2 points about which a triangle can be inverted into an equilateral triangle
Circles of Apollonius; isodynamic points S and S' at their intersections
Interior angle bisectors, used to construct the circles
Exterior angle bisectors, also used to construct the circles
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).[1]
^For the credit to Neuberg, see e.g. Casey (1893) and Eves (1995).
triangle into an equilateral triangle, and that the distances from the isodynamicpoint to the triangle vertices are inversely proportional to the opposite...
< 120°), (C < 120°). The isogonal conjugate of X(13) is the first isodynamicpoint, X(15): sin ( A + π 3 ) : sin ( B + π 3 ) : sin ( C + π 3 ) ...
been used to refer to the isodynamic points of a triangle. This usage could also be justified on the ground that the isodynamic points are related to the...
particular enzyme of a pathogenic bacterium. Apollonius point Apollonius' theorem Isodynamicpoint of a triangle Dörrie H (1965). "The Tangency Problem of...
dip, and an aclinic line is the isoclinic line of magnetic dip zero. An isodynamic line (from δύναμις or dynamis meaning 'power') connects points with the...
the line determined by the isodynamic points. Wikimedia Commons has media related to Circles of Apollonius. Apollonius point Ellipse Weintraub, Isaac;...
applied to a different pair of triangle centers, better known as the isodynamic points. Let △ABC be any given plane triangle. On the sides BC, CA, AB...
Each point in the list is identified by an index number of the form X(n)—for example, X(1) is the incenter. The information recorded about each point includes...
centroid G is (by definition) the symmedian point K. The isogonal conjugates of the Fermat points are the isodynamic points and vice versa. The Brocard points...
In 1878, German nutritionist Max Rubner crafted what he called the "isodynamic law". The law claims that the basis of nutrition is the exchange of energy...
cubic passes through the centroid, symmedian point, both Fermat points, both isodynamic points, the Parry point, other triangle centers, and the vertices...
points X(13) and X(14) in the Encyclopedia of Triangle Centers) The two isodynamic points (the points X(15) and X(16) in the Encyclopedia of Triangle Centers)...
operations (rotations, reflections, inversions) of rigid molecules, and (2) isodynamic operations, which take a nonrigid molecule into an energetically equivalent...
edges are perpendicular, it is called a semi-orthocentric tetrahedron. An isodynamic tetrahedron is one in which the cevians that join the vertices to the...
and three bimedians are all concurrent at a point called the centroid of the tetrahedron. An isodynamic tetrahedron is one in which the cevians that...
points that are the isodynamic points of the triangle formed by the three points of tangency. Inversive geometry Limiting point (geometry), the center...
In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example...
Trailhunter to be the halo model of the line. The TRD Pro trim level includes IsoDynamic seats for the driver and front passenger which control motion through...
the other functioning in gliding, anchorage, propulsion or "steering") isodynamic: flagella beating with the same patterns Other terms related to the flagellar...
Global Information