  We show that any embedding $\mathbb{R}^d \to \mathbb{R}^{2d+2^{\gamma(d)}-1}$
inscribes a trapezoid or maps three points to a line, where $2^{\gamma(d)}$ is
the smallest power of $2$ satisfying $2^{\gamma(d)} \geq \rho(d)$, and
$\rho(d)$ denotes the Hurwitz--Radon function. The proof is elementary and
includes a novel application of nonsingular bilinear maps. As an application,
we recover recent results on the nonexistence of affinely $3$-regular maps, for
infinitely many dimensions $d$, without resorting to sophisticated algebraic
techniques.
