Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Tangential trapezoid
Summary
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
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.