Modern triangle geometry

In mathematics, modern triangle geometry, or new triangle geometry, is the body of knowledge relating to the properties of a

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)^[5] Philip Davis has suggested several reasons for the decline of interest in triangle geometry.^[5] 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

For a given triangle ABC with centroid G, the symmedian through the vertex is the reflection of the line AG in the bisector of the angle A. 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 K, is called the Lemoine point or the symmedian point or the Grebe point of triangle ABC. If the sidelengths of triangle ABC are a, b, c the baricentric coordinates of the Lemoine point are a² : b² : c². It has been described as "one of the crown jewels of modern geometry".^[9] There are several earlier references to this point in the mathematical literature details of which are available in John Mackay' history of the symmedian point.^[10]

In fact, the concurrency of the symmedians is a special case of a more general result: For any point P in the plane of triangle ABC, the isogonals of the lines AP, BP, CP are concurrent, the isogonal of AP (respectively BP, CP) being the reflection of the line AP in the bisector of the angle A (respectively B, C). The point of concurrency is called the isogonal conjugate of P. 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 ABC parallel to the sides of the triangle lie on a circle called the first Lemoine circle of triangle ABC. 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

Lemoine axis

Any triangle ABC and its

• 
  A triangle with medians (black), angle bisectors (dotted) and symmedians (red). The symmedians intersect in the symmedian point (denoted by L in the figure), the angle bisectors in the incenter I and the medians in the centroid G.
• 
  First Lemoine circle of triangle ABC. The Lemoine point K, the incenter I, the centroid G and the lines through K parallel to the sides are also shown.
• 
  Second Lemoine Circle of triangle ABC. The Lemoine point K, the incenter I, the centroid G and the lines through K antiparrallel to the sides are also shown.
• 
  Lemoine axis of triangle ABC. The tangential triangle is also shown.

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^[13] published in 1910 and Roger A Johnsons' book^[14] 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 O and radius R and another circle with center I and radius r, there are an infinite number of triangles ABC with Circle O(R) as circumcircle and I(r) as incircle if and only if OI² = R² − 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.^[15]

The Simson line

For any point P on the circumcircle of triangle ABC, the feet of perpendiculars from P to the sides of triangle ABC are collinear and the line of collinearity is the well-known Simson line of P.^[16]

Pedal and antipedal triangles

Given a point P, let the feet of perpendiculars from P to the sides of the triangle ABC be D, E, F. The triangle DEF is called the

Given any line l, let P, Q, R be the feet of perpendiculars from the vertices A, B, C of triangle ABC to l. The lines through P. Q, R perpendicular respectively to the sides BC, CA, AB are concurrent and the point of concurrence is the orthopole of the line l with respect to the triangle ABC. In modern triangle geometry, there is a large body of literature dealing with properties of orthopoles.^[20][21]

The Brocard points

Let of circles be described on the sides BC, CA, AB of triangle ABC whose external segments contain the two triads of angles C, A, B and B, C, A 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

${\displaystyle \Omega ,\Omega ^{\prime }}$

${\displaystyle \omega }$

${\displaystyle \cot \omega =\cot A+\cot B+\cot C.}$

The Brocard points and the Brocard angles have several interesting properties.^{[22] [23]}

Some images

• 
  Two poristic triangles ABC and A'B'C' with respect to circles I(r) and O(R)
• 
  Simson line of P
• 
  Pedal triangle (DEF) and antipedal triangle (LMN) of P
• 
  Orthopole of line l
• 
  First Brocard point of triangle ABC

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.^[24] 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 f of three real variables a, b, c may have the following properties:

• Homogeneity: f(ta,tb,tc) = tⁿ f(a,b,c) for some constant n and for all t > 0.
• Bisymmetry in the second and third variables: f(a,b,c) = f(a,c,b).

If a non-zero f has both these properties it is called a triangle center function. If f is a triangle center function and a, b, c 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.

Clark Kimberling is maintaining a website devoted to a compendium of triangle centers. The website named Encyclopedia of Triangle Centers has definitions and descriptions of nearly 50,000 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 ABC be a plane triangle and let ( x : y : z ) be the trilinear coordinates of an arbitrary point in the plane of triangle ABC.

A straight line in the plane of triangle ABC 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 triangle ABC relative to the triangle ABC.^[25][26]

Geometrical construction of central line

Let X be any triangle center of the triangle ABC.

• Draw the lines AX, BX and CX and their reflections in the internal bisectors of the angles at the vertices A, B, C respectively.
• The reflected lines are concurrent and the point of concurrence is the isogonal conjugate Y of X.
• Let the cevians AY, BY, CY meet the opposite sidelines of triangle ABC at A' , B' , C' respectively. The triangle A'B'C' is the cevian triangle of Y.
• The triangle ABC and the cevian triangle A'B'C' are in perspective and let DEF be the axis of perspectivity of the two triangles. The line DEF is the trilinear polar of the point Y. The line DEF is the central line associated with the triangle center X.

Triangle conics

A

• 
  Steiner ellipse
• 
  Artz parabolas
• 
  Kiepert hyperbola

Triangle cubics

Cubic curves arise naturally in the study of triangles. For example, the locus of a point P in the plane of the reference triangle ABC such that, if the reflections of P in the sidelines of triangle ABC are P_a, P_b, P_c, then the lines AP_a, BP_b and CP_c are concurrent is a cubic curve named Neuberg cubic. It is the first cubic listed in Bernard Gilbert'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.

• 
  Neuberg cubic (K001)
• 
  McCay cubic with its three concurring asymptotes (K003)
• 
  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.^[5] Computers have been used to generate new results in triangle geometry.^[28] A survey article published in 2015 gives an account of some of the important new results discovered by the computer programme "Dircoverer".^[29] 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.^[30]

References