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Extremely clever, Chris, but then I would expect no less from you!Stretch the vertical axis by a factor $a/b$. Then the ellipse becomes a circle of radius $a$, and the area of the triangle is minimised when the triangle is equilateral with height $3a$. Now shrink the vertical axis back to where it started (by a factor $b/a$), and the minimising triangle then has height $3b.$