How could I apply basic linear algebra concepts, such as transformations and vector products, to find the area of a steiner inellipse (ellipse maximally inscribed in triangle) inside a scalene triangle?
e.g. if I have a triangle with sides 6, 8, 10, how could I find area of its steiner ellipse (without using upper-division proofs)?
Its area should be equal to Area(∆) * π/(3√3).
The Steiner inellipse is defined by the midpoints of the sides, that's an affine invariant. For an equilateral triangle, it's a circle due to symmetry. The ratio of the areas is an affine invariant, too, so the ratio is the same as for the equilateral triangle, having the value you mentioned (elementary calculation). So all you need is the area of your triangle, elementary, again (because it's a right triangle).