  We study the $G_2$ reflection equation for the three particles in $1+1$
dimension that undergo a special scattering/reflections described by the Pappus
theorem. It is a sixth order equation and serves as a natural $G_2$ analogue of
the Yang-Baxter and the reflection equations corresponding to the cubic and the
quartic Coxeter relations of type $A$ and $BC$, respectively. We construct
matrix product solutions to the $G_2$ reflection equation by exploiting a
connection to the representation theory of the quantized coordinate ring
$A_q(G_2)$.
