Triangle center

A triangle center is a point in a plane whose trilinear coordinates are functions of the side-lengths of the reference triangle having the properties of homogeneity, bisymmetry and cyclicity. A triangle center is in some sense a center of a triangle comparable to centers of squares and circles. The classic special points related to a triangle known since ancient Greek times - namely centroid, circumcenter, incenter and orthocenter - are all triangle centers. But the centers of the ex-circles and the Brocard points, named after Henri Brocard (1845 – 1922), are not triangle centers. After the ancient Greeks, several special points associated with a triangle like Fermat point, nine-point center, symmedian point, Gergonne point, and Feuerbach point were discovered. Even though the ancient Greeks discovered the classic centers of a triangle they had not formulated any definition of a triangle center. During of the great revival of interest in triangle geometry in 1980's, it was noticed that these special points share some general properties that now form the basis for a formal definition of triangle center. As on 31 October 2008 (last update), Clark Kimberling's Encyclopedia of Triangle Centers contains an annotated list of 3514 triangle centers. Each of these centers is assigned a unique name. In cases where no particular name arises from geometrical or historical considerations, the name of a star is used instead. For example the 770 th point in the list, is named point Acamar.

Formal definition

Let the side-lengths of the triangle of reference ABC be a, b and c. Let the trilinear coordinates of a point P in the plane of the triangle ABC be f(a,b,c) : g(a,b,c) : h(a,b,c). P is called a triangle center if the following conditions are satisfied.

• Homogeneity: f(ta,tb,tc) = tⁿ f(a,b,c) for some constant n and for all t.
• Bisymmetry in b and c: f(a,b,c) = f(a,c,b).
• Cyclicity: g(a,b,c) = f(b,c,a) and h(a,b,c) = f(c,a,b).

Examples

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).

That these coordinates satisfy the defining properties of the coordinates of a triangle center will now be verified.

For the circumcenter,

f(a,b,c) = a(b² + c² - a²)

g(a,b,c) = b(c² + a² - b²)

h(a,b,c) = c(a² + b² - c²)

and so,

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) = f(a,c,b) (bisymmetry)

g(a,b,c) = f(b,c,a) and h(a,b,c) = f(c,a,b) (cyclicity)

Thus, the circumcenter is a triangle center.

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 a triangle center. The trilinear coordinates of the second Brocard point are b/c : c/a : a/b. These coordinates also satisfy the properties of homogeneity and cyclicity but not bisymmetry. So the second Brocard point is also not a triangle center.

Some well known triangle centers

Classical triangle centers

Sl. No.	Position in       Encyclopedia of       Triangle Centers	Name	Notation	Trilinear       coordinates
1	X2	Centroid	G	bc : ca : ab
2	X1	Incenter	I	1 : 1 : 1
3	X3	Circumcenter	O	cos A : cos B : cos C
4	X4	Orthocenter	H	sec A : sec B : sec C
5	X13	Fermat point	X	csc(A + π/3) : csc(B + π/3) : csc(C + π/3)
6	X5	Nine-point center	N	cos(B - C) : cos(C - A) : cos(A - B)
7	X6	Symmedian point	K	a : b : c
8	X7	Gergonne point	Ge	bc/(b + c - a) : ca/(c + a - b) : ab/(a + b - c)
9	X8	Nagel point	Na	(b + c - a)/a : (c + a - b)/b : (a + b - c)/c
10	X9	Mittenpunkt	M	b + c - a : c + a - b : a + b - c
11	X10	Spieker center	Sp	bc(b + c) : ca(c + a) : ab(a + b)
12	X11	Feuerbach point	F	1 - cos(B - C) : 1 - cos(C - A) : 1 - cos(A - B)
13	X15      X16	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)
14	X17       X18	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)
15	X99	Steiner point	S	bc/(b2 - c2) : ca/(c2 - a2) : ab/(a2 - b2)

Recent triangle centers

In the following table of recent triangle centers, no specific notations are mentioned for the various points. Also for eah 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.

Sl. No.	Position in       Encyclopedia of       Triangle Centers	Name	Trilinear       coordinates       f(a,b,c)
1	X21	Schiffler point	1/(cos B + cos C)
2	X22	Exeter Ppint	a(b4 + c4 - a4)
3	X111	Parry point	a/(2a2 - b2 - c2)
4	X173	Congruent Isoscelizers point	tan(A/2) + sec(A/2)
5	X174	Yff center of congruence	sec A/2
6	X175	Isoperimetric point	-1 + sec A/2 cos B/2 cos C/2
7	X179	Ajima-Malfatti point	sec4(A/4)
8	X181	Apollonius point	a(b + c)2/(b + c - a)
10	X356	Morley center	cos A/3 + 2 cos B/3 cos C/3
11	X360	Hofstadter point	A/a
12	X192	Equal parallelians point	bc(ca + ab - bc)
13	X401	Bailey point	[sin 2B sin 2C - sin2(2A)](csc A)

See also

• Encyclopedia of Triangle Centers

References

• Weisstein, Eric W. "Triangle Center". MathWorld--A Wolfram Web Resource. Retrieved 2009-05-23.