Concept

Tangential trapezoid

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. Examples of tangential trapezoids are rhombi and squares. If the incircle is tangent to the sides AB and CD at W and Y respectively, then a tangential quadrilateral ABCD is also a trapezoid with parallel sides AB and CD if and only if and AD and BC are the parallel sides of a trapezoid if and only if 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, b, and any one of the other two sides has length c, then the area K is given by the formula (This formula can be used only in cases where the bases are parallel.) The area can be expressed in terms of the tangent lengths e, f, g, h as 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. The inradius can also be expressed in terms of the tangent lengths as Moreover, if the tangent lengths e, f, g, h emanate respectively from vertices A, B, C, D and AB is parallel to DC, then If the incircle is tangent to the bases at P, Q, then P, I, Q are collinear, where I is the incenter. The angles ∠ AID and ∠ BIC in a tangential trapezoid ABCD, with bases AB and DC, are right angles. The incenter lies on the median (also called the midsegment; that is, the segment connecting the midpoints of the legs). The median (midsegment) of a tangential trapezoid equals one fourth of the perimeter of the trapezoid. It also equals half the sum of the bases, as in all trapezoids.

Official source

https://en.wikipedia.org/wiki/Tangential_trapezoid

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In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers of these circles are called inradius and circumradius, and incenter and circumcenter respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent quadrilateral and inscribed and circumscribed quadrilateral.

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In geometry, a trapezoid (ˈtɹæpəzɔɪd) in USA and Canadian English, or trapezium (trəˈpiːziəm) in British and other forms of English, is a quadrilateral that has at least one pair of parallel sides. The parallel sides are called the bases of the trapezoid. The other two sides are called the legs (or the lateral sides) if they are not parallel; otherwise, the trapezoid is a parallelogram, and there are two pairs of bases. A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast with the special cases below.

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