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Question 3

In a quadrilateral $$ABCD$$, $$AB=AD=10$$, $$BD=12$$, $$CB=CD=13$$. Then

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Since $$AB=AD$$ and $$CB=CD$$, the quadrilateral is a kite with diagonal $$BD$$ as its symmetry axis. Also, $$AB+CD=AD+CB=23$$, which is exactly the condition for a convex quadrilateral to have an incircle. The angles at $$B$$ and $$D$$ are equal rather than supplementary, so the quadrilateral is not cyclic. Hence it has an in-circle but not a circum-circle.

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