Central eclipse contact-radius relations ($R_L\pm R_S=a\sin (\pi T_{\pm }/P)$)

For a circular edge-on eclipsing binary with central passage, define $T_{+}=t_4-t_1$ between external contacts and $T_{-}=t_3-t_2$ between internal contacts, with $R_L\ge R_S$. The projected separation from conjunction is $a|\sin [2\pi (t-t_c)/P]|$. At contact it equals $R_L+R_S$ or $R_L-R_S$, proving the title formula. Taking half the sum and difference recovers the radii. When durations are small compared with $P$, this becomes $R_{L,S}=v_{\mathrm{rel}}(T_{+}\pm T_{-})/4$ with $v_{\mathrm{rel}}=2\pi a/P$. A nonzero impact parameter changes the contact chords.