On the Poncelet triangle condition over finite fields

Let P2 denote the projective plane over a finite field Fq. A pair of nonsingular conics (A,B) in the plane is said to satisfy the Poncelet triangle condition if, considered as conics in P2(F‾q), they intersect transversally and there exists a triangle inscribed in A and circumscribed around B. It is shown in this article that a randomly chosen pair of conics satisfies the triangle condition with asymptotic probability 1/q. We also make a conjecture based upon computer experimentation which predicts this probability for tetragons, pentagons and so on up to enneagons.