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(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 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.

Angles

As well as being the intersections of the circles of Apollonius, each isodynamic point is the intersection points of another triple of circles. The first isodynamic point is the intersection of three circles through the pairs of points   and  where each of these circles intersects the circumcircle of triangle  to form a lens with apex angle 2π/3.  Similarly, the second isodynamic point is the intersection of three circles that intersect the circumcircle to form lenses with apex angle π/3.

The angles formed by the first isodynamic point with the triangle vertices satisfy the equations   and  Analogously, the angles formed by the second isodynamic point satisfy the equations  and 

The pedal triangle of an isodynamic point, the triangle formed by dropping perpendiculars from  to each of the three sides of triangle  is equilateral, as is the triangle formed by reflecting  across each side of the triangle. Among all the equilateral triangles inscribed in triangle  the pedal triangle of the first isodynamic point is the one with minimum area.

Additional properties
The isodynamic points are the isogonal conjugates of the two Fermat points of triangle  and vice versa.

The Neuberg cubic contains both of the isodynamic points.

If a circle is partitioned into three arcs, the first isodynamic point of the arc endpoints is the unique point inside the circle with the property that each of the three arcs is equally likely to be the first arc reached by a Brownian motion starting at that point. That is, the isodynamic point is the point for which the harmonic measure of the three arcs is equal.

Given a univariate polynomial  whose zeros are the vertices of a triangle  in the complex plane, the isodynamic points of  are the zeros of the polynomial  Note that  is a constant multiple of  where  is the degree of  This construction generalizes isodynamic points to polynomials of degree  in the sense that the zeros of the above discriminant are invariant under Möbius transformations. Here the expression  is the polar derivative of  with pole 

Equivalently, with  and  defined as above, the (generalized) isodynamic points of  are the critical values of  Here  is the expression that appears in the relaxed Newton’s method with relaxation parameter  A similar construction exists for rational functions instead of polynomials.

Construction

The circle of Apollonius through vertex  of triangle  may be constructed by finding the two (interior and exterior) angle bisectors of the two angles formed by lines  and  at vertex  and intersecting these bisector lines with line  The line segment between these two intersection points is the diameter of the circle of Apollonius. The isodynamic points may be found by constructing two of these circles and finding their two intersection points.

Another compass and straight-edge construction involves finding the reflection  of vertex  across line  (the intersection of circles centered at  and  through ), and constructing an equilateral triangle inwards on side  of the triangle (the apex  of this triangle is the intersection of two circles having  as their radius). The line  crosses the similarly constructed lines  and  at the first isodynamic point. The second isodynamic point may be constructed similarly but with the equilateral triangles erected outwards rather than inwards.

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

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

Notes

References

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. The definition of isodynamic points is in a footnote on page 204.
. The discussion of isodynamic points is on pp. 138–139. Rigby calls them "Napoleon points", but that name more commonly refers to a different triangle center, the point of concurrence between the lines connecting the vertices of Napoleon's equilateral triangle with the opposite vertices of the given triangle.
. See especially p. 498.

External links

 Isodynamic points X(15) and X(16) in the Encyclopedia of Triangle Centers, by Clark Kimberling

Triangle centers