What is the area of the smallest ellipse that can be circumscribed around a 3-4-5 triangle?

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What is the area of the largest ellipse that can be inscribed in a 3-4-5 triangle?

(In reply to

General Solution by Brian Smith)

Very interesting, and not that I disagree with it, but:

Given: that the minimized ellipse (a circle) in the transformed space is the answer in that space.

What guarantees that the corresponding ellipse in the inverted
transformation (in the original space) satisfies the requirements of
the problem?