The triangular flow in ultrarelativistic $^3$He-Au collisions at RHIC energies is enhanced due to the triangular arrangement of the nucleon configurations in $^3$He. We study the fireball eccentricities in the Glauber Monte Carlo model and find that since the configurations of the projectile $^3$He are elongated triangles, the created fireball has a large ellipticity and a smaller triangularity. The dependence of the triangularity on centrality is weak, so it cannot be extracted from the centrality dependence of the triangular flow $v_3$, as it is dominated by the centrality dependence of the hydrodynamic response. We propose to look at the centrality dependence of the ratio $v_n\{4\}/v_n\{2\}$, where the uncertainties from the hydrodynamic response cancel, and show that the basic signature of the geometry-driven collective flow is the raise of the ratio $v_3\{4\}/v_3\{2\}$ with the number of participant nucleons for centralities less than 10%.