Isodynamic Point - Transformations

Transformations

The isodynamic points and of a triangle may also defined by their properties with respect to transformations of the plane, and particularly with respect to inversions and Möbius transformations (products of multiple inversions). Inversion of the triangle with respect to an isodynamic point transforms the original triangle into an equilateral triangle. Inversion with respect to the circumcircle of triangle leaves the triangle invariant but transforms one isodynamic point into the other one. More generally, the isodynamic points are equivariant under Möbius transformations: the unordered pair of isodynamic points of a transformation of is equal to the same transformation applied to the pair . The individual isodynamic points are fixed by Möbius transformations that map the interior of the circumcircle of to the interior of the circumcircle of the transformed triangle, and swapped by transformations that exchange the interior and exterior of the circumcircle.