Basic Parameters of the Kimberling Center X(53)

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The angle bisectors of a general triangle intersect at the incenter . The isogonal conjugate of a point is found by reflecting the lines , , about the angle bisectors. The symmedian point of a triangle is the isogonal conjugate of the centroid. The orthic triangle consists of the feet of the altitudes of a general triangle.


The point is the symmedian point of the orthic triangle (shown in red) of [1].


, , be the side lengths,

, , be the circumradius, inradius and semiperimeter of ,


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

Then, with


we have


da=FractionBox[RowBox[{"S", RowBox[{"(", RowBox[{RowBox[{"b", " ", "c", " ", "S"}], "+", RowBox[{"a", " ", "R", RowBox[{"(", RowBox[{RowBox[{"4", SuperscriptBox["R", "2"]}], "-", RowBox[{"2", SubscriptBox["S", "A"]}], "-", SubscriptBox["S", "ω"]}], ")"}]}]}], ")"}]}], RowBox[{"2", "R", " ", "f"}]],


You can drag the vertices , , .


Contributed by: Minh Trinh Xuan (August 25)
Open content licensed under CC BY-NC-SA


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.


[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Jun 22, 2023)


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