page_title: Isodynamic point
section_title: Angles
hierarchical_section_title: Isodynamic point / Angles
context_page_description: 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; every non-equilateral triangle has two isodynamic points. Isodynamic points were first studied and named by Joseph Neuberg.
context_section_description: 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.
caption: Three circles, each making angles of π/3 with the circumcircle and each other, meet at the first isodynamic point.