I am having trouble with the following problem:
There are these points $A = (x_A, y_A)$ and $B = (x_B, y_B)$. $A$ is connected to $C$ by a rigid bar of length $L$ and a similar bar connects $B$ to $D$. A third bar, of length $d$, connects C to D. I need to find, for given $A$ and $B$, the position of $G = (x_G,y_G)$, which is the middle point of $\overline{CD}$, that minimizes $y_G$ (the vertical position of $G$), as well as the angle $\theta$ between $\overline{CD}$ and the horizontal axis.
I was able to establish the following system of equations:
$$(x_C - x_A)^2 + (y_C - y_A)^2 - L^2 = 0$$
$$(x_D - x_B)^2 + (y_D - y_B)^2 - L^2 = 0$$
$$(x_C - x_D)^2 + (y_C - y_D)^2 - d^2 = 0$$
$$y_G = \frac{y_C + y_D}{2}$$
With the above system in hands, I am able to find the minimal value for $y_G$ and then calculate $x_G$ and $\theta$, right? But how do I do it?
Thanks for reading.