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 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 a ...

that contains an

inscribed circle 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. ...

(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 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 each of the polygon's sides. The

dual polygon In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. Properties Regular polygons are self-dual. The dual of an isogonal (vertex-transitive) polygon is an isotoxal (ed ...

of a tangential polygon is a

cyclic polygon 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 ...

, which has a

circumscribed circle 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 ...

passing through each of its vertices. All

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

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 p ...

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

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 ...

s, which include the

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

kites A kite is a tethered heavier than air flight, heavier-than-air or lighter-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. ...

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" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

all of its internal

angle bisectors In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes throug ...

are

concurrent Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to: Law * Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea'' * Concurring opinion (also called a "concurrence"), a ...

. 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 (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 ...

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) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

of the incircle) is :

r=\frac=\frac

where ''K'' is 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 ...

of the polygon and ''s'' is the

semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...

. (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 surface of the figure. The same definition extends to any ob ...

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 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 ...

, 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 vertices. Thus the incircle of the ...

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 In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...

''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 δ ...

s ''AD'', ''BE'', and ''CF'' are

according to

Brianchon's theorem In geometry, Brianchon's theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. It is named after Charles Julien Brianchon (1 ...

References

{{Polygons Types of polygons