Summation of Forces through a Lever Linkage

I'm trying to brush up on some of my force and torque stuff so I can do some modelling in software. My problem is, it's been a while and I'm unsure of how correct my work is.

I have a system as shown in the picture. I want to find the torque around the point D, due to the force F acting at point A.

There's 3 members, AB which is a lever of length alpha, with the fulcrum point at alpha/2. Attached to the lever at point B is a link BC, of length = beta and angle phi. At the end of BC is attached a link CD, at an angle of omega and length = lambda.

My attempt has been to solve the force as a non-perpindicular force on a lever, so the moment around the fulcrum will be = F * alpha/2 * sin(theta). However, this is where I get stuck. How does that translate to the BC link? Is it equal, or does it too act at an angle?

If I define the force on BC as F_bc, a sum of moments around the fulcrum point gives F*alpha/2*sin(theta) = F_bc * alpha/2 *sin(phi). I can solve this for F_bc. Does this F_bc give a torque at D of F_bc * lambda * sin(omega) ?

Once force through BC is found, it'll be easy to find the torque around D, as it will just be the force from BC on CD, at an angle of omega and at a lever length of lambda.

Thanks!