Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

, a tangential trapezoid, also called a circumscribed trapezoid, is a

trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a Convex polygon, convex quadri ...

whose four sides are all

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...

to a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

within the trapezoid: the

incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

or ''inscribed circle''. It is the special case of a

tangential quadrilateral In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the ...

in which at least one pair of opposite sides are

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IBM ...

. 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.

Special cases

Examples of tangential trapezoids are

rhombi In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...

and

squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

Characterization

If the incircle is tangent to the sides and at and respectively, then a tangential quadrilateral is also a

with parallel sides and if and only if :

\overline \cdot \overline = \overline \cdot \overline

and and are the parallel sides of a trapezoid if and only if :

\overline \cdot \overline = \overline \cdot \overline.

Area

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 , and any one of the other two sides has length , then the area is given by the formulaH. Lieber and F. von Lühmann, ''Trigonometrische Aufgaben'', Berlin, Dritte Auflage, 1889, p. 154. (This formula can be used only in cases where the bases are parallel.) :

K=\frac\sqrt.

The area can be expressed in terms of the tangent lengths as. :

K=\sqrt e+f+g+h).

Inradius

Using the same notations as for the area, the radius in the incircle is :

r=\frac=\frac.

The

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...

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 :

r=\sqrt

Moreover, if the tangent lengths emanate respectively from vertices and is parallel to , then. :

r=\sqrt=\sqrt.

Properties of the incenter

If the incircle is tangent to the bases at , then are

collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...

, where is the incenter. The angles and in a tangential trapezoid , with bases and , are

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...

s. The incenter lies on the median (also called the midsegment; that is, the segment connecting the

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimens ...

s of the legs).

Other properties

The

median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...

(midsegment) of a tangential trapezoid equals one fourth of the

perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pract ...

of the trapezoid. It also equals half the sum of the bases, as in all trapezoids. If two circles are drawn, each with a diameter coinciding with the legs of a tangential trapezoid, then these two circles are

to each other.

Right tangential trapezoid

A right tangential trapezoid is a tangential trapezoid where two adjacent angles are

right angles In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. T ...

. If the bases have lengths , then the inradius is :

r=\frac.

Thus the

of the incircle is the

harmonic mean In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired. The harmonic mean can be expressed as the recipro ...

of the bases. The right tangential trapezoid has the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

\displaystyle K=ab

and its

is :

\displaystyle P=2(a+b).

Isosceles tangential trapezoid

An isosceles tangential trapezoid is a tangential trapezoid where the legs are equal. Since an

isosceles trapezoid In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined ...

cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in soc ...

, an isosceles tangential trapezoid is a

bicentric quadrilateral In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called ''inradius'' and ''circumradius'', and ''incenter'' and ''circumcenter'' r ...

. That is, it has both an incircle and a

circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

. If the bases are , then the inradius is given by :

r=\tfrac\sqrt.

To derive this formula was a simple

Sangaku Sangaku or San Gaku ( ja, 算額, lit=calculation tablet) are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period by members of all social classes ...

problem from

Japan Japan ( ja, 日本, or , and formally , ''Nihonkoku'') is an island country in East Asia. It is situated in the northwest Pacific Ocean, and is bordered on the west by the Sea of Japan, while extending from the Sea of Okhotsk in the north ...

. 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 In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...

of the product of the bases, an isosceles tangential trapezoid gives a nice geometric interpretation of the

arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...

and

geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...

of the bases as the length of a leg and the diameter of the incircle respectively. The area of an isosceles tangential trapezoid with bases is given byAbhijit Guha, ''CAT Mathematics'', PHI Learning Private Limited, 2014, p. 7-73. :

K=\tfrac\sqrt(a+b).

References

{{Polygons Types of quadrilaterals