De Longchamps point

In geometry, the de Longchamps point of a triangle is a triangle center named after French mathematician Gaston Albert Gohierre de Longchamps. It is the reflection of the orthocenter of the triangle around the circumcenter.

Let the given triangle have vertices ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$, opposite the respective sides ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$, as is the standard notation in triangle geometry. In the 1886 paper in which he introduced this point, de Longchamps initially defined it as the center of a circle ${\displaystyle \Delta }$ orthogonal to the three circles ${\displaystyle \Delta _{a}}$, ${\displaystyle \Delta _{b}}$, and ${\displaystyle \Delta _{C}}$, where ${\displaystyle \Delta _{a}}$ is centered at ${\displaystyle A}$ with radius ${\displaystyle a}$ and the other two circles are defined symmetrically. De Longchamps then also showed that the same point, now known as the de Longchamps point, may be equivalently defined as the orthocenter of the anticomplementary triangle of ${\displaystyle ABC}$, and that it is the reflection of the orthocenter of ${\displaystyle ABC}$ around the circumcenter.

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