  We study the number of points in the family of plane curves defined by a
trinomial \[
  \mathcal{C}(\alpha,\beta)=
  \{(x,y)\in\mathbb{F}_q^2\,:\,\alpha x^{a_{11}}y^{a_{12}}+\beta
x^{a_{21}}y^{a_{22}}=x^{a_{31}}y^{a_{32}}\} \] with fixed exponents (not
collinear) and varying coefficients over finite fields. We prove that each of
these curves has an almost predictable number of points, given by a closed
formula that depends on the coefficients, exponents, and the field, with a
small error term $N(\alpha,\beta)$ that is bounded in absolute value by
$2\tilde{g}q^{1/2}$, where $\tilde{g}$ is a constant that depends only on the
exponents and the field. A formula for $\tilde{g}$ is provided, as well as a
comparison of $\tilde{g}$ with the genus $g$ of the projective closure of the
curve over $\overline{\mathbb{F}_q}$. We also give several linear and quadratic
identities for the numbers $N(\alpha,\beta)$ that are strong enough to prove
the estimate above, and in some cases, to characterize them completely.
