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Basic Parameters of the Far-Out Point

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Given a triangle , let be its circumcenter and be its tangential triangle (shown in purple). Then the circumcircles of the three triangles , , intersect at the "far-out point" (Randy Hutson, October 13, 2015) [1].

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

Let

, , be the side lengths,

, , be the circumradius, inradius and semiperimeter of ,

and introduce the parameters , , , in Conway notation, where is the Brocard angle.

Then

,

,

.

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 have a sum of 1.

Reference

[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Sep 27, 2022) faculty.evansville.edu/ck6/encyclopedia.

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