Editing Concyclic points (section)

==Integer area and side lengths==
Some cyclic polygons have the property that their area and all of their side lengths are positive integers.  Triangles with this property are called [[Heronian triangle]]s; cyclic quadrilaterals with this property (and that the diagonals that connect opposite vertices have integer length) are called [[Brahmagupta quadrilateral]]s; cyclic pentagons with this property are called [[Robbins pentagon]]s.  More generally, versions of these cyclic polygons scaled by a [[rational number]] will have area and side lengths that are rational numbers.

Let {{math|''θ''<sub>1</sub>}} be the angle spanned by one side of the cyclic polygon as viewed from the center of the circumscribing circle.  Similarly define the [[central angle]]s {{math|''θ''<sub>2</sub>, ..., ''θ''<sub>''n''</sub>}} for the remaining {{math|''n'' − 1}} sides.  Every Heronian triangle and every Brahmagupta quadrilateral has a rational value for the tangent of the quarter angle, {{math|tan ''θ''<sub>''k''</sub>/4}}, for every value of {{mvar|k}}.  Every known Robbins pentagon (has diagonals that have rational length and) has this property, though it is an unsolved problem whether every possible Robbins pentagon has this property.

The reverse is true for all cyclic polygons with any number of sides; if all such central angles have rational tangents for their quarter angles then the implied cyclic polygon circumscribed by the unit circle will simultaneously have rational side lengths and rational area.  Additionally, each diagonal that connects two vertices, whether or not the two vertices are adjacent, will have a rational length.  Such a cyclic polygon can be scaled so that its area and lengths are all integers.

This reverse relationship gives a way to generate cyclic polygons with integer area, sides, and diagonals.  For a polygon with {{mvar|n}} sides, let {{math|0 < ''c''<sub>1</sub> < ... < ''c''<sub>''n''−1</sub> < +∞}} be rational numbers.  These are the tangents of one quarter of the cumulative angles {{math|''θ''<sub>1</sub>}}, {{math|''θ''<sub>1</sub> + ''θ''<sub>2</sub>}}, ..., {{math|''θ''<sub>1</sub> + ... + ''θ''<sub>''n''−1</sub>}}.  Let {{math|1=''q''<sub>1</sub> = ''c''<sub>1</sub>}}, let {{math|1=''q''<sub>''n''</sub> = 1 / ''c''<sub>''n''−1</sub>}}, and let {{math|1=''q''{{sub|''k''}} = (''c''{{sub|''k''}} − ''c''{{sub|''k''−1}}) / (1 + ''c''{{sub|''k''}}''c''{{sub|''k''−1}})}} for {{math|1=''k'' = 2, ..., ''n''−1}}.  These rational numbers are the tangents of the individual quarter angles, using the formula for the tangent of the difference of angles.  Rational side lengths for the polygon circumscribed by the unit circle are thus obtained as {{math|1=''s''<sub>''k''</sub> = 4''q''<sub>''k''</sub> / (1 + ''q''<sub>''k''</sub><sup>2</sup>)}}.  The rational area is {{math|1=''A'' = ∑{{sub|''k''}} 2''q''{{sub|''k''}}(1 − ''q''{{sub|''k''}}{{sup|2}}) / (1 + ''q''{{sub|''k''}}{{sup|2}}){{sup|2}}}}.  These can be made into integers by scaling the side lengths by a shared constant.