page_title: Tangential trapezoid
section_title: Isosceles tangential trapezoid
hierarchical_section_title: Tangential trapezoid / Isosceles tangential trapezoid
context_page_description: 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, but they don't have to be.
context_section_description: 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
caption: Every isosceles tangential trapezoid is bicentric.