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

These polygons are respective generalizations of isosceles trapezia and kites, and defined by the following two 'side-angle' dual results.
Dynamic hexagon examples are given below.

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

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

Explore
1) 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.
2) Drag the first figure until all the angles are equal to obtain a semi-regular angle-gon.
3) Drag the second figure until all the sides are equal to obtain a semi-regular side-gon.

Challenge
4) Can you explain why (prove that) the two dual results above are true?

References
De Villiers, M. (2011). Equi-angular cyclic and equilateral circumscribed polygons. The Mathematical Gazette, 95(532), pp. 102-106.
De Villiers, M. (2011). Feedback: Equi-angular cyclic and equilateral circumscribed polygons. The Mathematical Gazette, July, 95(533), p. 361.

Explore More
5) What other general properties do these two types of polygons have? Explore on your own by constructing your own dynamic sketches.
6) Can you explain why (prove that) your experimentally observed properties are true?

Other Generalizations
New mathematical objects are often defined by modifying or extending the definitions or properties of known objects in mathematics. This is called 'constructive' defining.
7) Can you think of other ways in which to generalize the dual concepts of isosceles trapezia and kites to 2n-gons while maintaining the 'angle-side' duality between them?
8) Can you explain why (prove that) the properties of your defined generalizations are true?
Hint: While there are many possibilities, a particularly useful approach is to define 'isosceles 2n-gons' and 'kite 2n-gons' by using symmetry as discussed in my book Some Adventures in Euclidean Geometry (free to download) - for example, see these two short EXCERPTS from the book on p. 126 and p. 166. Also compare with p. 146 of my book Rethinking Proof with Geometer's Sketchpad (free to download).

External Link
Isosceles trapezoid (Wikipedia)
Isosceles trapezoid Kite duality (Wikipedia)
Kite (geometry) (Wikipedia)
Cyclic quadrilateral (Wikipedia)
Tangential quadrilateral (Wikipedia)
SA Mathematics Olympiad
Questions and worked solutions for past South African Mathematics Olympiad papers as well as books on problem solving can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

***************

Free Download of Geometer's Sketchpad

***************

Back to "Dynamic Geometry Sketches"

Back to "Student Explorations"

Michael de Villiers, created 2011 with JavaSketchpad, updated to WebSketchpad 13 July 2018; updated 3 April 2025; 12 May 2025.

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

.sketch_canvas { border: medium solid lightgray; display: inline-block; }