Triangle center – Wikipedia

Posted on March 28, 2017 by lordneo

Point in a triangle that can be seen as its middle under some criteria

In geometry, a triangle center or triangle centre is a point in the triangle’s plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions.

Each of these classical centers has the property that it is invariant (more precisely equivariant) under similarity transformations. In other words, for any triangle and any similarity transformation (such as a rotation, reflection, dilation, or translation), the center of the transformed triangle is the same point as the transformed center of the original triangle.
This invariance is the defining property of a triangle center. It rules out other well-known points such as the Brocard points which are not invariant under reflection and so fail to qualify as triangle centers.

For an equilateral triangle, all triangle centers coincide at its centroid. However the triangle centers generally take different positions from each other on all other triangles. The definitions and properties of thousands of triangle centers have been collected in the Encyclopedia of Triangle Centers.

Table of Contents

History[edit]

Even though the ancient Greeks discovered the classic centers of a triangle they had not formulated any definition of a triangle center. After the ancient Greeks, several special points associated with a triangle like the Fermat point, nine-point center, Lemoine point, Gergonne point, and Feuerbach point were discovered.

During the revival of interest in triangle geometry in the 1980s it was noticed that these special points share some general properties that now form the basis for a formal definition of triangle center.^[1]^[2] As of 17 June 2022^[update], Clark Kimberling’s Encyclopedia of Triangle Centers contains an annotated list of 50,730 triangle centers.^[3] Every entry in the Encyclopedia of Triangle Centers is denoted by

{displaystyle X(n)}

${displaystyle X(n)}$ or

{displaystyle X_{n}}

$X_{n}$ where

{displaystyle n}

$n$ is the positional index of the entry. For example, the centroid of a triangle is the second entry and is denoted by

{displaystyle X(2)}

${displaystyle X(2)}$ or

{displaystyle X_{2}}

$X_{2}$ .

Formal definition[edit]

A real-valued function f of three real variables a, b, c may have the following properties:

Homogeneity: f(ta,tb,tc) = tⁿf(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.

This definition ensures that triangle centers of similar triangles meet the invariance criteria specified above. By convention only the first of the three trilinear coordinates of a triangle center is quoted since the other two are obtained by cyclic permutation of a, b, c. This process is known as cyclicity.^[4]^[5]

Every triangle center function corresponds to a unique triangle center. This correspondence is not bijective. Different functions may define the same triangle center. For example, the functions f₁(a,b,c) = 1/a and f₂(a,b,c) = bc both correspond to the centroid.
Two triangle center functions define the same triangle center if and only if their ratio is a function symmetric in a, b and c.

Even if a triangle center function is well-defined everywhere the same cannot always be said for its associated triangle center. For example, let f(a, b, c) be 0 if a/b and a/c are both rational and 1 otherwise. Then for any triangle with integer sides the associated triangle center evaluates to 0:0:0 which is undefined.

Default domain[edit]

In some cases these functions are not defined on the whole of ℝ³. For example, the trilinears of X₃₆₅ which is the 365th entry in the Encyclopedia of Triangle Centers, are a^1/2 : b^1/2 : c^1/2 so a, b, c cannot be negative. Furthermore, in order to represent the sides of a triangle they must satisfy the triangle inequality. So, in practice, every function’s domain is restricted to the region of ℝ³ where a ≤ b + c, b ≤ c + a, and c ≤ a + b. This region T is the domain of all triangles, and it is the default domain for all triangle-based functions.

Other useful domains[edit]

There are various instances where it may be desirable to restrict the analysis to a smaller domain than T. For example:

The centers X₃, X₄, X₂₂, X₂₄, X₄₀ make specific reference to acute triangles, namely that region of T where a² ≤ b² + c², b² ≤ c² + a², c² ≤ a² + b².
When differentiating between the Fermat point and X₁₃ the domain of triangles with an angle exceeding 2π/3 is important, in other words triangles for which a² > b² + bc + c² or b² > c² + ca + a² or c² > a² + ab + b².
A domain of much practical value since it is dense in T yet excludes all trivial triangles (i.e. points) and degenerate triangles (i.e. lines) is the set of all scalene triangles. It is obtained by removing the planes b = c, c = a, a = b from T.

Domain symmetry[edit]

Not every subset D ⊆ T is a viable domain. In order to support the bisymmetry test D must be symmetric about the planes b = c, c = a, a = b. To support cyclicity it must also be invariant under 2π/3 rotations about the line a = b = c. The simplest domain of all is the line (t,t,t) which corresponds to the set of all equilateral triangles.

Examples[edit]

Circumcenter[edit]

The point of concurrence of the perpendicular bisectors of the sides of triangle ABC is the circumcenter. The trilinear coordinates of the circumcenter are

a(b² + c² − a²) : b(c² + a² − b²) : c(a² + b² − c²).

Let f(a,b,c) = a(b² + c² − a²). Then

f(ta,tb,tc) = (ta) ( (tb)² + (tc)² − (ta)² ) = t³ ( a( b² + c² − a²) ) = t³f(a,b,c) (homogeneity)

f(a,c,b) = a(c² + b² − a²) = a(b² + c² − a²) = f(a,b,c) (bisymmetry)

so f is a triangle center function. Since the corresponding triangle center has the same trilinears as the circumcenter it follows that the circumcenter is a triangle center.

1st isogonic center[edit]

Let A’BC be the equilateral triangle having base BC and vertex A’ on the negative side of BC and let AB’C and ABC’ be similarly constructed equilateral triangles based on the other two sides of triangle ABC. Then the lines AA’, BB’ and CC’ are concurrent and the point of concurrence is the 1st isogonal center. Its trilinear coordinates are

csc(A + π/3) : csc(B + π/3) : csc(C + π/3).

Expressing these coordinates in terms of a, b and c, one can verify that they indeed satisfy the defining properties of the coordinates of a triangle center. Hence the 1st isogonic center is also a triangle center.

Fermat point[edit]

Let

{displaystyle f(a,b,c)={begin{cases}1&quad {text{if }}a^{2}>b^{2}+bc+c^{2}&({text{equivalently }}A>2pi /3),\0&quad {text{if }}b^{2}>c^{2}+ca+a^{2}{text{ or }}c^{2}>a^{2}+ab+b^{2}&({text{equivalently }}B>2pi /3{text{ or }}C>2pi /3),\csc(A+pi /3)&quad {text{otherwise }}&({text{equivalently no vertex angle exceeds }}2pi /3).end{cases}}}

Encyclopedia of Triangle Centers reference	Name	Standard symbol	Trilinear coordinates	Description
$X 1$	Incenter	$I$	1 : 1 : 1	Intersection of the angle bisectors. Center of the triangle’s inscribed circle.
$X 2$	Centroid	$G$	bc : ca : ab	Intersection of the medians. Center of mass of a uniform triangular lamina.
$X 3$	Circumcenter	$O$	cos A : cos B : cos C	Intersection of the perpendicular bisectors of the sides. Center of the triangle’s circumscribed circle.
$X 4$	Orthocenter	$H$	sec A : sec B : sec C	Intersection of the altitudes.
$X 5$	Nine-point center	$N$	cos(B − C) : cos(C − A) : cos(A − B)	Center of the circle passing through the midpoint of each side, the foot of each altitude, and the midpoint between the orthocenter and each vertex.
$X 6$	Symmedian point	$K$	a : b : c	Intersection of the symmedians – the reflection of each median about the corresponding angle bisector.
$X 7$	Gergonne point	$G e$	$bc /(b + c - a) : ca /(c + a - b) : ab /(a + b - c)$	Intersection of the lines connecting each vertex to the point where the incircle touches the opposite side.
$X 8$	Nagel point	$N a$	(b + c − a)/a : (c + a − b)/b: (a + b − c)/c	Intersection of the lines connecting each vertex to the point where an excircle touches the opposite side.
$X 9$	Mittenpunkt	$M$	(b + c − a) : (c + a − b) : (a + b − c)	Symmedian point of the excentral triangle (and various equivalent definitions).
$X 10$	Spieker center	$S p$	bc(b + c) : ca(c + a) : ab(a + b)	Incenter of the medial triangle. Center of mass of a uniform triangular wireframe.
$X 11$	Feuerbach point	$F$	1 − cos(B − C) : 1 − cos(C − A) : 1 − cos(A − B)	Point at which the nine-point circle is tangent to the incircle.
$X 13$	Fermat point	$X$	$csc(A + π/3) : csc(B + π/3) : csc(C + π/3)$ (*)	Point that is the smallest possible sum of distances from the vertices.
$X 15 X 16$	Isodynamic points	$S S'$	$sin(A + π/3) : sin(B + π/3) : sin(C + π/3) sin(A - π/3) : sin(B - π/3) : sin(C - π/3)$	Centers of inversion that transform the triangle into an equilateral triangle.
$X 17 X 18$	Napoleon points	$N N'$	$sec(A - π/3) : sec(B - π/3) : sec(C - π/3) sec(A + π/3) : sec(B + π/3) : sec(C + π/3)$	Intersection of the lines connecting each vertex to the center of an equilateral triangle pointed outwards (first Napoleon point) or inwards (second Napoleon point), mounted on the opposite side.
$X 99$	Steiner point	$S$	bc/(b² − c²) : ca/(c² − a²) : ab/(a² − b²)	Various equivalent definitions.

Triangle center – Wikipedia

History[edit]