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[[File:Tangential trapezoid.svg|thumb|right|300px|A 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 a special case of a [[tangential quadrilateral]], where at least one pair of opposite sides are [[parallel (geometry)|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 [[Tangential trapezoid#Isosceles tangential trapezoid|isosceles tangential trapezoid]] below), but they don't have to be.

==Special cases==
Examples of tangential trapezoids are [[rhombus|rhombi]] and [[square (geometry)|squares]].

==Characterization==
A convex quadrilateral is a tangential trapezoid [[if and only if]] opposite sides satisfy [[Pitot's theorem]] (so it is tangential) ''and'' it has two adjacent angles that are [[supplementary angles|supplementary]] (then this is also true for the other two angles) (so it is a trapezoid). Hence ''AB'' and ''CD'' are the bases in a tangential trapezoid ''ABCD'' if and only if
:<math>
\begin{cases}
AB+CD=BC+DA\\
A+D=B+C=\pi.
\end{cases}
</math>

==Area==
The formula for the [[Trapezoid#Area|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{{citation needed|date=May 2012}}
:<math>K=\frac{a+b}{|b-a|}\sqrt{ab(a-c)(c-b)}.</math>

The area can be expressed in terms of the [[Quadrilateral#Special line segments|tangent lengths]] ''e'', ''f'', ''g'', ''h'' as<ref name=Josefsson>{{citation
|last=Josefsson |first=Martin
|journal=Forum Geometricorum
|pages=119–130
|title=Calculations concerning the tangent lengths and tangency chords of a tangential quadrilateral
|url=http://forumgeom.fau.edu/FG2010volume10/FG201013.pdf
|volume=10
|year=2010}}.</ref>{{rp|p.129}}
:<math>K=\sqrt[4]{efgh}(e+f+g+h).</math>

==Inradius==
Using the same notations as for the area, the radius in the incircle is{{citation needed|date=May 2012}}
:<math>r=\frac{K}{a+b}=\frac{\sqrt{ab(a-c)(c-b)}}{|b-a|}.</math>

The [[diameter]] of the incircle is equal to the height of the tangential trapezoid.

The inradius can also be expressed in terms of the [[Quadrilateral#Special line segments|tangent lengths]] as<ref name=Josefsson/>{{rp|p.129}}
:<math>r=\sqrt[4]{efgh}.</math>

==Properties of the incenter==
If the incircle is tangent to the bases at ''P'' and ''Q'', then ''P'', ''I'' and ''Q'' are [[Line (geometry)#Collinear points|collinear]], where ''I'' is the incenter.<ref name=Wilson>J. Wilson, ''Problem Set 2.2'', The University of Georgia, 2010, [http://jwilson.coe.uga.edu/MATH7200/ProblemSet2.2.html].</ref>

The angles ''AID'' and ''BIC'' in a tangential trapezoid ''ABCD'', with bases ''AB'' and ''DC'', are [[right angle]]s.<ref name=Wilson/>

The incenter lies on the median (also called the midsegment; that is, the segment connecting the [[midpoint]]s of the legs).<ref name=Wilson/>

==Other properties==
The [[Trapezoid#Midsegment and height|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.

If two circles are drawn, each with a diameter coinciding with the legs of a tangential trapezoid, then these two circles are [[tangent]] to each other.<ref>Chernomorsky Lyceum, ''Inscribed and circumscribed quadrilaterals'', 2010, [http://math.chernomorsky.com/index.php?option=com_content&view=article&id=120:inscribed-and-circumscribed-quadrilaterals&catid=62:geometry-homework-8th].</ref>

==Right tangential trapezoid==
[[File:Right tangential trapezoid 001.svg|thumb|right|250px|A right tangential trapezoid.]]
A '''right tangential trapezoid''' is a tangential trapezoid where two adjacent angles are [[right angles]]. If the bases has lengths ''a'' and ''b'', then the inradius is<ref name=AoPS>Circle inscribed in a trapezoid, ''Art of Problem Soving'', 2011,
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=399084]</ref>
:<math>r=\frac{ab}{a+b}.</math>

Thus the [[diameter]] of the incircle is the [[harmonic mean]] of the bases.

The right tangential trapezoid has the [[area]]<ref name=AoPS/>
:<math>\displaystyle K=ab</math>

and its [[perimeter]] ''P'' is<ref name=AoPS/>
:<math>\displaystyle P=2(a+b).</math>

==Isosceles tangential trapezoid==
[[File:Bicentric isosceles trapezoid 001.svg|thumb|right|250px|Every [[Isosceles trapezoid|isosceles]] tangential trapezoid is [[Bicentric quadrilateral|bicentric]].]]
An '''isosceles tangential trapezoid''' is a tangential trapezoid where the legs are equal. Since an [[isosceles trapezoid]] is [[cyclic quadrilateral|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<ref>MathDL, ''Inscribed circle and trapezoid'', The Mathematical Association of America, 2012, [http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=2287&pf=1].</ref>
:<math>r=\tfrac{1}{2}\sqrt{ab}.</math>

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.

==References==
{{reflist}}

[[Category:Quadrilaterals]]

Decoupling (analysis) - Revision history

en>David Eppstein: unreferenced, stub sort