the ellipse y² – zx and the hyperbola y² + zx

This ellipse has so many striking properties that it has changed my thinking about many things.

1. It is congruent to the Steiner ellipse.

2. It's center is on the Steiner ellipse.

3. The isotomic and b-harmonic conjugates of points on the curve are on the curve.

4. If : m : is on the curve, so are all powers : mⁱ : .

5. It goes through A, C, G, and B^G, the B-harmonic associate of G. It is tangent to edges a and c at C and A.

This picture shows the two graphs along with the orbit of an arbitrary point P under the group of triangle centers under barycentric multiplication. more later i have to go home.