## Alternate sides cyclic-2*n*-gons and Alternate angles circum-2*n*-gons

These polygons are respective generalizations of isosceles trapezia and kites, and defined by the following two dual results.

**Alternate sides cyclic-2***n*-gons |
**Alternate angles circum-2***n*-gons |

A cyclic 2*n*-gon has *n* distinct pairs of adjacent angles equal, if and only if, a set of alternate sides are equal. |
A circumscribed 2*n*-gon has *n *distinct pairs of adjacent sides equal, if and only if, a set of alternate angles are equal. |

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Alternate sides cyclic-2*n*-gons and Alternate angles circum-2*n*-gons

In the first figure, drag vertices *A*, *B*, *D* or *F* and in the second one, drag *P*, *Q*, *R* or *S* to dynamically change the figures.

**Reference**: De Villiers, M. (2011). Feedback: Equi-angular cyclic and equilateral circumscribed polygons. *Mathematical Gazette*, July, 95(533), p. 361.

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Michael de Villiers, 16 July 2011.