# Basic Parameters of the Feuerbach Point

Initializing live version

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

Given a triangle , Feuerbach's theorem states that the nine-point circle is tangent to the three excircles and the incircle of . The point of tangency of with the incircle is called the Feuerbach point, [1].

[more]

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

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

Let , , be the Conway parameters, with .

Then, it can be shown that

,

,

.

You can drag the vertices , and .

[less]

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 are normalized to a sum of 1.

Reference

[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Aug 15, 2022) faculty.evansville.edu/ck6/encyclopedia.

## Permanent Citation

Minh Trinh Xuan

 Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send