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Basic Parameters of the First Napoleon Point

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Given a triangle , construct the three exterior equilateral triangles on its sides with centers , , . Then the lines , , intersect at the first Napoleon point [1].

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Let , , be the exact trilinear coordinates of with respect to ; ; , , be the side lengths of ; , , be the circumradius, inradius and semiperimeter of ; ; and set , , , to be the 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." (Aug 25, 2022) faculty.evansville.edu/ck6/encyclopedia.

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