Isodynamic point

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 Joseph Neuberg (1885).

[1] The isodynamic points were originally defined from certain equalities of ratios (or equivalently of products) of distances between pairs of points.

[2] Equivalently to the product formula, the distances

are inversely proportional to the corresponding triangle side lengths

the three circles that each pass through one vertex of the triangle and maintain a constant ratio of distances to the other two vertices.

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.

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).

leaves the triangle invariant but transforms one isodynamic point into the other one.

[3] More generally, the isodynamic points are equivariant under Möbius transformations: the unordered pair of isodynamic points of a transformation of

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.

[6] As well as being the intersections of the circles of Apollonius, each isodynamic point is the intersection points of another triple of circles.

Similarly, the second isodynamic point is the intersection of three circles that intersect the circumcircle to form lenses with apex angle π/3.

[6] The angles formed by the first isodynamic point with the triangle vertices satisfy the equations

Analogously, the angles formed by the second isodynamic point satisfy the equations

is equilateral,[5] as is the triangle formed by reflecting

the pedal triangle of the first isodynamic point is the one with minimum area.

This construction generalizes isodynamic points to polynomials of degree

in the sense that the zeros of the above discriminant are invariant under Möbius transformations.

A similar construction exists for rational functions instead of polynomials.

The line segment between these two intersection points is the diameter of the circle of Apollonius.

[3] Another compass and straight-edge construction involves finding the reflection

), and constructing an equilateral triangle inwards on side

The second isodynamic point may be constructed similarly but with the equilateral triangles erected outwards rather than inwards.

[12] Alternatively, the position of the first isodynamic point may be calculated from its trilinear coordinates, which are[13]

The second isodynamic point uses trilinear coordinates with a similar formula involving