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This is a direct consequence of the inscribed angle theorem and the exterior angle theorem. There are no cyclic quadrilaterals with rational area and with unequal rational sides in either arithmetic or geometric progression. [26] If a cyclic quadrilateral has side lengths that form an arithmetic progression the quadrilateral is also ex-bicentric.
As a consequence of the theorem, opposite angles of cyclic quadrilaterals sum to 180°; conversely, any quadrilateral for which this is true can be inscribed in a circle. As another example, the inscribed angle theorem is the basis for several theorems related to the power of a point with respect to a circle.
Ptolemy's Theorem yields as a corollary a pretty theorem [2] regarding an equilateral triangle inscribed in a circle. Given An equilateral triangle inscribed on a circle, and a point on the circle. The distance from the point to the most distant vertex of the triangle is the sum of the distances from the point to the two nearer vertices.
This more general formula is known as Bretschneider's formula. It is a property of cyclic quadrilaterals (and ultimately of inscribed angles) that opposite angles of a quadrilateral sum to 180°. Consequently, in the case of an inscribed quadrilateral, θ is 90°, whence the term
In geometry, Brahmagupta's theorem states that if a cyclic quadrilateral is orthodiagonal (that is, has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. [1] It is named after the Indian mathematician Brahmagupta (598-668). [2]
[1] [2] Due to the risk of confusion with a quadrilateral that has a circumcircle, which is called a cyclic quadrilateral or inscribed quadrilateral, it is preferable not to use any of the last five names. [1] All triangles can have an incircle, but not all quadrilaterals do.
Hadwiger's theorem (geometry, measure theory) Helly's theorem (convex sets) Holditch's theorem (plane geometry) John ellipsoid ; Jung's theorem ; Kepler conjecture (discrete geometry) Kirchberger's theorem (discrete geometry) Krein–Milman theorem (mathematical analysis, discrete geometry) Minkowski's theorem (geometry of numbers)
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).