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Basic Parameters of the Triangle Centroid

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Given a triangle , let , , be the midpoints of the sides , , . Then the three lines , , are called the medians and they intersect at a point called the centroid of [1].

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Let , , be the exact trilinear coordinates of and let .

Let , , be the side lengths opposite the corresponding vertices of and let , , , be the circumradius, inradius, exradius for and semiperimeter of .

Let and let be the foot of the perpendicular from to .

It can be shown that

,

,

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You can drag the vertices , and .

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Contributed by: Minh Trinh Xuan (August 2022)
Open content licensed under CC BY-NC-SA

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A triangle center is said to be even when its barycentric coordinates can be expressed as a function of three variables , , that all occur with even exponents. If the center of a triangle has constant barycentric coordinates, it is called a neutral center (the centroid is the only neutral center). A triangle center is said to be odd if it is neither even nor neutral.

Standard barycentric coordinates of a point with respect to a reference triangle are normalized to a sum of 1.

Reference

[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Aug 9, 2022) faculty.evansville.edu/ck6/encyclopedia.

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