View source for Isodynamic point

{{short description|2 points about which a triangle can be inverted into an equilateral triangle}}
[[File:Isodynamic Point.svg|thumb|upright=1.35|
{{legend-line|solid lime|[[Circles of Apollonius]]; isodynamic points {{mvar|S}} and {{mvar|S'}} at their intersections}}
{{legend-line|solid blue|Interior [[angle bisectors]], used to construct the circles}}
{{legend-line|solid red|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 [[Inversive geometry|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 [[Similarity (geometry)|similar]] to each other have isodynamic points in corresponding locations in the plane, so the isodynamic points are [[triangle center]]s, and unlike other triangle centers the isodynamic points are also invariant under [[Möbius transformation]]s. 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 {{harvs|first=Joseph|last=Neuberg|authorlink=Joseph Jean Baptiste Neuberg|year=1885|txt}}.<ref>For the credit to Neuberg, see e.g. {{harvtxt|Casey|1893}} and {{harvtxt|Eves|1995}}.</ref>

==Distance ratios==
The isodynamic points were originally defined from certain equalities of ratios (or equivalently of products) of distances between pairs of points. If <math>S</math> and <math>S'</math> are the isodynamic points of a triangle <math>ABC,</math> then the three products of distances <math>AS\cdot BC=BS\cdot AC=CS\cdot AB</math> are equal. The analogous equalities also hold for <math>S'.</math><ref>{{harvtxt|Neuberg|1885}} states that this property is the reason for calling these points "isodynamic".</ref> Equivalently to the product formula, the distances <math>AS,</math> <math>BS,</math> and <math>CS</math> are inversely proportional to the corresponding triangle side lengths <math>BC,</math> <math>AC,</math> and <math>AB.</math>

<math>S</math> and <math>S'</math> are the common intersection points of the three [[circles of Apollonius]] associated with triangle of a triangle <math>ABC,</math> the three circles that each pass through one vertex of the triangle and maintain a constant ratio of distances to the other two vertices.<ref name="botjo"/> Hence, line <math>SS'</math> is the common [[radical axis]] for each of the three pairs of circles of Apollonius. The perpendicular bisector of line segment <math>SS'</math> is the [[circles of Apollonius|Lemoine line]], which contains the three centers of the circles of Apollonius.<ref name="wildberger"/>

==Transformations==
The isodynamic points <math>S</math> and <math>S'</math> of a triangle <math>ABC</math> may also be defined by their properties with respect to transformations of the plane, and particularly with respect to [[Inversion in a point|inversion]]s and [[Möbius transformation]]s (products of multiple inversions).
Inversion of the triangle <math>ABC</math> with respect to an isodynamic point transforms the original triangle into an [[equilateral]] [[triangle]].<ref name="casjo"/>
Inversion with respect to the [[circumcircle]] of triangle <math>ABC</math> leaves the triangle invariant but transforms one isodynamic point into the other one.<ref name="botjo">{{harvtxt|Bottema|2008}}; {{harvtxt|Johnson|1917}}.</ref>
More generally, the isodynamic points are [[equivariant]] under [[Möbius transformation]]s: the [[unordered pair]] of isodynamic points of a transformation of <math>ABC</math> is equal to the same transformation applied to the pair <math>\{S,S'\}.</math> The individual isodynamic points are fixed by Möbius transformations that map the interior of the circumcircle of <math>ABC</math> to the interior of the circumcircle of the transformed triangle, and swapped by transformations that exchange the interior and exterior of the circumcircle.<ref name="rigby"/>

==Angles==
[[File:Isodynamic-lenses.svg|thumb|Three circles, each making angles of π/3 with the circumcircle and each other, meet at the first isodynamic point.]]
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 <math>AB,</math> <math>AC,</math> and <math>BC,</math> where each of these circles intersects the [[circumcircle]] of triangle <math>ABC</math> to form a [[Lens (geometry)|lens]] 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.<ref name="rigby">{{harvtxt|Rigby|1988}}.</ref>

The angles formed by the first isodynamic point with the triangle vertices satisfy the equations <math>ASB = ACB + \pi/3,</math> <math>ASC = ABC + \pi/3,</math> and <math>BSC = BAC + \pi/3.</math> Analogously, the angles formed by the second isodynamic point satisfy the equations<math>AS'B = ACB - \pi/3,</math> <math>AS'C = ABC - \pi/3,</math> and <math>BS'C = BAC - \pi/3.</math><ref name="rigby"/>

The [[pedal triangle]] of an isodynamic point, the triangle formed by dropping perpendiculars from <math>S</math> to each of the three sides of triangle <math>ABC,</math> is equilateral,<ref name="casjo">{{harvtxt|Casey|1893}}; {{harvtxt|Johnson|1917}}.</ref> as is the triangle formed by reflecting <math>S</math> across each side of the triangle.<ref>{{harvtxt|Carver|1956}}.</ref> Among all the equilateral triangles inscribed in triangle <math>ABC,</math> the pedal triangle of the first isodynamic point is the one with minimum area.<ref>{{harvtxt|Moon|2010}}.</ref>

==Additional properties==
The isodynamic points are the [[isogonal conjugate]]s of the two [[Fermat point]]s of triangle <math>ABC,</math> and vice versa.<ref>{{harvtxt|Eves|1995}}; {{harvtxt|Wildberger|2008}}.</ref>

The [[Neuberg cubic]] contains both of the isodynamic points.<ref name="wildberger">{{harvtxt|Wildberger|2008}}.</ref>

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]] starting at that point. That is, the isodynamic point is the point for which the [[harmonic measure]] of the three arcs is equal.<ref>{{harvtxt|Iannaccone|Walden|2003}}.</ref>

Given a univariate polynomial <math>P(z) = z^3+az^2+bz+c</math> whose zeros are the vertices of a triangle <math>T</math> in the complex plane, the isodynamic points of <math>T</math> are the zeros of the polynomial <math>I(z) = (a^2-3b)z^2 + (ab-9c)z + b^2 - 3ac.</math> Note that <math>I(z)</math> is a constant multiple of <math>\mathrm{Discriminant}_u(nP(u) + (z-u)P'(u)) ,</math> where <math>n</math> is the degree of <math>P.</math> This construction generalizes isodynamic points to polynomials of degree <math>n\ge 3</math> in the sense that the zeros of the above discriminant are invariant under Möbius transformations. Here the expression <math>nP(u) + (z-u)P'(u)</math> is the polar derivative of <math>P(u)</math> with pole <math>z.</math><ref name="bss">{{harvtxt|Hägg|Shapiro|Shapiro|2023}}.</ref>

Equivalently, with <math>P</math> and <math>n</math> defined as above, the (generalized) isodynamic points of <math>P</math> are the [[Complex quadratic polynomial#Critical value|critical values]] of <math>f(z) = z-nP(z)/P'(z).</math> Here <math>f(z)</math> is the expression that appears in the relaxed Newton’s method with relaxation parameter <math>n.</math> A similar construction exists for rational functions instead of polynomials.<ref name="bss"/>

==Construction==
[[File:Isodynamic from reflections.svg|thumb|Construction of the isodynamic point from reflected copies of the given triangle and inwards-pointing equilateral triangles.]]
The circle of Apollonius through vertex <math>A</math> of triangle <math>ABC</math> may be constructed by finding the two (interior and exterior) [[Bisection|angle bisector]]s of the two angles formed by lines <math>AB</math> and <math>AC</math> at vertex <math>A,</math> and intersecting these bisector lines with line <math>BC.</math> 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.<ref name="botjo"/>

Another compass and straight-edge construction involves finding the reflection <math>A'</math> of vertex <math>A</math> across line <math>BC</math> (the intersection of circles centered at <math>B</math> and <math>C</math> through <math>A</math>), and constructing an equilateral triangle inwards on side <math>BC</math> of the triangle (the apex <math>A''</math> of this triangle is the intersection of two circles having <math>BC</math> as their radius). The line <math>A'A''</math> crosses the similarly constructed lines <math>B'B''</math> and <math>C'C''</math> at the first isodynamic point. The second isodynamic point may be constructed similarly but with the equilateral triangles erected outwards rather than inwards.<ref>{{harvtxt|Evans|2002}}.</ref>

Alternatively, the position of the first isodynamic point may be calculated from its [[trilinear coordinate]]s, which are<ref>{{harvtxt|Kimberling|1993}}.</ref>
<math display="block">\sin(A + \pi/3) : \sin(B + \pi/3) : \sin(C + \pi/3).</math>
The second isodynamic point uses trilinear coordinates with a similar formula involving <math>-\pi/3</math> in place of <math>\pi/3.</math>

==Notes==
{{reflist|colwidth=30em}}

==References==
{{refbegin|colwidth=30em}}
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*{{citation
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*{{citation
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*{{citation
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*{{citation
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*{{citation
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 }}. The discussion of isodynamic points is on pp. 138–139. Rigby calls them "[[Napoleon points]]", but that name more commonly refers to a different triangle center, the point of concurrence between the lines connecting the vertices of [[Napoleon's theorem|Napoleon's equilateral triangle]] with the opposite vertices of the given triangle.
*{{citation
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 }}. See especially [https://books.google.com/books?id=bKUQ-JpsbKEC&pg=PA498 p. 498].
{{refend}}

==External links==
{{commonscat|Isodynamic points}}

* [https://web.archive.org/web/20120419171900/http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X15#X15 Isodynamic points X(15) and X(16)] in the [[Encyclopedia of Triangle Centers]], by [[Clark Kimberling]]
*{{mathworld|title=Isodynamic Points|urlname=IsodynamicPoints}}

[[Category:Triangle centers]]