Example 4.1.5 Let $\Omega$ be a given ellipse. Find (1) the maximum value of the area of an inscribed triangle in $\Omega$; 
(2) the minimum value of the area of a circumscribed triangle around $\Omega$.
(1) When $\Omega$ is a circle with radius $r$, it can be proven that the area of the inscribed triangle is maximized when the triangle is equilateral. The maximum area is $\frac{3 \sqrt{3}}{4} r^{2}$, which is $\frac{3 \sqrt{3}}{4 \pi}$ times the area of the circle.

Since the area ratio and the cross ratio of collinear points are affine invariants, when this figure is affinely transformed into an ellipse, the area of the inscribed triangle is maximized only when the centroid of the inscribed triangle coincides with the center of $\Omega$. The maximum area is $\frac{3 \sqrt{3}}{4 \pi}$ times the area of the ellipse.
(2) Similarly, the area of the circumscribed triangle is minimized only when the centroid of the circumscribed triangle coincides with the center of $\Omega$. The minimum area is $\frac{3 \sqrt{3}}{4 \pi}$ times the area of the ellipse.