The Steiner Ellipses
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Each triangle has a unique circle, the circumcircle, through its vertices. Less well known is the unique ellipse, the Steiner ellipse, also through the vertices.
Steiner resoned that each triangle shape is a projection of an equilateral triangle. The projection turns the equilateral triangle with its circumcircle and incircle into an arbitrary triangle and the circles into ellipses of the same shape.
These ellipses, one inscribed, and one circumscribed, are the Steiner ellipses and are just as fundamental to the triangle as the circumcircle. The projection that turns an equilateral triangle into a triangle of arbitrary shape is called an affine triansformation. The Steiner ellipses are preserved under affine triansformations.
Although as important as the circumcircle, the Steiner ellipse is less well known because a circle is easy to draw but an ellipse is not. We are not taught it in school; it was the most important thing you were never taught, about geometry at least.
Important features of the Steiner ellipses:
The affine plane includes the line at infinity. Add triangle ABC and much triangle structure appears. The most important is the istomic conjugate (an affine invariant opertion) of the line at infinity. The conjugate of a line is a circumconic. The conjugate of this particular line is the Steiner ellipse.
The isotomic conjugate of any point on the Steiner circumellipse is on the line at infinity.
The circumellipse has the smallest area of any circumscribed conic and the inellipse the largest area of any inellipse.
These ellipses are strong objects.
The inscribed ellipse meets the sides at their midpoints mA, mB, mC. The circumscribed ellipse has tangents at the vertices parallel to the corresponding side.
The inellipse is the medial of the circumellipse, i.e., the circumellipse of the medial triangle.
The Steiner circumellipse and the circumcircle meet four times, three of which are A, B, and C. The fourth intersection S is known as the Steiner point, itself an important and diagnostic feature of the triangle.
The axes of the Steiner ellipses meet at the centroid G, which is the center of each ellipse.
The dual of each point on a Steiner axis is perpendicular to that axis.
Conic variables for the Steiner ellipse
The following picture shows the Euclidean variables for the Steiner ellipse. It was worked out by John Conway for the forthcoming The Triangle Book. Ellipse parameters such as a, referring to the length of the semi-major axis, are written in Greek letters to avoid conflict with letters usually reserved for the triangle edges.
The Steiner ellipse is a strong object. John Conway calls this a "Brocardian object" because its parameters depend on SW and Q, both of which depend on the Brocard angle W of the triangle. Here
SW = S cot W = (a2 + b2 + c2)/2, where S is twice the triangle area.
Q = a2 + b2 + c2 – bc – ca – ab
The relation between Q, SW, and S is Q2 = SW2 – 3 S2.
The area of the Steiner ellipse = πab = π 4/3√(SW2 – Q2) = 4π √3∆/3, and is slightly less than twice as big as the area of the triangle. The inellipse is 1/4 as big.
Figure: Here are the variables [computed by John Conway] of the Steiner circumellipse in terms of the basic triangle parameter SW and the Steiner ellipse parameter Q, defined above. Both of these are functions of the Brocard angle. The ellipse parameters usually written as a, b, c, and e are written with greek letters to avoid resonance with the symbols usually reserved for the triangle edges. d (or delta) is the radius of the director circle of the Steiner ellipse, which is shown in blue. The director circle can be constructed as the circumcircle of the bounding rectangle of the ellipse.
General Principles
A more complete version of these general affine principles is found here.
For almost all P in the triangle plane, there are two natural lines: G—P and ~P where the "~" symbol means "dual."
Note: I prefer the affine invariant dual to the projectively invariant tripolar. The dual of (l:m:n) is lx + my + nz = 0. Note: "natural" in this context means "leading to structure which is affinely preserved."
The isotomic conjugates of the infinite points on these two lines are antipodal on the Steiner ellipse. This allows us to associate a particular point in the affine plane with antipodal points on the Steiner ellipse.
The dual of each point on G—P is parallel to ~P.
P(l:m:n) –> P2–(l2–mn : m2–nl : n2–lm) is inversion in the Steiner ellipse. Both points are on G—P.
G—P and ~P intersect at mP2–, the medial (complement) of P2–.
Figure: the two conspicuous lines are G—P and ~P, the dual of P. The conjugates of the infinite points on the two lines are shown and are antipodal on the Steiner ellipse.
E+ = P2– + Z P and E– = P2– – Z P . where Z = √( l2 + m2 + n2 – mn – nl – lm) are the intersections of G—P with the Steiner ellipse.
The conjugates of E± are at infinity. That of E+ is the infinite point on E–. These are the infinite points on the axes of the conic that is the isotomic of G—P. This statement can be checked by computation.
The dual of the isotomic of E+ is the line through G with the correct infinite point. The axes through the conic center will be parallel to these lines. The particular case we are interested in has P = K, the symmedian point in which case Z is Q = √( a4 + b4 + c4 – a2b2 – b2c2 – c2a2). The relation between Q, SW , and S, twice the area, is Q2 = SW2 – 3S2 .
The Steiner axes and the Kiepert asymptotes
The axes of a circumconic, perspector P, are constructed from the intersections of the line dP—dK with the Steiner ellipse, where "d" is the dilated, or anticomplementary operation. Since dK is always inside this ellipse, this construction is always possible. The asymptotes for a circumconic, perspector P, are constructed from the interesections of ~P, if any, with the Steiner ellipse.
For the circumconic with perspector G, the Steiner ellipse, the line dP—dK is the same as G—K. The Kiepert hyperbola asymptotes are also generated from G—K, the dual of the Kiepert perspector. This insures that the Kiepert asymptotes and the Steiner axes go through the same points at infinity. Hence the asymptotes of the Kiepert hyperbola and the axes of the Steiner ellipse are parallel.
Now let P = K, then the points named above become
~K•∞ = : c2 – a2 : the isotomic Steiner point tS
(G—K)•∞ = : c2 + a2 – 2 b2 :, X(524)
t (~K•∞) = : 1/ (c2–a2) : = S, the Steiner point.
t((G—K)•∞) = :1/(c2 + a2 – 2 b2 ): , antipodal to the above, X(671).
E+ = K2– + Q K = :(b4–c2a2 + b2Q:) and E– = K2– – Q K = :(b4–c2a2 – b2Q:)
E+ and E– are the isotomic conjugates of X530 and X531, the infinite points on the Kiepert hyperbola because the Kiepert hyperbola is the istomic of the line G—K.
The duals of X530 and X531 are the Steiner axes since they have the same directions as the Kiepert hyperbola asymptotes. The equations of the axes thus can easily be found as ~tE1 and ~tE2
My formulas for the axes of the Steiner ellipse are
... + y / (b4–c2a2 + b2Q) + .... = 0,
with Q being oppositely signed for the other axis.
The endpoint of this line is : 1/ (b4 – c2a2 – b2Q) : which has the opposite sign as the axis so that the endpoint of an axis is the dual of the other axis. Since the line goes through the centroid, the point must be on the line at infinity, which is the dual of the centroid.
The relation between the Steiner ellipse and the Circumcircle.
The pro operation p = gt, the isogonal conjugate of the isotomic conjugate, is the projective operation that, among other things, takes G to K and the Steiner ellipse to the circumcircle. The line through corresponding points goes through S. Hence one can construct the corresponding point using this line. This picture is taken from this GSP document.
