Is Steiner ellipse defining a unique triangle?

Question

Steiner ellipse is a unique ellipse that touches the triangle at its vertices and whose centre is the triangle's centroid.

Similar is the case for the Steiner inellipse. However, this one seems to be uniquely linked to the Steiner ellipse, i.e. it carries no additional info.

So my question is whether these Steiner ellipses define a unique triangle? It seems to me that I still have an extra degree of freedom to fix the triangle. What is this degree of freedom once I have fixed the Steiner ellipse?

In other words, once I have a Steiner ellipse how can I parametrised all triangles associated with it?

Answer 1

Different congruent equilateral triangles with the same centroid clearly have the same Steiner circumellipse and inellipse

This will then be true in every other case too: start with a triangle and its Steiner ellipses, affine transform the ellipses to circles (the triangle becoming equilateral), rotate the triangle, and then undo the transformation to recover the original ellipses: you will have a new triangle with the original Steiner ellipses

So perhaps you could parametrise by the orientation of the transformed equilateral triangle or some function of this

Answer 2

In fact, this is related to a classical issue, called "Poncelet porism" or "Poncelet closure theorem" (See here). This theorem asserts that if 2 ellipses included one in the other have a triangle inscribed in one and circumscribed to the other, one can build an infinity of such triangles; more precisely, any point taken on the exterior ellipse can be a vertex of such a triangle.

This theorem can be generalized in two different ways: to other conic curves, and to any $n$-sided polygon.

Take a look at this presentation for a very enlarged point of view... and nice graphics.

For a connection between this theorem and another famous one, Pascal's theorem, under the condition to be a little familiar with projective geometry, see this publication.