Another way to see the non-uniqueness of the triangle is to take any triangle containing the ellipse, and move its sides towards the ellipse until each one is first tangent to it. --Dan Rich wrote: << answering one question ...

Given an ellipse, is the triangle that the ellipse is inscribed within unique?

No. Start with an equilateral triangle, and the inscribed circle. Choose any direction, and stretch the diagram in that direction, while leaving perpendicular distances alone. This operation preserves some geometric properties, and mangles others. Tangency and midpoints are preserved. The inscribed circle becomes an inscribed ellipse. Foci are not preserved. We can get an ellipse of prescribed shape, in any direction. Working backwards, knowing an ellipse alone doesn't determine the orientation of the circumscribed "midpoint triangle".

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