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(as well as its orthocenter, its incenter, and its circumcenter, which are concurrent); every non-equilateral triangle has two isodynamic points. Isodynamic points were first studied and named by . Distance ratios The isodynamic points were originally defined from certain equalities of ratios (or equivalently of products) of distances between pairs of points. If and are the isodynamic points of a triangle , then the three products of distances are equal. The analogous equalities also hold for . Equivalently to the product formula, the distances , , and are inversely proportional to the corresponding triangle side lengths , , and . and are the common intersection points of the three circles of Apollonius associated with triangle of a triangle , the three circles that each pass through one vertex of the triangle and maintain a constant ratio of distances to the other two vertices. Hence, line is the common radical axis for each of the three pairs of circles of Apollonius. The perpendicular bisector of line segment is the Lemoine line, which contains the three centers of the circles of Apollonius. Transformations The isodynamic points and of a triangle may also be 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