge of interest in investigating the properties of the triangle during the last quarter of the nineteenth century and the early years of the twentieth century. A hundred-page article on triangle geometry in Klein's Encyclopedia of Mathematical Sciences published in 1914 bears witness to this upsurge of interest in triangle geometry.

In the early days, the expression "new triangle geometry" referred to only the set of interesting objects associated with a triangle like the Lemoine point, Lemoine circle, Brocard circle and the Lemoine line. Later the theory of correspondences which was an offshoot of the theory of geometric transformations was developed to give coherence to the various isolated results. With its development, the expression "new triangle geometry" indicated not only the many remarkable objects associated with a triangle but also the methods used to study and classify these objects. Here is a definition of triangle geometry from 1887: "Being given a point M in the plane of the triangle, we can always find, in an infinity of manners, a second point M' that corresponds to the first one according to an imagined geometrical law; these two points have between them geometrical relations whose simplicity depends on the more or less the lucky choice of the law which unites them and each geometrical law gives rise to a method of transformation a mode of conjugation which it remains to study." (See the conference paper titled "Teaching new geometrical methods with an ancient figure in the nineteenth and twentieth centuries: the new triangle geometry in textbooks in Europe and USA (1888–1952)" by Pauline Romera-Lebret presented in 2009.)

However, this escalation of interest soon collapsed and triangle geometry was completely neglected until the closing years of the twentieth century. In his "Development of Mathematics", Eric Temple Bell offers his judgement on the status of modern triangle geometry in 1940 thus: "The geometers of the 20th Century have long since piously removed all these treasures to the museum of geometry where the dust of history quickly dimmed their luster." (The Development of Mathematics, p. 323) Philip Davis has suggested several reasons for the decline of interest in triangle geometry. These include:

• The feeling that the subject is elementary and of low professional status.
• The exhaustion of its methodologic possibilities.
• The visual complexity of the so-called deeper results of the subject.
• The downgrading of the visual in favor of the algebraic.
• A dearth of connections to other fields.
• Competition with other topics with a strong visual content like tessellations, fractals, graph theory, etc.

A further revival of interest was witnessed with the advent of the modern electronic computer. The triangle geometry has again become an active area of research pursued by a group of dedicated geometers. As epitomizing this revival, one can point out the formulation of the concept of a "triangle centre" and the compilation by Clark Kimberling of an Encyclopedia of Triangle Centers containing a listing of nearly 70,000 triangle centers and their properties and also the compilation of a catalogue of triangle cubics with detailed descriptions of several properties of more than 1200 triangle cubics. The open access journal Forum Geometricorum founded by Paul Yiu of Florida Atlantic University in 2001 also provided a tremendous impetus in furthering this new found enthusiasm for triangle geometry. Unfortunately, since 2019, the journal is not accepting submissions although back issues are still available online.

The Lemoine geometry

Lemoine (symmedian) point Template:Mvar of triangle Template:Math. Template:Legend-line Template:Legend-line Template:Legend-line

Lemoine axis of triangle Template:Math. Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line

Lemoine point

For a given triangle Template:Math with centroid Template:Mvar, the symmedian through the vertex is the reflection of the line Template:Mvar in the bisector of the angle Template:Mvar. There are three symmedians for a triangle one passing through each vertex. The three symmedians are concurrent and the point of concurrency, commonly denoted by Template:Mvar, is called the Lemoine point or the symmedian point or the Grebe point of Template:Math. If the sidelengths of Template:Math are Template:Mvar, Template:Mvar, Template:Mvar the barycentric coordinates of the Lemoine point are Template:Math. It has been described as "one of the crown jewels of modern geometry". There are several earlier references to this point in the mathematical literature, details of which are available in John Mackay's history of the symmedian point.

In fact, the concurrency of the symmedians is a special case of a more general result: For any point Template:Mvar in the plane of Template:Math, the isogonals of the lines Template:Mvar, Template:Mvar, Template:Mvar are concurrent, the isogonal of Template:Mvar (respectively Template:Mvar, Template:Mvar) being the reflection of the line Template:Mvar in the bisector of the angle Template:Mvar (respectively Template:Mvar, Template:Mvar). The point of concurrency is called the isogonal conjugate of Template:Mvar. In this terminology, the Lemoine point is the isogonal conjugate of the centroid.

Lemoine circles

The points of intersections of the lines through the Lemoine point of a triangle Template:Math parallel to the sides of the triangle lie on a circle called the first Lemoine circle of Template:Math. The center of the first Lemoine circle lies midway between the circumcenter and the Lemoine point of the triangle.

The points of intersections of the antiparallels to the sides of Template:Math through the Lemoine point of a Template:Math lie on a circle called the second Lemoine circle or the cosine circle of Template:Math. The name "cosine circle" is due to the property of the second Lemoine circle that the lengths of the segments intercepted by the circle on the sides of the triangle proportional to the cosines of the angles opposite to the sides. The center of the second Lemoine circle is the Lemoine point.

Lemoine axis

Any triangle Template:Math and its tangential triangle are in perspective and the axis of perspectivity is called the Lemoine axis of Template:Math. It is the trilinear polar of the symmedian point of Template:Math and also the polar of Template:Mvar with regard to the circumcircle of Template:Math.

Early modern triangle geometry

A quick glance into the world of modern triangle geometry as it existed during the peak of interest in triangle geometry subsequent to the publication of Lemoine's paper is presented below. This presentation is largely based on the topics discussed in William Gallatly's book published in 1910 and Roger A. Johnsons' book first published in 1929.

Poristic triangles

Two triangles are said to be poristic triangles if they have the same incircle and circumcircle. Given a circle with center Template:Mvar and radius Template:Mvar and another circle with center Template:Mvar and radius Template:Mvar, there are an infinite number of triangles Template:Math with circle Template:Math as circumcircle and Template:Math as incircle if and only if $${\overline{OI}}^2=R^2-2Rr.$$ These triangles form a poristic system of triangles. The loci of certain special points like the centroid as the reference triangle traces the different triangles poristic with it turn out to often be circles and points.

The Simson line

For any point Template:Mvar on the circumcircle of triangle Template:Math, the feet of perpendiculars from Template:Mvar to the sides of Template:Math are collinear and the line of collinearity is the well-known Simson line of Template:Mvar.

Pedal and antipedal triangles

Given a point Template:Mvar, let the feet of perpendiculars from Template:Mvar to the sides of the triangle Template:Math be Template:Mvar, Template:Mvar, Template:Mvar. The triangle Template:Math is called the pedal triangle of P. The antipedal triangle of Template:Mvar is the triangle formed by the lines through Template:Mvar, Template:Mvar, Template:Mvar perpendicular to Template:Mvar, Template:Mvar, Template:Mvar respectively. Two points Template:Mvar and Template:Mvar are called counter points if the pedal triangle of Template:Mvar is homothetic to the antipedal triangle of Template:Mvar and the pedal triangle of Template:Mvar is homothetic to the antipedal triangle of Template:Mvar.

The orthopole

Given any line Template:Mvar, let Template:Mvar, Template:Mvar, Template:Mvar be the feet of perpendiculars from the vertices Template:Mvar, Template:Mvar, Template:Mvar of triangle Template:Math to Template:Mvar. The lines through Template:Mvar, Template:Mvar, Template:Mvar perpendicular respectively to the sides Template:Mvar, Template:Mvar, Template:Mvar are concurrent and the point of concurrence is the orthopole of the line Template:Mvar with respect to the Template:Math. In modern triangle geometry, there is a large body of literature dealing with properties of orthopoles.

The Brocard points

Let of circles be described on the sides Template:Mvar, Template:Mvar, Template:Mvar of triangle Template:Math whose external segments contain the two triads of angles Template:Mvar, Template:Mvar, Template:Mvar and Template:Mvar, Template:Mvar, Template:Mvar respectively. Each triad of circles determined by a triad of angles intersect at a common point thus yielding two such points. These points are called the Brocard points of Template:Math and are usually denoted by Template:Math. If Template:Mvar is the first Brocard point (which is the Brocard point determined by the first triad of circles) then the angles Template:Mvar, Template:Mvar and Template:Mvar are equal to each other and the common angle is called the Brocard angle of Template:Math and is commonly denoted by Template:Mvar. The Brocard angle is given by $$\cot \omega =\cot A+\cot B+\cot C.$$ The Brocard points and the Brocard angles have several interesting properties.

Some images

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  Two poristic triangles Template:Math and Template:Math with respect to circles Template:Math and Template:Math
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  Simson line of Template:Mvar
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  Pedal triangle (Template:Math) and antipedal triangle (Template:Math) of Template:Mvar
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  Orthopole of line Template:Mvar
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  First Brocard point of triangle Template:Math

Contemporary modern triangle geometry

Triangle center

One of the most significant ideas that has emerged during the revival of interest in triangle geometry during the closing years of twentieth century is the notion of triangle center. This concept introduced by Clark Kimberling in 1994 unified in one notion the very many special and remarkable points associated with a triangle. Since the introduction of this idea, nearly no discussion on any result associated with a triangle is complete without a discussion on how the result connects with the triangle centers.

Definition of triangle center

A real-valued function Template:Mvar of three real variables Template:Mvar, Template:Mvar, Template:Mvar may have the following properties:

• Homogeneity: $$f(ta,tb,tc)=t^{n}f(a,b,c)$$ for some constant Template:Mvar and for all Template:Math.
• Bisymmetry in the second and third variables: $$f(a,b,c)=f(a,c,b).$$

If a non-zero Template:Mvar has both these properties it is called a triangle center function. If Template:Mvar is a triangle center function and Template:Mvar, Template:Mvar, Template:Mvar are the side-lengths of a reference triangle then the point whose trilinear coordinates are $$f(a,b,c):f(b,c,a):f(c,a,b)$$ is called a triangle center.

American mathematician Clark Kimberling maintains a website devoted to a compendium of triangle centers, the Encyclopedia of Triangle Centers; it contains definitions and descriptions of over 70,000 triangle centers.

Central line

Another unifying notion of contemporary modern triangle geometry is that of a central line. This concept unifies the several special straight lines associated with a triangle. The notion of a central line is also related to the notion of a triangle center.

Definition of central line

Let Template:Math be a plane triangle and let Template:Math be the trilinear coordinates of an arbitrary point in the plane of Template:Math.

A straight line in the plane of triangle Template:Math whose equation in trilinear coordinates has the form $$f(a,b,c)x+g(a,b,c)y+h(a,b,c)z=0$$ where the point with trilinear coordinates $$f(a,b,c):g(a,b,c):h(a,b,c)$$ is a triangle center, is a central line in the plane of Template:Math relative to Template:Math.

Geometrical construction of central line

Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line

Let Template:Mvar be any triangle center of the triangle Template:Math.

• Draw the lines Template:Mvar, Template:Mvar and Template:Mvar and their reflections in the internal bisectors of the angles at the vertices Template:Mvar, Template:Mvar, Template:Mvar respectively.
• The reflected lines are concurrent and the point of concurrence is the isogonal conjugate Template:Mvar of Template:Mvar.
• Let the cevians Template:Mvar, Template:Mvar, Template:Mvar meet the opposite sidelines of Template:Math at Template:Mvar, Template:Mvar, Template:Mvar respectively. The triangle Template:Math is the cevian triangle of Template:Mvar.
• Template:Math and the cevian triangle Template:Math are in perspective and let Template:Mvar be the axis of perspectivity of the two triangles. The line Template:Mvar is the trilinear polar of the point Template:Mvar. The line Template:Mvar is the central line associated with the triangle center Template:Mvar.

Triangle conics

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. Some recently studied triangle conics include Hofstadter ellipses and yff conics. However, there is no formal definition of the terminology of triangle conic in the literature; that is, the relations a conic should have with the reference triangle so as to qualify it to be called a triangle conic have not been precisely formulated. WolframMathWorld has a page titled "Triangle conics" which gives a list of 42 items (not all of them are conics) without giving a definition of triangle conic.

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  Steiner ellipse
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  Artzt parabolas
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  Kiepert hyperbola

Triangle cubics

Cubic curves arise naturally in the study of triangles. For example, the locus of a point Template:Mvar in the plane of the reference triangle Template:Math such that, if the reflections of Template:Mvar in the sidelines of Template:Math are Template:Mvar, Template:Mvar, Template:Mvar, then the lines Template:Mvar, Template:Mvar and Template:Mvar are concurrent is a cubic curve named Neuberg cubic. It is the first cubic listed in Bernard Gibert's Catalogue of Triangle Cubics. This Catalogue lists more than 1200 triangle cubics with information on each curve such as the barycentric equation of the curve, triangle centers which lie on the curve, locus properties of the curve and references to literature on the curve.

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  Neuberg cubic (K001)
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  McCay cubic with its three concurring asymptotes (K003)
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  Tucker cubic (K011)

Computers in triangle geometry

The entry of computers had a deciding influence on the course of development in the interest in triangle geometry witnessed during the closing years of the twentieth century and the early years of the current century. Some of the ways in which the computers had influenced this course have been delineated by Philip Davis. Computers have been used to generate new results in triangle geometry. A survey article published in 2015 gives an account of some of the important new results discovered by the computer programme "Discoverer". The following sample of theorems gives a flavor of the new results discovered by Discoverer.

• Theorem 6.1 Let P and Q are points, neither lying on a sideline of triangle ABC. If P and Q are isogonal conjugates with respect to ABC, then the Ceva product of their complements lies on the Kiepert hyperbola.
• Theorem 9.1. The Yff center of congruence is the internal center of similitude of the incircle and the circumcircle with respect to the pedal triangle of the incenter.
• The Lester circle is the circle which passes through the circumcenter, the nine-point center and the outer and inner Fermat points. A generalised Lester circle is a circle which passes through at least four triangle centers. Discoverer has discovered several generalized Lester circles.

Sava Grozdev, Hiroshi Okumura, Deko Dekov are maintaining a web portal dedicated to computer discovered encyclopedia of Euclidean geometry.

Additional reading

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