Smallest Triplet of Primitive Pythagorean Triangles with Same Area

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Theorem

The smallest set of $3$ primitive 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:

\(\ds A_1\) \(=\) \(\ds \dfrac {4485 \times 5852} 2\)
\(\ds \) \(=\) \(\ds \dfrac {\paren {3 \times 5 \times 13 \times 23} \times \paren {2^2 \times 7 \times 11 \times 19} } 2\)
\(\ds \) \(=\) \(\ds 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 19 \times 23\)
\(\ds \) \(=\) \(\ds 13 \, 123 \, 110\)


\(\ds A_2\) \(=\) \(\ds \dfrac {3059 \times 8580} 2\)
\(\ds \) \(=\) \(\ds \dfrac {\paren {7 \times 19 \times 23} \times \paren {2^2 \times 3 \times 5 \times 11 \times 13} } 2\)
\(\ds \) \(=\) \(\ds 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 19 \times 23\)
\(\ds \) \(=\) \(\ds 13 \, 123 \, 110\)


\(\ds A_3\) \(=\) \(\ds \dfrac {1380 \times 19 \, 019} 2\)
\(\ds \) \(=\) \(\ds \dfrac {\paren {2^2 \times 3 \times 5 \times 23} \times \paren {7 \times 11 \times 13 \times 19} } 2\)
\(\ds \) \(=\) \(\ds 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 19 \times 23\)
\(\ds \) \(=\) \(\ds 13 \, 123 \, 110\)




Historical Note

The smallest triplet of primitive Pythagorean triangles with the same area was reported by Martin Gardner as having been discovered by Charles L. Shedd in $1945$.


Sources