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.

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Challenge
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.