Hello, atreyyu!
My approach is the same as Plato's . . .
The vertices are: .Straight line passes through the center of a square with area 1.
Let denote the shortest distances between line and vertices resp.
Prove: .
The line through center (½, ½) with slope m has the equation:
. .
Formula: The distance from point to line
. . . . . . . . is given by: .
We have: .
The distance is: .
For
For
For
For
Hence: .
. .
. .