2 points about which a triangle can be inverted into an equilateral triangle
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 Joseph Neuberg (1885).
Distance ratios.
The isodynamic points were originally defined from certain equalities of ratios (or equivalently of products) of distances between pairs of points. If formula_1 and formula_2 are the isodynamic points of a triangle formula_3 then the three products of distances formula_4 are equal. The analogous equalities also hold for formula_5 Equivalently to the product formula, the distances formula_6 formula_7 and formula_8 are inversely proportional to the corresponding triangle side lengths formula_9 formula_10 and formula_11
formula_1 and formula_2 are the common intersection points of the three circles of Apollonius associated with triangle of a triangle formula_3 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 formula_15 is the common radical axis for each of the three pairs of circles of Apollonius. The perpendicular bisector of line segment formula_15 is the Lemoine line, which contains the three centers of the circles of Apollonius.
Transformations.
The isodynamic points formula_1 and formula_2 of a triangle formula_19 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 formula_19 with respect to an isodynamic point transforms the original triangle into an equilateral triangle.
Inversion with respect to the circumcircle of triangle formula_19 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 formula_19 is equal to the same transformation applied to the pair formula_23 The individual isodynamic points are fixed by Möbius transformations that map the interior of the circumcircle of formula_19 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 formula_25 formula_10 and formula_9 where each of these circles intersects the circumcircle of triangle formula_19 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 formula_29 formula_30 and formula_31 Analogously, the angles formed by the second isodynamic point satisfy the equationsformula_32 formula_33 and formula_34
The pedal triangle of an isodynamic point, the triangle formed by dropping perpendiculars from formula_1 to each of the three sides of triangle formula_3 is equilateral, as is the triangle formed by reflecting formula_1 across each side of the triangle. Among all the equilateral triangles inscribed in triangle formula_3 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 formula_3 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 formula_40 whose zeros are the vertices of a triangle formula_41 in the complex plane, the isodynamic points of formula_41 are the zeros of the polynomial formula_43 Note that formula_44 is a constant multiple of formula_45 where formula_46 is the degree of formula_47 This construction generalizes isodynamic points to polynomials of degree formula_48 in the sense that the zeros of the above discriminant are invariant under Möbius transformations. Here the expression formula_49 is the polar derivative of formula_50 with pole formula_51
Equivalently, with formula_52 and formula_46 defined as above, the (generalized) isodynamic points of formula_52 are the critical values of formula_55 Here formula_56 is the expression that appears in the relaxed Newton’s method with relaxation parameter formula_57 A similar construction exists for rational functions instead of polynomials.
Construction.
The circle of Apollonius through vertex formula_58 of triangle formula_19 may be constructed by finding the two (interior and exterior) angle bisectors of the two angles formed by lines formula_60 and formula_61 at vertex formula_62 and intersecting these bisector lines with line formula_63 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 formula_64 of vertex formula_58 across line formula_66 (the intersection of circles centered at formula_67 and formula_68 through formula_58), and constructing an equilateral triangle inwards on side formula_66 of the triangle (the apex formula_71 of this triangle is the intersection of two circles having formula_66 as their radius). The line formula_73 crosses the similarly constructed lines formula_74 and formula_75 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
formula_76
The second isodynamic point uses trilinear coordinates with a similar formula involving formula_77 in place of formula_78
Notes.
 ;templatestyles src="Reflist/styles.css" / ;
References.
 ;templatestyles src="Refbegin/styles.css" / ;