Isodynamic point

In Euclidean geometry, the isodynamic points of a triangle are points associated with the triangle, with the properties that an inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the distances from the isodynamic point to the triangle vertices are inversely proportional to the opposite side lengths of the triangle. Triangles that are similar to each other have isodynamic points in corresponding locations in the plane, so the isodynamic points are triangle centers, and unlike other triangle centers the isodynamic points are also invariant under Möbius transformations. A triangle that is itself equilateral has a unique isodynamic point, at its centroid; every non-equilateral triangle has two isodynamic points. Isodynamic points were first studied and named by Joseph Neuberg (1885).

The isodynamic points were originally defined from certain equalities of ratios (or equivalently of products) of distances between pairs of points. If ${\displaystyle S}$ and ${\displaystyle S'}$ are the isodynamic points of a triangle ${\displaystyle ABC}$, then the three products of distances ${\displaystyle AS\cdot BC=BS\cdot AC=CS\cdot AB}$ are equal. The analogous equalities also hold for ${\displaystyle S'}$. Equivalently to the product formula, the distances ${\displaystyle AS}$, ${\displaystyle BS}$, and ${\displaystyle CS}$ are inversely proportional to the corresponding triangle side lengths ${\displaystyle BC}$, ${\displaystyle AC}$, and ${\displaystyle AB}$.

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