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

, a tangential polygon, also known as a circumscribed polygon, is a

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is ...

that contains an inscribed circle (also called an ''incircle''). This is a circle that is

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

to each of the polygon's sides. The dual polygon of a tangential polygon is a

cyclic polygon In geometry, a set (mathematics), set of point (geometry), points are said to be concyclic (or cocyclic) if they lie on a common circle. A polygon whose vertex (geometry), vertices are concyclic is called a cyclic polygon, and the circle is cal ...

, which has a circumscribed circle passing through each of its vertices. All

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

s are tangential, as are all

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

s with any number of sides. A well-studied group of tangential polygons are the tangential quadrilaterals, which include the

rhombi In plane Euclidean geometry, a rhombus (: 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 rhom ...

and

kites A kite is a tethered heavier than air flight, heavier-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. Kites often have ...

Characterizations

A convex polygon has an incircle

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

all of its internal angle bisectors are concurrent. This common point is the ''incenter'' (the center of the incircle). There exists a tangential polygon of ''n'' sequential sides ''a''₁, ..., ''a''_''n'' if and only if the system of equations :

x_1+x_2=a_1,\quad x_2+x_3=a_2,\quad \ldots,\quad x_n+x_1=a_n

has a solution (''x''₁, ..., ''x''_''n'') in positive reals. If such a solution exists, then ''x''₁, ..., ''x''_''n'' are the ''tangent lengths'' of the polygon (the lengths from the vertices to the points where the incircle is

to the sides).

Uniqueness and non-uniqueness

If the number of sides ''n'' is odd, then for any given set of sidelengths

a_1, \dots , a_n

satisfying the existence criterion above there is only one tangential polygon. But if ''n'' is even there are an infinitude of them.. For example, in the quadrilateral case where all sides are equal we can have a

rhombus In plane Euclidean geometry, a rhombus (: 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 rhom ...

with any value of the acute angles, and all rhombi are tangential to an incircle.

Inradius

If the ''n'' sides of a tangential polygon are ''a''₁, ..., ''a''_''n'', the inradius (

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

of the incircle) is :

r=\frac=\frac

where ''K'' is the

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

of the polygon and ''s'' is the semiperimeter. (Since all

s are tangential, this formula applies to all triangles.)

Other properties

*For a tangential polygon with an odd number of sides, all sides are equal if and only if all angles are equal (so the polygon is regular). A tangential polygon with an even number of sides has all sides equal if and only if the alternate angles are equal (that is, angles ''A'', ''C'', ''E'', ... are equal, and angles ''B'', ''D'', ''F'', ... are equal). *In a tangential polygon with an even number of sides, the sum of the odd numbered sides' lengths is equal to the sum of the even numbered sides' lengths.Dušan Djukić, Vladimir Janković, Ivan Matić, Nikola Petrović, ''The IMO Compendium'', Springer, 2006, p. 561. *A tangential polygon has a larger area than any other polygon with the same perimeter and the same interior angles in the same sequence. *The

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...

of any tangential polygon, the centroid of its boundary points, and the center of the inscribed circle are

collinear In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...

, with the polygon's centroid between the others and twice as far from the incenter as from the boundary's centroid.

Tangential triangle

While all triangles are tangential to some circle, a triangle is called the

tangential triangle In geometry, the tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at the reference triangle's vertex (geometry), vertices. Thus ...

of a reference triangle if the tangencies of the tangential triangle with the circle are also the vertices of the reference triangle.

Tangential quadrilateral

Tangential hexagon

*In a tangential hexagon ''ABCDEF'', the main

diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek � ...

s ''AD'', ''BE'', and ''CF'' are concurrent according to Brianchon's theorem.

References

{{Polygons Types of polygons