Smallest Pythagorean Quadrilateral with Integer Sides

Theorem
The smallest Pythagorean quadrilateral in which the sides of the $4$ right triangles formed by its sides and perpendicular diagonals are all integers has an area of $21 \, 576$.

The sides of the right triangles in question are:
 * $25, 60, 65$
 * $91, 60, 109$
 * $91, 312, 325$
 * $25, 312, 313$

Proof

 * SmallestPythagoreanQuadrilateral.png

The $4$ right triangles are inspected:

The area of each right triangle is calculated:

Thus the total area is:
 * $750 + 2730 + 14 \, 196 + 3900 = 21 \, 576$