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# Isodynamic Points

The first and second isodynamic points of a triangle 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 and , for , 2, 3. The three circles having , , and as diameters are the Apollonius circles , , and . The points and in which the three Apollonius circles intersect are the first and second isodynamic points, respectively.

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

and have triangle center functions

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

where 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.

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

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## References

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." http://faculty.evansville.edu/ck6/tcenters/class/isodyn.html.

## Referenced on Wolfram|Alpha

Isodynamic Points

## Cite this as:

Weisstein, Eric W. "Isodynamic Points." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IsodynamicPoints.html