Basic Parameters of the Kimberling Center X(47)

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In the triangle , let
be the incenter,
be the orthocenter,
be the Schiffler point,
be the
-beth conjugate of
(see the glossary at [1]) and
be the
-Ceva conjugate of
. Then the center
is the intersection of the lines
and
.
Contributed by: Minh Trinh Xuan (January 2023)
Open content licensed under CC BY-NC-SA
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Details
A triangle center is said to be "even center" if 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 barycentric coordinates as a constant, it is called a "neutral center" (The centroid
is the only "neutral center".) Conversely, a triangle center is said to be "odd center" if it is neither even nor neutral.
Standard barycentric coordinates of a point with respect to a reference triangle have a sum of 1.
Classification: odd center
Reference
[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Dec 13, 2022) faculty.evansville.edu/ck6/encyclopedia.
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