The first and second isodynamic points of a triangle angle
bisectors and exterior angle bisectors .
Each pair of bisectors intersects a side of the
triangle (or its extension) in two points circles having
diameters are the
Apollonius circles Apollonius
circles intersect are the first and second isodynamic
points, respectively.
The two isodynamic points of a reference triangle inverse
with respect to the circumcircle of
The first isodynamic point lies inside the circumcircle. If
triangle center functions
respectively. The antipedal triangles of both
points are equilateral and have areas
where 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 ,
Cyclologic Triangles ,
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. Johnson, R. A. Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. 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." https://faculty.evansville.edu/ck6/tcenters/class/isodyn.html . Rabinowitz,
S. "Catalog of Properties of the First Isodynamic Point of a Triangle."
Int. J. Comput. Discovered Math. 6 , 108-136, 2021. https://www.journal-1.eu/2021/Stanley%20Rabinowitz.%20Catalog%20of%20Properties%20of%20the%20First%20Isodynamic%20Point%20of%20a%20Triangle%2C%20pp.%20108-136..pdf . Rabinowitz,
S. and Suppa, E. "Cyclologic Triangles Formed by Three Isodynamic Points."
To appear in Int. J. Comput. Discovered Math. , 2026. https://www.researchgate.net/publication/411174028_Cyclologic_Triangles_Formed_by_Three_Isodynamic_Points . Referenced
on Wolfram|Alpha Isodynamic Points
Cite this as:
Weisstein, Eric W. "Isodynamic Points."
From MathWorld https://mathworld.wolfram.com/IsodynamicPoints.html
Subject classifications
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.
are mutually inverse
with respect to the circumcircle of 
(Gallatly 1913, p. 103).
is the first isodynamic point of 
, then 
is the second isodynamic point of 
(Rabinowitz 2021, Properties 1.1.3 and 1.4.1). Analogous
relations hold under cyclic permutation of 
, 
, and 
.
and 
have triangle center functions
![Delta^'=2Delta[cotomegacot(1/3pi)],](/https/mathworld.wolfram.com/images/equations/IsodynamicPoints/NumberedEquation2.svg)
is the Brocard angle.
