Wikipedia / Isodynamic point
Wikipedia / 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
are the common intersection points of the three circles of Apollonius 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 is the Lemoine line, which contains the three centers of the circles of Apollonius.
Transformations
may also be defined by their properties with respect to transformations of the plane, and particularly with respect to inversionss and Möbius transformations with respect to an isodynamic point transforms the original triangle into an equilateral triangle. Inversion with respect to the circumcircle 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 to the interior of the circumcircle of the transformed triangle, and swapped by transformations that exchange the interior and exterior of the circumcircle.
Angles
where each of these circles intersects the circumcircle 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 pedal triangle is equilateral, across each side of the 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 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.
in the complex plane
are the critical values A similar construction exists for rational functions instead of polynomials.
Construction
may be constructed by finding the two (interior and exterior) angle bisectors 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.
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
Notes
References
.
.
.
.
.
.
.
.
.
.
. 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.