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

, 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 (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. Mo ...

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 (edg ...

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 poly ...

passing through each of its vertices. All

triangle A triangle is a polygon with three edges and three 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, any three points, when non- colli ...

s are tangential, as are all

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

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

kites A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the face ...

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 bi ...

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 (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. Th ...

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 or planar lamina, while ''surface area'' refers to the area of an open su ...

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

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 o ...

, 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 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 Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A ''regular hexagon'' h ...

''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 Gree ...

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

References

{{Polygons Types of polygons