Dual polygon

In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other.

Regular polygons are self-dual.

The dual of an isogonal (vertex-transitive) polygon is an isotoxal (edge-transitive) polygon. For example, the (isogonal) rectangle and (isotoxal) rhombus are duals.

In a cyclic polygon, longer sides correspond to larger exterior angles in the dual (a tangential polygon), and shorter sides to smaller angles. Further, congruent sides in the original polygon yields congruent angles in the dual, and conversely. For example, the dual of a highly acute isosceles triangle is an obtuse isosceles triangle.

In the Dorman Luke construction, each face of a dual polyhedron is the dual polygon of the corresponding vertex figure.

As an example of the side-angle duality of polygons we compare properties of the cyclic and tangential quadrilaterals.

This duality is perhaps even more clear when comparing an isosceles trapezoid to a kite.

The simplest qualitative construction of a dual polygon is a rectification operation, where the edges of a polygon are truncated down to vertices at the center of each original edge. New edges are formed between these new vertices.

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