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