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I think your diagram is wrong because you are assuming that the head of the vector of T2 hangs from the intersection formed by the horizontal line and the vertical line. But the vector head of T2 just hangs from the vertical line. It doesn't touch the ceiling. Sorry for the bad drawing.
I am working on a method that replaces the use of vector components (as Lami's theorem does for three-force systems) for four-force based systems, both in 2D and 3D. So any approach based on vector components falls outside my purposes.
I have edited my question to include a generalization for a general quad where I remove the angles restriction. Are cranes designed using quadrilateral of forces? Because all cranes I have seen use a triangle of forces.