Cyclic polygon

Short description: Geometry of polygons

Cyclic polygon is a polygon whose vertices all lie on a single common circle, known as its circumscribed circle or circumcircle.

As a corollary of the annulus chord formula, the area bounded by the circumcircle and incircle of every unit regular n-gon is π/4

More generally, a polygon in which all vertices are concyclic is called a cyclic polygon. A polygon is cyclic if and only if the perpendicular bisectors of its edges are concurrent. Every regular polygon is a cyclic polygon.

For a cyclic polygon with an odd number of sides, all angles are equal if and only if the polygon is regular. A cyclic polygon with an even number of sides has all angles equal if and only if the alternate sides are equal (that is, sides 1, 3, 5, ... are equal, and sides 2, 4, 6, ... are equal).

A cyclic pentagon with rational sides and area is known as a Robbins pentagon. In all known cases, its diagonals also have rational lengths, though whether this is true for all possible Robbins pentagons is an unsolved problem.

In any cyclic n-gon with even n, the sum of one set of alternate angles (the first, third, fifth, etc.) equals the sum of the other set of alternate angles. This can be proven by induction from the n = 4 case, in each case replacing a side with three more sides and noting that these three new sides together with the old side form a quadrilateral which itself has this property; the alternate angles of the latter quadrilateral represent the additions to the alternate angle sums of the previous n-gon.

A tangential polygon is one having an inscribed circle tangent to each side of the polygon; these tangency points are thus concyclic on the inscribed circle. Let one n-gon be inscribed in a circle, and let another n-gon be tangential to that circle at the vertices of the first n-gon. Then from any point P on the circle, the product of the perpendicular distances from P to the sides of the first n-gon equals the product of the perpendicular distances from P to the sides of the second n-gon.

Point on the circumcircle

Let a cyclic n-gon have vertices A₁ , ..., A_n on the unit circle. Then for any point M on the minor arc A₁A_n, the distances from M to the vertices satisfy

$$\begin{array}{rlrl}M{A}_1+M{A}_3+\cdots +M{A}_{n-2}+M{A}_n& <\frac{n}{\sqrt{2}}& & \text{if }n\text{ is odd};\\ M{A}_1+M{A}_3+\cdots +M{A}_{n-3}+M{A}_{n-1}& \le \frac{n}{\sqrt{2}}& & \text{if }n\text{ is even}.\end{array}$$

For a regular n-gon, if MA_i are the distances from any point M on the circumcircle to the vertices A_i, then

$$3{(M{A}_1^2+M{A}_2^2+\dots +M{A}_n^2)}^2=2n(M{A}_1^4+M{A}_2^4+\dots +M{A}_n^4).$$