If C is a circumellipse, with centre C and perspector P, then
A point of the form P with U+V+W = 2Kπ lies inside the Steiner
Thus C is a circumellipse if and only if the centre C has the form
C = [sin(U),sin(V),sin(W)], P = [sin2(U/2),sin2(V/2),sin2(W/2)],
where U+V+W = 0.
Inellipse, so is the
perspector of a circumellipse.
[sin(U),sin(V),sin(W)] with U+V+W = Kπ.
The real circumellipse meets the line at infinity in conjugate imaginary points.
The tangents are therefore conjugate imaginary asymptotes of C.
But we know that the asymptotes have the form xf+yg+zh = 0, and
x/f+y/g+z/h = 0. Since these are conjugate, we must have, for some λ,
f* = λ/f, g* = λ/g, h* = λ/h, so that |f| = |g| =|h|.
We may as well scale so that the common modulus is 1. Then we have
f = eiu, g = eiv, h = eiw for some real u,v,w.
The centre of C is the intersection of the asymptotes. The x-coordinate
is ei(v-w) - ei(w-v) = 2isin(v-w). This gives the stated form for C once we
write
U = v-w, V = w-u, W = u-v, so U+V+W = 0.
The infinite point on the first asymptote has x-coordinate eiv - eiw.
This can be rewritten as 2iei(v-w)/2sin((v-w)/2).
From earlier results, we know that the perspector is the barycentric product
of the two points at infinity. Since here these are conjugate, we get the
x-coordinate of P as required. The others are similarly proved.
For the last part, we begin by writing
F(x,y,z) = x2 + y2 + z2 - 2xy - 2yz - 2zx.
Then the Steiner Inellipse S' has equation F(x,y,z) = 0.
We can easily verify that
F(r2,s2,t2) = (r+s+t)(r+s-t)(r-s+t)(r-s-t).
It follows that the points of S' are the barycentric squares of points on
the line at infinity.
If we use the coordinates of P, we get, after quite a lot of trigonometry
F(P) = -4(sin(U/2)sin(V/2)sin(W/2))2, which is always negative.
Any circumconic C is the isotomic conjugate of a line L. The infinite points
on C arise from intersections of L with the Steiner Ellipse S : yz+zx+xy = 0.
If C has perspector P = [p,q,r] the line L has equation px+qy+rz = 0, we
eliminate z and get a quadratic for x, y. The discriminant is F(p,q,r).
Thus C is a circumellipse if and only if F(p,q,r) < 0.
It follows that a perspector of the given form corresponds to a circumellipse.
Now, G = [1,1,1] is inside S', and F(1,1,1) = -3. Thus the condition for a
circumellipse is that the perspector lies inside S'.