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
The $4$ right triangles are inspected:
| \(\ds 25^2 + 60^2\) | \(=\) | \(\ds 625 + 3600\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 4225\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 65^2\) |
| \(\ds 91^2 + 60^2\) | \(=\) | \(\ds 8281 + 3600\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 11 \, 881\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 109^2\) |
| \(\ds 91^2 + 312^2\) | \(=\) | \(\ds 8281 + 97 \, 344\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 105 \, 625\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 325^2\) |
| \(\ds 25^2 + 312^2\) | \(=\) | \(\ds 625 + 97 \, 344\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 97 \, 969\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 313^2\) |
The area of each right triangle is calculated:
| \(\ds \dfrac {25 \times 60} 2\) | \(=\) | \(\ds 750\) | ||||||||||||
| \(\ds \dfrac {91 \times 60} 2\) | \(=\) | \(\ds 2730\) | ||||||||||||
| \(\ds \dfrac {91 \times 312} 2\) | \(=\) | \(\ds 14 \, 196\) | ||||||||||||
| \(\ds \dfrac {25 \times 312} 2\) | \(=\) | \(\ds 3900\) |
Thus the total area is:
- $750 + 2730 + 14 \, 196 + 3900 = 21 \, 576$
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Historical Note
David Wells reports in his Curious and Interesting Numbers, 2nd ed. of $1997$ that this result was published in Journal of Recreational Mathematics Volume $21$ no. $9$ by Hugh ApSimon, but this has not been corroborated.
It is also uncertain at this stage exactly what a Pythagorean quadrilateral actually is.
Sources
- 1989: Andy Pepperdine: Pythagorean Quadrilaterals (J. Recr. Math. Vol. 21: pp. 8 – 12)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $21,576$