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Tangential Quadrilateral

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TangentialQuadrilateral

A quadrilateral which has an incircle, i.e., one for which a single circle can be constructed which is tangent to all four sides. Opposite sides of such a quadrilateral satisfy

s=a+c=b+d,
(1)

where

s=1/2(a+b+c+d)
(2)

is the semiperimeter, and the area is

A=rs,
(3)

where r is the inradius. Using Bretschneider's formula together with (1) and (3) then gives the beautiful formula

r=(sqrt(4p^2q^2-(a^2-b^2+c^2-d^2)^2))/(2(a+b+c+d))
(4)
=(sqrt(p^2q^2-(a-b)^2(a+b-s)^2))/(2s),
(5)

where p and q are the diagonal lengths.

A rhombus is a special case of a tangential quadrilateral.

See also

Bicentric Quadrilateral, Cyclic Quadrilateral, Incircle, Quadrilateral, Rhombus, Tangential Triangle

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References

Harris, J. W. and Stocker, H. "Quadrilateral of Tangents." §3.6.8 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 86, 1998.

Referenced on Wolfram|Alpha

Tangential Quadrilateral

Cite this as:

Weisstein, Eric W. "Tangential Quadrilateral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TangentialQuadrilateral.html

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