1. History

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][3] (As of June 2019), Clark Kimberling's Encyclopedia of Triangle Centers contains an annotated list of 32,784 triangle centers.^[4]

2. Formal Definition

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.

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.^[5][6]

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.

2.1. Default Domain

In some cases these functions are not defined on the whole of ℝ³. For example the trilinears of X₃₆₅ 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.

2.2. Other Useful Domains

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 (ie points) and degenerate triangles
  (ie lines) is the set of all scalene triangles. It is obtained by removing the planes b = c, c = a, a = b from T.

2.3. Domain Symmetry

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.

3. Examples

A triangle (ΔABC) with centroid (G), incenter (I), circumcenter (O), orthocenter (H) and nine-point center (N). https://handwiki.org/wiki/File:Trigonometric_centres.png

3.1. Circumcenter

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) ) = t³ f(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.

3.2. 1st isogonic Center

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.

3.3. Fermat Point

Let

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

Then f is bisymmetric and homogeneous so it is a triangle center function. Moreover the corresponding triangle center coincides with the obtuse angled vertex whenever any vertex angle exceeds 2π/3, and with the 1st isogonic center otherwise. Therefore this triangle center is none other than the Fermat point.

4. Non-Examples

4.1. Brocard Points

The trilinear coordinates of the first Brocard point are c/b : a/c : b/a. These coordinates satisfy the properties of homogeneity and cyclicity but not bisymmetry. So the first Brocard point is not (in general) a triangle center. The second Brocard point has trilinear coordinates b/c : c/a : a/b and similar remarks apply.

The first and second Brocard points are one of many bicentric pairs of points,^[7] pairs of points defined from a triangle with the property that the pair (but not each individual point) is preserved under similarities of the triangle. Several binary operations, such as midpoint and trilinear product, when applied to the two Brocard points, as well as other bicentric pairs, produce triangle centers.

5. Position Vectors

Triangle centers can be written as following

[math]\displaystyle{ P=\frac{w_A A + w_B B + w_C C} {w_A + w_B + w_C}. }[/math]

Here, [math]\displaystyle{ P,A,B,C }[/math] are position vectors, and, coordinates [math]\displaystyle{ w_A, w_B, w_C }[/math] are scalars whose definition corresponds each center instances can be seen in the following table, where, [math]\displaystyle{ a, b, c }[/math] are side lengths, and, [math]\displaystyle{ S }[/math] is area of the triangle that Heron's formula can be utilized to get.

[math]\displaystyle{ a\equiv\overline{BC}=\sqrt{(\vec{BC},\vec{BC})}, }[/math]

[math]\displaystyle{ b\equiv\overline{CA}=\sqrt{(\vec{CA},\vec{CA})}, }[/math]

[math]\displaystyle{ c\equiv\overline{AB}=\sqrt{(\vec{AB},\vec{AB})}, }[/math]

[math]\displaystyle{ 16S^2=(a^2 + b^2+c^2)^2-2(a^4 + b^4+c^4). }[/math]

6. Some Well-Known Triangle Centers

6.1. Classical Triangle Centers

6.2. Recent Triangle Centers

In the following table of more recent triangle centers, no specific notations are mentioned for the various points. Also for each center only the first trilinear coordinate f(a,b,c) is specified. The other coordinates can be easily derived using the cyclicity property of trilinear coordinates.

7. General Classes of Triangle Centers

7.1. Kimberling Center

In honor of Clark Kimberling who created the online encyclopedia of more than 32,000 triangle centers, the triangle centers listed in the encyclopedia are collectively called Kimberling centers.^[8]

7.2. Polynomial Triangle Center

A triangle center P is called a polynomial triangle center if the trilinear coordinates of P can be expressed as polynomials in a, b and c.

7.3. Regular Triangle Center

A triangle center P is called a regular triangle point if the trilinear coordinates of P can be expressed as polynomials in Δ, a, b and c, where Δ is the area of the triangle.

7.4. Major Triangle Center

A triangle center P is said to be a major triangle center if the trilinear coordinates of P can be expressed in the form f(A) : f(B) : f(C) where f(A) is a function of the angle A alone and does not depend on the other angles or on the side lengths.^[9]

7.5. Transcendental Triangle Center

A triangle center P is called a transcendental triangle center if P has no trilinear representation using only algebraic functions of a, b and c.

8. Miscellaneous

8.1. Isosceles and Equilateral Triangles

Let f be a triangle center function. If two sides of a triangle are equal  (say a = b)  then

[math]\displaystyle{ f(a,b,c) = f(b,a,c) }[/math] (since a = b)

[math]\displaystyle{ \quad\quad\quad\quad= f(b,c,a) }[/math] (by bisymmetry)

so two components of the associated triangle center are always equal. Therefore all triangle centers of an isosceles triangle must lie on its line of symmetry. For an equilateral triangle all three components are equal so all centers coincide with the centroid. So, like a circle, an equilateral triangle has a unique center.

8.2. Excenters

Let

[math]\displaystyle{ f(a, b, c) = \begin{cases} -1 & \quad \text{if } a \ge b \text{ and } a \ge c, \\ \;\;\; 1 & \quad \text{otherwise}. \end{cases} }[/math]

This is readily seen to be a triangle center function and (provided the triangle is scalene) the corresponding triangle center is the excenter opposite to the largest vertex angle. The other two excenters can be picked out by similar functions. However as indicated above only one of the excenters of an isosceles triangle and none of the excenters of an equilateral triangle can ever be a triangle center.

8.3. iantisymmetric Functions

A function f is biantisymmetric if f(a,b,c) = −f(a,c,b) for all a,b,c. If such a function is also non-zero and homogeneous it is easily seen that the mapping (a,b,c) → f(a,b,c)² f(b,c,a) f(c,a,b) is a triangle center function. The corresponding triangle center is f(a,b,c) : f(b,c,a) : f(c,a,b). On account of this the definition of triangle center function is sometimes taken to include non-zero homogeneous biantisymmetric functions.

8.4. New Centers from Old

Any triangle center function f can be normalized by multiplying it by a symmetric function of a,b,c so that n = 0. A normalized triangle center function has the same triangle center as the original, and also the stronger property that f(ta,tb,tc) = f(a,b,c) for all t > 0 and all (a,b,c). Together with the zero function, normalized triangle center functions form an algebra under addition, subtraction, and multiplication. This gives an easy way to create new triangle centers. However distinct normalized triangle center functions will often define the same triangle center, for example f and (abc)⁻¹(a+b+c)³f .

8.5. Uninteresting Centers

Assume a,b,c are real variables and let α,β,γ be any three real constants. Let

[math]\displaystyle{ f(a, b, c) = \begin{cases} \alpha & \quad \text{ if } a \lt b \text{ and } a \lt c \quad \text{(equivalently the first variable is the smallest)}, \\ \gamma & \quad \text{ if } a \gt b \text{ and } a \gt c \quad \text{(equivalently the first variable is the largest)}, \\ \beta & \quad \; \text{otherwise} \quad \; \quad \quad \, \quad \text{(equivalently the first variable is in the middle)}. \end{cases} }[/math]

Then f is a triangle center function and α : β : γ is the corresponding triangle center whenever the sides of the reference triangle are labelled so that a < b < c. Thus every point is potentially a triangle center. However the vast majority of triangle centers are of little interest, just as most continuous functions are of little interest. The Encyclopedia of Triangle Centers is an ever-expanding list of interesting ones.

8.6. Barycentric Coordinates

If f is a triangle center function then so is af and the corresponding triangle center is af(a,b,c) : bf(b,c,a) : cf(c,a,b). Since these are precisely the barycentric coordinates of the triangle center corresponding to f it follows that triangle centers could equally well have been defined in terms of barycentrics instead of trilinears. In practice it isn't difficult to switch from one coordinate system to the other.

8.7. Binary Systems

There are other center pairs besides the Fermat point and the 1st isogonic center. Another system is formed by X₃ and the incenter of the tangential triangle. Consider the triangle center function given by:

[math]\displaystyle{ f(a, b, c) = \begin{cases} \cos(A) \quad \; \quad \; \quad \; \quad \; \quad \; \quad \;\,\, \text{if the triangle is acute}, \\ \cos(A) + \sec(B)\sec(C) \quad \text{if the vertex angle at } A \text{ is obtuse}, \\ \cos(A) - \sec(A) \quad \; \quad \; \quad \;\, \text{if either of the angles at } B \text{ or } C \text{ is obtuse}. \end{cases} }[/math]

For the corresponding triangle center there are four distinct possibilities:

•   cos(A) : cos(B) : cos(C)     if the reference triangle is acute (this is also the circumcenter).
•   [cos(A) + sec(B)sec(C)] : [cos(B) − sec(B)] : [cos(C) − sec(C)]     if the angle at A is obtuse.
•   [cos(A) − sec(A)] : [cos(B) + sec(C)sec(A)] : [cos(C) − sec(C)]     if the angle at B is obtuse.
•   [cos(A) − sec(A)] : [cos(B) − sec(B)] : [cos(C) + sec(A)sec(B)]     if the angle at C is obtuse.

Routine calculation shows that in every case these trilinears represent the incenter of the tangential triangle. So this point is a triangle center that is a close companion of the circumcenter.

8.8. Bisymmetry and Invariance

Reflecting a triangle reverses the order of its sides. In the image the coordinates refer to the (c,b,a) triangle and (using "|" as the separator) the reflection of an arbitrary point α : β : γ is γ | β | α. If f is a triangle center function the reflection of its triangle center is f(c,a,b) | f(b,c,a) | f(a,b,c) which, by bisymmetry, is the same as f(c,b,a) | f(b,a,c) | f(a,c,b). As this is also the triangle center corresponding to f relative to the (c,b,a) triangle, bisymmetry ensures that all triangle centers are invariant under reflection. Since rotations and translations may be regarded as double reflections they too must preserve triangle centers. These invariance properties provide justification for the definition.

8.9. Alternative Terminology

Some other names for dilation are uniform scaling, isotropic scaling, homothety, and homothecy.

9. Hyperbolic Triangle Centers

The study of triangle centers traditionally is concerned with Euclidean geometry, but triangle centers can also be studied in hyperbolic geometry. Using gyrotrigonometry, expressions for trigonometric barycentric coordinates can be calculated that have the same form for both Euclidean and hyperbolic geometry. In order for the expressions to coincide, the expressions must not encapsulate the specification of the angle sum being 180 degrees.^[10][11][12]

10. Tetrahedron Centers and N-simplex Centers

A generalization of triangle centers to higher dimensions is centers of tetrahedrons or higher-dimensional simplices.^[12]