Questions tagged [triangle-centres]

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. 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.

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.

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

• Homogeneity: $f(ta,tb,tc) = t^n f(a,b,c)$ 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.

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 $f_1(a,b,c) = \frac 1 a$ 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 $\frac a b$ and $\frac a c$ 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.

Source: Wikipedia