In Euclidean geometry, a tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the incircle or inscribed circle. It is the special case of a tangential quadrilateral in which at least one pair of opposite sides are parallel. As for other trapezoids, the parallel sides are called the bases and the other two sides the legs. The legs can be equal (see isosceles tangential trapezoid below), but they don't have to be.
In turn, a tangential quadrilateral is a trapezoid if and only if either of the following two properties hold (in which case they both do):
- It has two adjacent angles that are supplementary (then this is also true for the other two angles). Specifically, a tangential quadrilateral ABCD is a trapezoid with parallel bases AB and CD if and only if
- The product of two adjacent tangent lengths equals the product of the other two tangent lengths. Specifically, if e, f, g, h are the tangent lengths emanating from A, B, C, D respectively in a tangential quadrilateral ABCD, then AB and CD are the parallel bases of a trapezoid if and only if:Thm. 2
The formula for the area of a trapezoid can be simplified using Pitot's theorem to get a formula for the area of a tangential trapezoid. If the bases have lengths a and b, and any one of the other two sides has length c, then the area K is given by the formula
The area can be expressed in terms of the tangent lengths e, f, g, h as:p.129
Using the same notations as for the area, the radius in the incircle is
The diameter of the incircle is equal to the height of the tangential trapezoid.
Moreover, if the tangent lengths e, f, g, h emanate respectively from vertices A, B, C, D and AB is parallel to DC, then
Properties of the incenter
Right tangential trapezoid
Isosceles tangential trapezoid
An isosceles tangential trapezoid is a tangential trapezoid where the legs are equal. Since an isosceles trapezoid is cyclic, an isosceles tangential trapezoid is a bicentric quadrilateral. That is, it has both an incircle and a circumcircle.
If the bases are a and b, then the inradius is given by
To derive this formula was a simple Sangaku problem from Japan. From Pitot's theorem it follows that the lengths of the legs are half the sum of the bases. Since the diameter of the incircle is the square root of the product of the bases, an isosceles tangential trapezoid gives a nice geometric interpretation of the arithmetic mean and geometric mean of the bases as the length of a leg and the diameter of the incircle respectively.
The area K of an isosceles tangential trapezoid with bases a and b is given by
- Josefsson, Martin (2014), "The diagonal point triangle revisited" (PDF), Forum Geometricorum, 14: 381–385.
- H. Lieber and F. von Lühmann, Trigonometrische Aufgaben, Berlin, Dritte Auflage, 1889, p. 154.
- Josefsson, Martin (2010), "Calculations concerning the tangent lengths and tangency chords of a tangential quadrilateral" (PDF), Forum Geometricorum, 10: 119–130.
- J. Wilson, Problem Set 2.2, The University of Georgia, 2010, .
- Chernomorsky Lyceum, Inscribed and circumscribed quadrilaterals, 2010, .
- Circle inscribed in a trapezoid, Art of Problem Soving, 2011
- MathDL, Inscribed circle and trapezoid, The Mathematical Association of America, 2012, .
- Abhijit Guha, CAT Mathematics, PHI Learning Private Limited, 2014, p. 7-73.