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.
Read more about this topic: Isodynamic Point