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Brahmagupta's Trapezium

A quadrilateral whose consecutive sides have the lengths a_1b_3, a_3b_2, a_2b_3, a_3b_1, where

a_1^2+a_2^2=a_3^2
(1)

and

b_1^2+b_2^2=b_3^2.
(2)

Brahmagupta's trapezium is a cyclic quadrilateral with perpendicular diagonals.

It has area

A=1/2(a_1a_2b_3^2+b_1b_2a_3^2),
(3)

circumradius,

R=1/2a_3b_3,
(4)

and the diagonal lengths

p=a_1b_2+a_2b_1
(5)
q=a_1b_1+a_2b_2.
(6)

All these values are rational if a_1,a_2,a_3 and b_1,b_2,b_3 are. In particular, if a_1,a_2,a_3 and b_1,b_2,b_3 are Pythagorean triples, the area, circumdiameter, the lengths of the diagonals are all integers.

See also

Brahmagupta's Formula, Brahmagupta's Theorem, Cyclic Quadrilateral, Trapezium

This entry contributed by Margherita Barile

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References

Eves, H. An Introduction to the History of Mathematics, 3rd ed. New York: Holt, Rinehart and Winston, 1969.

Referenced on Wolfram|Alpha

Brahmagupta's Trapezium

Cite this as:

Barile, Margherita. "Brahmagupta's Trapezium." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BrahmaguptasTrapezium.html

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