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In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. 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. All centers of an equilateral triangle coincide at its centroid, but they generally differ from each other on scalene triangles. The definitions and properties of thousands of triangle centers have been collected in the Encyclopedia of Triangle Centers.
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]}
A real-valued function f of three real variables a, b, c may have the following properties:
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 f_{1}(a,b,c) = 1/a and f_{2}(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.
In some cases these functions are not defined on the whole of ℝ^{3}. For example the trilinears of X_{365} 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 ℝ^{3} 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.
There are various instances where it may be desirable to restrict the analysis to a smaller domain than T. For example:
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.
The point of concurrence of the perpendicular bisectors of the sides of triangle ABC is the circumcenter. The trilinear coordinates of the circumcenter are
Let f(a,b,c) = a(b^{2} + c^{2} − a^{2}). Then
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.
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
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.
Let
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.
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.
Triangle centers can be written as following
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{ w_A }[/math] | [math]\displaystyle{ w_B }[/math] | [math]\displaystyle{ w_C }[/math] | [math]\displaystyle{ w_A+w_B+w_C }[/math] | |
---|---|---|---|---|
Incenter | [math]\displaystyle{ a }[/math] | [math]\displaystyle{ b }[/math] | [math]\displaystyle{ c }[/math] | [math]\displaystyle{ a+b+c }[/math] |
Excenter | [math]\displaystyle{ -a }[/math] | [math]\displaystyle{ b }[/math] | [math]\displaystyle{ c }[/math] | [math]\displaystyle{ b+c-a }[/math] |
[math]\displaystyle{ a }[/math] | [math]\displaystyle{ -b }[/math] | [math]\displaystyle{ c }[/math] | [math]\displaystyle{ c+a-b }[/math] | |
[math]\displaystyle{ a }[/math] | [math]\displaystyle{ b }[/math] | [math]\displaystyle{ -c }[/math] | [math]\displaystyle{ a+b-c }[/math] | |
Centroid | [math]\displaystyle{ 1 }[/math] | [math]\displaystyle{ 1 }[/math] | [math]\displaystyle{ 1 }[/math] | [math]\displaystyle{ 3 }[/math] |
Circumcenter | [math]\displaystyle{ a^2(b^2 + c^2 - a^2) }[/math] | [math]\displaystyle{ b^2(c^2 + a^2 - b^2) }[/math] | [math]\displaystyle{ c^2(a^2 + b^2 - c^2) }[/math] | [math]\displaystyle{ 16S^2 }[/math] |
Orthocenter | [math]\displaystyle{ a^4 - (b^2 - c^2)^2 }[/math] | [math]\displaystyle{ b^4 - (c^2 - a^2)^2 }[/math] | [math]\displaystyle{ c^4 - (a^2 - b^2)^2 }[/math] | [math]\displaystyle{ 16S^2 }[/math] |
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 | 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/(b^{2} − c^{2}) : ca/(c^{2} − a^{2}) : ab/(a^{2} − b^{2}) | Various equivalent definitions. |
(*) : actually the 1st isogonic center, but also the Fermat point whenever A,B,C ≤ 2π/3
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.
Encyclopedia of Triangle Centers reference |
Name | Center function f(a,b,c) |
Year described |
---|---|---|---|
X_{21} | Schiffler point | 1/(cos B + cos C) | 1985 |
X_{22} | Exeter point | a(b^{4} + c^{4} − a^{4}) | 1986 |
X_{111} | Parry point | a/(2a^{2} − b^{2} − c^{2}) | early 1990s |
X_{173} | Congruent isoscelizers point | tan(A/2) + sec(A/2) | 1989 |
X_{174} | Yff center of congruence | sec(A/2) | 1987 |
X_{175} | Isoperimetric point | − 1 + sec(A/2) cos(B/2) cos(C/2) | 1985 |
X_{179} | First Ajima-Malfatti point | sec^{4}(A/4) | |
X_{181} | Apollonius point | a(b + c)^{2}/(b + c − a) | 1987 |
X_{192} | Equal parallelians point | bc(ca + ab − bc) | 1961 |
X_{356} | Morley center | cos(A/3) + 2 cos(B/3) cos(C/3) | |
X_{360} | Hofstadter zero point | A/a | 1992 |
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]}
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.
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.
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]}
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.
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.
Let
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.
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.
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)^{3}f .
Assume a,b,c are real variables and let α,β,γ be any three real constants. Let
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.
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.
There are other center pairs besides the Fermat point and the 1st isogonic center. Another system is formed by X_{3} and the incenter of the tangential triangle. Consider the triangle center function given by:
For the corresponding triangle center there are four distinct possibilities:
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.
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.
Some other names for dilation are uniform scaling, isotropic scaling, homothety, and homothecy.
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]}
A generalization of triangle centers to higher dimensions is centers of tetrahedrons or higher-dimensional simplices.^{[12]}