Calculus How To

Center Function (Triangle)

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Types of Functions >

What is a Center Function?

A center function (also called a triangle center function, symmetric triangle center function or simply a center) gives the trilinear coordinates of a triangle’s center. The function’s three variables {a, b, c} or {α, β γ} correspond to angles or sides.

What is a Triangle Center?

A triangle center is a point defined in terms of a triangle’s side lengths and angles and for which a center function exists. A major triangle center is one where the function α = f(A, B, C) is only a function of angle A.

It sounds simple, but don’t be fooled into thinking there’s just one center to a triangle. A “center” can be defined in many ways. For example, the centroid is the intersection of the triangle medians, and the circumcenter is the center of the circle inscribing the triangle.
triangle center function

In fact, there are thousands of different triangle centers including the Fermat point, which has the center function α = csc(A + ⅓π) and the Far-Out point, with function α = a(b4 + c4 – a4 – b2c2).

Properties of the Triangle Center Function

The triangle center function is a homogeneous function (i.e. with variables that increase by the same proportion). In notation, that’s:
f ( t a, t b, t c) = tn f (a, b, c).
Where n is a constant.

The function is also nonzero, meaning that it can’t have values equal to zero. This makes sense, because a non-zero value of a triangle would result in a non-triangle shape: a line, a point, or no shape at all.

The triangle center function has bisymmetry (two planes of symmetry at right angles) in the second and third variables. In notation:
f(a, b, c) = f(a, c, b).

References

Kimblerling, C. (2020). Encyclopedia of Triangle Centers. Retrieved September 4, 2020 from: https://faculty.evansville.edu/ck6/encyclopedia/ETC.html
Reinhart, C. et al. Triangle Centers. Retrieved September4, 2020 from: https://brightspace.uakron.edu/d2l/common/viewFile.d2lfile/Database/MTQzMzU4MQ/Presentation_Group6_Reinhart_Kuzas_Burke.pptx?ou=6605&contextId=14871,13579


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