Basic Parameters of the Symmedian Point

The angle bisectors of a triangle intersect at the incenter . The isogonal conjugate of a point is found by reflecting the lines , , about the angle bisectors. The symmedian point [1] of is the isogonal conjugate of the centroid .
Let
, , be the exact trilinear coordinates of with respect to ,
,
, , be the side lengths opposite the corresponding vertices and let be the semiperimeter of ,
,
, , be the Conway parameters with ,
be the Brocard angle.
Then, it can be shown that
,
,
.
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 a sum of 1.
Reference
[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Aug 15, 2022) faculty.evansville.edu/ck6/encyclopedia.
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