Extangential Quadrilateral

An extangential quadrilateral is a quadrilateral with an excircle as shown below. In other words, the extensions of all four of its sides are tangent to a circle exterior to the quadrilateral. Hence, the exterior angle bisectors at A, B and D are concurrent at the excentre.
An interesting property of the extangential quadrilateral is that it has two (distinct) sums of adjacent sides equal. For example, as illustrated in the sketch: a + b = c + d.

1) Can you prove that if a quadrilateral has an excircle that it has two (distinct) sums of adjacent sides equal?
2) Can you prove the converse, namely that if a quadrilateral has two (distinct) sums of adjacent sides equal then it has an excircle?
3) Apart from the 3 exterior angle bisectors at A, B and D concurrent at the excentre, there are 3 more angle bisectors concurrent at the excentre. Can you identify them?

I first learnt of this very interesting quadrilateral from Martin Josefsson (Sweden) who attributes the first proof of this result in 1846 to the famous Swiss mathematician Jakob Steiner (1796-1863). Read Josefsson's excellent 2012 Forum Geometricorum paper at Similar Metric Characterizations of Tangential and Extangential Quadrilaterals.

More information about extangential quadrilaterals is also available at Wikipedia at the link here.

Of further interest is that according to the side-angle duality/analogy already observed by Coolidge in 1916, and also explored in De Villiers (1996) in relation to the classification of quadrilaterals, the extangential quadrilateral is the dual of the trapezium (trapezoid). The side-angle duality is clearly apparent in that a trapezium has two (distinct) sums of adjacent angles equal (e.g. ∠A + ∠B = ∠C + ∠D) while an extangential quadrilateral has two (distinct) sums of adjacent sides equal (e.g. a + b = c + d).
Coolidge, J.L. (1916). A Treatise on the Circle and the Sphere (pp 53-57). Bronx, NY: Chelsea Publishing Company.
De Villiers, M. (1996). Some Adventures in Euclidean Geometry. Lulu Publishers.

Some readers may also be interested in a dynamic online version of a side-angle duality classification of quadrilaterals.

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Created by Michael de Villiers, 25 July 2020.