Orbital quadrature and relative velocities

Assume for simplicity that Earth ($E$) and Mars ($M$) orbit the Sun ($S$) on circular orbits in the same plane, with radii $r_E$ and $r_M$, respectively. Call $alpha$ the angle $widehat{SEM}$ and $beta$ the angle $widehat{SME}$. The law of sines in the triangle $Delta SEM$ gives us the relation $$ r_Esinalpha = r_Msinbeta. $$ The velocity of the Earth is perpendicular to the line $SE$, so the velocity of Earth along the line $EM$ is given by $$ bar{v}_E = v_Ecos(pi/2-alpha) = v_Esinalpha. $$ Likewise, the velocity of Mars along the line $EM$ is $$ bar{v}_M = v_Mcos(pi/2-beta) = v_Msinbeta. $$ Therefore, the relative velocity between Earth and Mars along the line $EM$ is $$ bar{v} = bar{v}_E - bar{v}_M = left(v_E - frac{r_E}{r_M}v_Mright)sinalpha, $$ which is maximal if $alpha= pi/2$.