Basic Parameters of the de Longchamps Point

Let and be the orthocenter and circumcenter of a triangle . The de Longchamps point is the reflection of in [1]. The point is also the orthocenter of the anticomplementary triangle (shown in purple) of .
Let , , be the exact trilinear coordinates of with respect to , and .
Let , , be the side lengths; , , be the circumradius, inradius and semiperimeter of ; and .
Let the parameters , , , be the Conway notation, where is the Brocard angle.
Then
,
,
.
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.
Reference
[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Sep 7, 2022) faculty.evansville.edu/ck6/encyclopedia.
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