Basic Parameters of the Second Napoleon Point

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].
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 .

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.