The physics
Every string pulls the ring along the string. The ring stays still only if all the pulls add up to zero.
ΣFx = 0 ΣFy = 0
Two forces A and B with angle θ between them add to:
R = √(A² + B² + 2AB cos θ) tan α = B sin θ / (A + B cos θ)
Lami's theorem is for three balanced forces. Divide each force by the sine of the angle between the other two. All three give the same number.
F₁ / sin α = F₂ / sin β = F₃ / sin γ
With many forces, add the x-parts and the y-parts separately. Both totals must be zero.