# Basic Parameters of the Second Napoleon Point

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Construct the three interior equilateral triangles with centers , , on the sides of a given triangle . Then the lines , , intersect at the second 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 express , , , 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

## Details

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 31, 2022) faculty.evansville.edu/ck6/encyclopedia.

## Permanent Citation

Minh Trinh Xuan

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