Bicentric quadrilateral
The radii and centers of these circles are called inradius and circumradius, and incenter and circumcenter respectively.
Examples of bicentric quadrilaterals are squares, right kites, and isosceles tangential trapezoids.
)[5] There is a simple method for constructing a bicentric quadrilateral: It starts with the incircle Cr around the centre I with the radius r and then draw two to each other perpendicular chords WY and XZ in the incircle Cr.
It can also be derived directly from the trigonometric formula for the area of a tangential quadrilateral.
If M, N are the midpoints of the diagonals, and E, F are the intersection points of the extensions of opposite sides, then the area of a bicentric quadrilateral is given by where I is the center of the incircle.
[13] If M, N are the midpoints of the diagonals, and E, F are the intersection points of the extensions of opposite sides, then the area can also be expressed as where Q is the foot of the perpendicular to the line EF through the center of the incircle.
[9] If r and R are the inradius and the circumradius respectively, then the area K satisfies the inequalities[14] There is equality on either side only if the quadrilateral is a square.
A similar inequality giving a sharper upper bound for the area than the previous one is[13] with equality holding if and only if the quadrilateral is a right kite.
In addition, with sides a, b, c, d and semiperimeter s: If a, b, c, d are the length of the sides AB, BC, CD, DA respectively in a bicentric quadrilateral ABCD, then its vertex angles can be calculated with the tangent function:[9] Using the same notations, for the sine and cosine functions the following formulas holds:[16] The angle θ between the diagonals can be calculated from[10] The inradius r of a bicentric quadrilateral is determined by the sides a, b, c, d according to[7] The circumradius R is given as a special case of Parameshvara's formula.
The four sides a, b, c, d of a bicentric quadrilateral are the four solutions of the quartic equation where s is the semiperimeter, and r and R are the inradius and circumradius respectively.[18]: p.
Moreover,[15]: p.39, #1203 and Fuss' theorem gives a relation between the inradius r, the circumradius R and the distance x between the incenter I and the circumcenter O, for any bicentric quadrilateral.
In fact the converse also holds: given two circles (one within the other) with radii R and r and distance x between their centers satisfying the condition in Fuss' theorem, there exists a convex quadrilateral inscribed in one of them and tangent to the other[23] (and then by Poncelet's closure theorem, there exist infinitely many of them).
to the expression of Fuss's theorem for x in terms of r and R is another way to obtain the above-mentioned inequality
A generalization is[19]: p.5 Another formula for the distance x between the centers of the incircle and the circumcircle is due to the American mathematician Leonard Carlitz (1907–1999).
For the tangent lengths e, f, g, h the following inequalities holds:[19]: p.3 and where r is the inradius, R is the circumradius, and x is the distance between the incenter and circumcenter.
The sides a, b, c, d satisfy the inequalities[19]: p.5 and The circumcenter, the incenter, and the intersection of the diagonals in a bicentric quadrilateral are collinear.
[25] There is the following equality relating the four distances between the incenter I and the vertices of a bicentric quadrilateral ABCD:[26] where r is the inradius.
