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:
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:
\(\ds A_1\) | \(=\) | \(\ds \dfrac {4485 \times 5852} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {3 \times 5 \times 13 \times 23} \times \paren {2^2 \times 7 \times 11 \times 19} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 19 \times 23\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 13 \, 123 \, 110\) |
\(\ds A_2\) | \(=\) | \(\ds \dfrac {3059 \times 8580} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {7 \times 19 \times 23} \times \paren {2^2 \times 3 \times 5 \times 11 \times 13} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 19 \times 23\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 13 \, 123 \, 110\) |
\(\ds A_3\) | \(=\) | \(\ds \dfrac {1380 \times 19 \, 019} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {2^2 \times 3 \times 5 \times 23} \times \paren {7 \times 11 \times 13 \times 19} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 19 \times 23\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 13 \, 123 \, 110\) |
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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
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $13,123,110$