Isodynamic Points


The first and second isodynamic points of a triangle DeltaABC can be constructed by drawing the triangle's angle bisectors and exterior angle bisectors. Each pair of bisectors intersects a side of the triangle (or its extension) in two points D_(i1) and D_(i2), for i=1, 2, 3. The three circles having D_(11)D_(12), D_(21)D_(22), and D_(31)D_(32) as diameters are the Apollonius circles C_1, C_2, and C_3. The points S and S^' in which the three Apollonius circles intersect are the first and second isodynamic points, respectively.


The two isodynamic points of a reference triangle DeltaABC are mutually inverse with respect to the circumcircle of DeltaABC (Gallatly 1913, p. 103).

S and S^' have triangle center functions


respectively. The antipedal triangles of both points are equilateral and have areas


where omega is the Brocard angle.

The isodynamic points are isogonal conjugates of the Fermat points. They lie on the Brocard axis. The distances from either isodynamic point to the polygon vertices are inversely proportional to the sides. The pedal triangle of either isodynamic point is an equilateral triangle. An inversion with either isodynamic point as the inversion center transforms the triangle into an equilateral triangle.

The circle that passes through both the isodynamic points and the triangle centroid of a triangle is known as the Parry circle.

See also

Apollonius Circle, Brocard Axis, Fermat Points, First Isodynamic Point, Parry Circle, Second Isodynamic Point, Triangle Centroid

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Gallatly, W. "The Isodynamic Points." §149 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 106, 1913.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 295-297, 1929.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Isodynamic Points."

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Isodynamic Points

Cite this as:

Weisstein, Eric W. "Isodynamic Points." From MathWorld--A Wolfram Web Resource.

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