This puzzle is solved using coordinate geometry and invoking the
properties of the AFFINE transformation.
X = a*x + b*y + c
Y = d*x + e*y + f
This transformation has the property that the ratio of any two areas is unchanged by transformation. Furthermore:
Straight lines transform to straight lines
Ellipses (and circles) transform to ellipses
The transformation has 6 parameters (a,b,c,d,e,f) and this means that any three points can be transformed into any other three points. Thus, any triangle can be transformed into any other triangle.
Consider an equilateral triangle of side 1 unit.
By symmetry, the largest inscribed ellipse is a circle and the smallest ellipse through the vertices is a circle.
The area of the inscribed circle is Pi/12
The area of the triangle is a quarter of the Square root of 3. (approx 0.4330)
The area of the circumcircle circle is Pi/3.
There is an affine transformation which will transform an equilateral triangle into a 5,12,13 triangle and the ratios of these areas will not be altered by this transformation. The area of a 5,12,13 triangle is 30 square units. Hence, the area of the smallest circumscribing ellipse is 72.55 to 4 significant figures,
and the area of the largest inscribed ellipse is 18.14 to 4 significant figures.
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