Things like this seem to work
Solve b=l sin(alpha)+a,B=l cos(alpha)+A,d=a+k(b-a),D=A+k(B-A),e=b+k(c-b) for b,B,k
Maybe you can use this as a starting point for what you are trying to do. Notice I inserted a space in front of sin and cos.
If it is essential that your solution not contain any of d,D,e,E then there is a function Eliminate that will sometimes create a more complicated set of equations, but which do not contain particular variables. If you start typing Eliminate into Wolfram|Alpha then you might see after a few characters that it does recognize the name of the Eliminate function, but I have not been able to get it to eliminate your four variables from your eight equations.
If it is really essential to eliminate those four variables and you just cannot get Wolfram|Alpha to understand and cooperate then, since they appear in fairly simple form in the rest of your equations, you might be able to do manual substitutions to get rid of those four variables, just watch out for any possible hidden division by zero if you do that.
