Bravo, John!
It is worth remarking that your two (Law of Sines) equations are also
obtainable by resolving the forces on each mass along, and perpendicular
to, its supporting string. Setting the perpendicular (to strings)
components to zero directly gives your equations: mg*sinT1 = F*sinT3 and
mg*sinT2=F*sinT3.
So the requirement that these "perpendicular to the string" components
cancel is alone sufficient to force T1=T2.
This is a case where resolving force components along horizontal and
vertical directions is not (at least to me) algebraically useful! By the
way, I apologize for inadvertently using the symbols T1 & T2 for both
tensions and angles in my previous posts! I just noticed that, and hope
some were not unduly confused.