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Basic Parameters of the Gergonne Point

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Let the triangle have contact triangle , whose vertices are the tangent points of the incircle. Then the lines , , intersect at the Gergonne point [1].

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

Let , , be the exact trilinear coordinates of and be the sum of the exact trilinear coordinates of the point .

Then

,

,

.

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 have a sum of 1.

Reference

[1] Encyclopedia of Triangle Centers (ETC). https://faculty.evansville.edu/ck6/encyclopedia/etc.html.

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