We define two types of second-order polynomial sequences. A sequence is of Fibonacci-type (Lucas-type) if its Binet formula is similar in structure to the Binet formula for the Fibonacci (Lucas) numbers. Familiar examples are Fibonacci polynomials, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell polynomials, Fermat polynomials, Jacobsthal polynomials, Vieta polynomials and other known sequences of polynomials.

We generalize the numerical recurrence relation given by Hosoya to polynomials by constructing a Hosoya triangle for polynomials where each entry is either a product of two polynomials of Fibonacci-type or a product of two polynomials of Lucas-type. For every such choice of polynomial sequence we obtain a triangular array of polynomials. In this paper we extend the star of David property, also called the Hoggatt-Hansell identity, to these types of triangles. In addition, we study other geometric patterns in these triangles and as a consequence we obtain geometric interpretations for the Cassini identity, the Catalan identity, and other identities for Fibonacci polynomials.