Delaunay Triangulation and smallest area triangle

Question

I have read that the Delaunay Triangulation maximizes the minimum angle meaning that the smallest angle in the triangulation will be at least as large as the smallest angle in any other triangulation. Does this mean that the smallest area triangle is also maximized?

From this post I gathered that the area of a triangle can be maximized by fixing one side and increasing the angle between it and another side up to 90 degrees and also that Delaunay Triangulation tend to avoid skinny triangles. However, I'm not convinced that this means the smallest area is maximized although I can't find a counter example.

Answer 1

This is false. Consider the counterexample below. Note that the Delaunay triangulation includes the convex hull and will either use the short or the long diagonal. The circle drawn here shows that the short diagonal is Delaunay. Clearly this yields a small area triangle at the top, whereas a triangulation using the long diagonal would yield $2$ equal area triangles, both with areas larger than that of the small one.