In
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, the isodynamic points of a
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
are points associated with the triangle, with the properties that an
inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
centered at one of these points transforms the given triangle into an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
, 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 center
In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example ...
s, and unlike other triangle centers the isodynamic points are also invariant under
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...
s. A triangle that is itself equilateral has a unique isodynamic point, at its
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
(as well as its
orthocenter
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...
, its
incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
, and its
circumcenter
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
, which are concurrent); every non-equilateral triangle has two isodynamic points. Isodynamic points were first studied and named by .
Distance ratios
The isodynamic points were originally defined from certain equalities of ratios (or equivalently of products) of distances between pairs of points. If
and
are the isodynamic points of a triangle
, then the three products of distances
are equal. The analogous equalities also hold for
. Equivalently to the product formula, the distances
,
, and
are inversely proportional to the corresponding triangle side lengths
,
, and
.
and
are the common intersection points of the three
circles of Apollonius
The circles of Apollonius are any of several sets of circles associated with Apollonius of Perga, a renowned Greek geometer. Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for examp ...
associated with triangle of a triangle
, the three circles that each pass through one vertex of the triangle and maintain a constant ratio of distances to the other two vertices.
Hence, line
is the common
radical axis
In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose Power of a point, power with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of ...
for each of the three pairs of circles of Apollonius. The perpendicular bisector of line segment
is the
Lemoine line
In geometry, central lines are certain special straight lines that lie in the plane of a triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This s ...
, which contains the three centers of the circles of Apollonius.
Transformations
The isodynamic points
and
of a triangle
may also be defined by their properties with respect to transformations of the plane, and particularly with respect to
inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
s and
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...
s (products of multiple inversions).
Inversion of the triangle
with respect to an isodynamic point transforms the original triangle into an
equilateral
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
.
Inversion with respect to the
circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
of triangle
leaves the triangle invariant but transforms one isodynamic point into the other one.
[; .]
More generally, the isodynamic points are
equivariant
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...
under
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...
s: the
unordered pair In mathematics, an unordered pair or pair set is a set of the form , i.e. a set having two elements ''a'' and ''b'' with no particular relation between them, where = . In contrast, an ordered pair (''a'', ''b'') has ''a'' as its first ...
of isodynamic points of a transformation of
is equal to the same transformation applied to the pair
. The individual isodynamic points are fixed by Möbius transformations that map the interior of the circumcircle of
to the interior of the circumcircle of the transformed triangle, and swapped by transformations that exchange the interior and exterior of the circumcircle.
Angles
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
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
of triangle
to form a
lens
A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements''), ...
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
In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle.
More specifically, consider a triangle ''ABC'', and a point ''P'' that is not one of the vertices ''A, B, C''. Drop perpendiculars from ''P'' to the th ...
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.
Additional properties
The isodynamic points are the
isogonal conjugate __notoc__
In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (Th ...
s of the two
Fermat point
In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest ...
s of triangle
, and vice versa.
The
Neuberg cubic
In Euclidean geometry, the Neuberg cubic is a special cubic plane curve associated with a reference triangle with several remarkable properties. It is named after Joseph Jean Baptiste Neuberg (30 October 1840 – 22 March 1926), a Luxembourger mat ...
contains both of the isodynamic points.
[.]
If a circle is partitioned into three arcs, the first isodynamic point of the arc endpoints is the unique point inside the circle with the property that each of the three arcs is equally likely to be the first arc reached by a
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
starting at that point. That is, the isodynamic point is the point for which the
harmonic measure In mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical Dirichlet problem. In probability theory, the harmonic measure of a subset of the ...
of the three arcs is equal.
Construction
The circle of Apollonius through vertex
of triangle
may be constructed by finding the two (interior and exterior)
angle bisector
In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
s of the two angles formed by lines
and
at vertex
, and intersecting these bisector lines with line
. The line segment between these two intersection points is the diameter of the circle of Apollonius. The isodynamic points may be found by constructing two of these circles and finding their two intersection points.
Another compass and straight-edge construction involves finding the reflection
of vertex
across line
(the intersection of circles centered at
and
through
), and constructing an equilateral triangle inwards on side
of the triangle (the apex
of this triangle is the intersection of two circles having
as their radius). The line
crosses the similarly constructed lines
and
at the first isodynamic point. The second isodynamic point may be constructed similarly but with the equilateral triangles erected outwards rather than inwards.
Alternatively, the position of the first isodynamic point may be calculated from its
trilinear coordinate
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
s, which are
[.]
:
The second isodynamic point uses trilinear coordinates with a similar formula involving
in place of
.
Notes
References
*.
*.
*.
*.
*.
*.
*.
*.
*.
*. The definition of isodynamic points is in a footnote on page 204.
*. The discussion of isodynamic points is on pp. 138–139. Rigby calls them "
Napoleon points
In geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of ...
", but that name more commonly refers to a different triangle center, the point of concurrence between the lines connecting the vertices of
Napoleon's equilateral triangle with the opposite vertices of the given triangle.
*. See especiall
p. 498
External links
in the
Encyclopedia of Triangle Centers
The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville.
, the l ...
, by
Clark Kimberling
Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer seq ...
*{{mathworld, title=Isodynamic Points, urlname=IsodynamicPoints
Triangle centers