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Basic Parameters of the Feuerbach Point

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Given a triangle , Feuerbach's theorem states that the nine-point circle is tangent to the three excircles and the incircle of . The point of tangency of with the incircle is called the Feuerbach point, [1].

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Let , , be the exact trilinear coordinates of with respect to and let .

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

Let , , be the Conway parameters, with .

Then, it can be shown that

,

,

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You can drag the vertices , and .

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Contributed by: Minh Trinh Xuan (January 2023)
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 15, 2022) faculty.evansville.edu/ck6/encyclopedia.

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