an anon posted that the first step to solving pic related is:

$A_{1}B_{1}=A_{1}C*cos(\alpha)+B_{1}C*cos(\beta)$

which I confirmed by doing law of cosine on alpha and beta (focusing on the smaller triangle A1B1C), then adding the equations and solving for AB. Never seen this trig identity before, makes me wonder how many other basic trig identities I don't know

but now I want to figure out how to turn it into the final equation. times by AB and somehow the cosines cancel out somehow?