Basic Parameters of the Nine-Point Center

For a triangle , nine points lie on its nine-point circle : the midpoints , , ; the feet of the altitudes , , ; and the midpoints , , of the line segments joining the vertices to the orthocenter .
The nine-point center of is the center of its nine-point circle [1].
Let , , be the exact trilinear coordinates of with respect to and let .
Let , , be the side lengths opposite the corresponding vertices and , , be the circumradius, inradius and semiperimeter of .
Let , set , , to be the Conway notation, and let be the foot of the perpendicular from to .
You can drag the vertices , and .


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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 are normalized to have a sum of 1.
[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Aug 25, 2022)
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