Steiner Inellipse is Unique

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Theorem

For every triangle, there is one and only one Steiner inellipse.


Proof (outline)

Take your triangle and apply an affine transformation to map it to an equilateral triangle.

Inscribe an incircle in that equilateral triangle.

Perform the inverse affine transformation to map the equilateral triangle (complete with incircle) back to the triangle you started with.

By properties of affine transformations the circle will be a Steiner inellipse.