Basic Parameters of the Incenter of a Triangle

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The incenter of a triangle is the center of the incircle of that triangle [1].


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

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

The triangle created by the points of tangency of the incircle to the sides of is called the contact triangle (or the intouch triangle or the pedal triangle of ).

It can be shown that




You can drag the vertices , and .


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



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.


[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Aug 9, 2022)

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