Smallest Triplet of Primitive Pythagorean Triangles with Same Area

Theorem
The smallest set of $3$ Pythagorean triangles which all have the same area are:


 * the $4485-5852-7373$ triangle


 * the $3059-8580-9109$ triangle


 * the $1380-19 \, 019-19 \, 069$ triangle.

That area is $13 \, 123 \, 110$.

Proof
We have that:
 * the $4485-5852-7373$ triangle $T_1$ is Pythagorean
 * the $3059-8580-9109$ triangle $T_2$ is Pythagorean
 * the $1380-19 \, 019-19 \, 069$ triangle $T_3$ is Pythagorean.

Then from Area of Triangle, their areas $A_1$, $A_2$ and $A_3$ respectively are given by: