Definition:Pythagorean Quadrilateral

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Definition

A Pythagorean quadrilateral is a convex quadrilateral whose diagonals intersect at right angles, and is formed by fitting four right triangles with integer-valued sides.

The interest is finding solutions where no two triangles are similar.

The smallest Pythagorean quadrilateral is formed by the triangles with side lengths:

$\tuple {25, 60, 65}, \tuple {91, 60, 109}, \tuple {91, 312, 325}, \tuple {25, 312, 313}$

The smallest primitive Pythagorean quadrilateral, where each Pythagorean triple is primitive is:

$\tuple {28435, 20292, 34933}, \tuple {284795, 20292, 285517}, \tuple {284795, 181908, 337933}, \tuple {28435, 181908, 184117}$

The smallest anti-primitive Pythagorean quadrilateral, where no Pythagorean triples are primitive is:

$\tuple {1209, 6188, 6305}, \tuple {10659, 6188, 12325}, \tuple {10659, 23560, 25859}, \tuple {1209, 23560, 23591}$

with common divisors:

$13, 17, 19, 31$

Sources

• 1989: Andy Pepperdine: Pythagorean Quadrilaterals (J. Recr. Math. Vol. 21: pp. 8 – 12)