This question is somewhat dual to my previously stated question about https://mathoverflow.net/questions/282195, where the triangle and parallelogram each is assumed to be of unit area.

Now I ask: 

> **Question.** Suppose a parallelogram is of area 1 (say, the unit square if you prefer) and a triangle is of area $1$ as well, then what is the minimum area of the convex hull of the union of the parallelogram and the triangle?

It is obvious that the minimum exists and is greater than $1$. The best I can do is $\sqrt{2}$ $-$ is this the minimum? 

My example is: the unit square and the right triangle with legs of length $\sqrt2$ each, the triangle's right angle coinciding with the square's corner. (Added by Joe's request.)