Optimization problem of angles for classical mechanics problem

Question

I come from civil engineering background but the problem I came across is more mathematical, so I seek your help here.

Imagine there is force to be supported using truss structure. Given is the location of the force in terms of L3 , the value of a force F . From mechanics and trigonometry, it is possible to set up some restraints and force-balance relations which are shown in below figure.

Image for Illustration [not able to embed due to reputation score]

The goal is to find the optimal angle alpha and theta (sum of absolute values of forces in member is minimum, so there is less net force to carry), which satisfies all equations. I struggle to find solution, only solution I can think of is trial and error but I think there has to be mathematical solution. Any suggestions are welcome.

UPDATE 1: After some time thinking about the problem, I have come with four equations, out of which two are restraints, in particular, force-balance gives:

(1) $-F+\frac{F*tan(\alpha)}{cot(\theta)+tan(\alpha)}+\frac{F*cot(\theta)}{cot(\theta)+tan(\alpha)}=0$

(2) $\frac{2*F}{cot(\theta)+tan(\alpha)}=0$

On the other hand, forces in members are given by $F$ and geometry as follows:

(3) $F_{BC}=\frac{F}{(cot(\theta)+tan(\alpha))*cos(\alpha)}$

(4) $F_{AC}=\frac{-F}{(cot(\theta)+tan(\alpha))*sin(\theta)}$

So basically, I need all solutions of equations (1) & (2), for which $|F_{BC}+F_{AC}|$ is minimum. In some sense it is optimization problem and I think only way to solve this is numerically, but unfortunately I am no expert in that field, so I need your help and explanation. This is as far as I can go with my current knowledge.