InEuclidean geometry, atangential polygon, also known as acircumscribed polygon, is aconvex polygon that contains aninscribed circle (also called anincircle). This is a circle that istangent to each of the polygon's sides. Thedual polygon of a tangential polygon is acyclic polygon, which has acircumscribed circle passing through each of itsvertices.

Alltriangles are tangential, as are allregular polygons with any number of sides. A well-studied group of tangential polygons are thetangential quadrilaterals, which include therhombi andkites.

A convex polygon has an incircleif and only if all of its internalangle bisectors areconcurrent. This common point is theincenter (the center of the incircle).^[1]

There exists a tangential polygon ofn sequential sides of lengthsa₁, ...,a_n if and only if thesystem of equations

${\displaystyle x_1+x_2=a_1,x_2+x_3=a_2,\dots ,x_n+x_1=a_n}$

has a solution (x₁, ...,x_n) in positivereals.^[2] If such a solution exists, thenx₁, ...,x_n are thetangent lengths of the polygon (the lengths from thevertices to the points where the incircle istangent to the sides).

If the number of sidesn is odd, then for any given set of sidelengths${\displaystyle a_1,\dots ,a_n}$ satisfying the existence criterion above there is only one tangential polygon. But ifn is even there are an infinitude of them.^{[3]: p. 389} For example, in the quadrilateral case where all sides are equal we can have arhombus with any value of the acute angles, and all rhombi are tangential to an incircle.

If then sides of a tangential polygon area₁, ...,a_n, the inradius (radius of the incircle) is^[4]

${\displaystyle r=\frac{K}{s}=\frac{2K}{\sum _{i=1}^n{a}_i}}$

whereK is thearea of the polygon ands is thesemiperimeter. (Since alltriangles are tangential, this formula applies to all triangles.)

• 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, anglesA,C,E, ... are equal, and anglesB,D,F, ... are equal).^[5]
• 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.^[2]
• A tangential polygon has a larger area than any other polygon with the same perimeter and the same interior angles in the same sequence.^{[6]: p. 862 [7]}
• Thecentroid of any tangential polygon, the centroid of its boundary points, and the center of the inscribed circle arecollinear, with the polygon's centroid between the others and twice as far from the incenter as from the boundary's centroid.^{[6]: pp. 858–9}

While all triangles are tangential to some circle, a triangle is called thetangential triangle of a reference triangle if the tangencies of the tangential triangle with the circle are also the vertices of the reference triangle.

• In a tangentialhexagonABCDEF, the maindiagonalsAD,BE, andCF areconcurrent according toBrianchon's theorem.

• Circumgon

• Types of polygons

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