The image above is of a root locus diagram (plotted in the complex plane) from a control systems problem. The diagram is formed from the transfer function by plotting the gain $K$ as a function of the open-loop pole (denominator roots) and open-loop zero (numerator roots) positions, although this is not necessary to know to answer this question. In control systems the roots of the denominator are referred to as poles and the numerator roots referred to as zeros. The transfer function is
$$G(s) = \frac{K(s+2)(s+1)}{(s-2)(s-1)}$$
I'd basically like to find the point where the blue dotted line intersects the circle. (Through simulation I know that it lies at -1+1$i$ but I am trying to do so without the aid of software). I know only that this point is at some position $(x,y)$ and that the blue dotted line is at an angle of $\theta$ which happens to be $45^{\circ}$. I also know through some posture relevant to control theory that in order for a point to lie on the root locus (the circle) the sum of the zero angles (the angles from the roots at -2 and -1 to the point) and the sum of the negative of the pole angles (the angles from the roots at 1 and 2) must equal to $-180^{\circ}$. That is:
$\theta 1 + \theta 2 - \theta 3 - \theta 4 = -180^{\circ}$
I am able to express $y$ as $y=-tan(\theta)x$ through the tangent rule which becomes $y=-x$ as $\theta=45^{\circ}$
then writing the angles above in terms of x:
$tan^{-1}(\frac{-x}{2-x}) + tan^{-1}(\frac{-x}{1-x}) - (180^{\circ} - tan^{-1}(\frac{-x}{1+x})) - (180^{\circ} - tan^{-1}(\frac{-x}{2+x})) = -180^{\circ}$
When trying to solve this on my calculator I keep not being able to solve the equation. I know that solving for $x$ must give a value of 1 because I have verified so graphically. I suspect that the reason I am unable to solve the equation above is that I am making some mistake in the geometry and hence when I try solve on my calculator (which employs the Newton Raphson method) it is unable to converge on a solution.
edit: I can arrive at an approximate solution using some "guess work" by substituting $R\angle{(180^{\circ}-45^{\circ})}$ into the transfer function given above. I can then vary the value of R until I find a result which gives an angle of $180^\circ$ however I am seeking a geometric solution to the problem as shown above which will give an exact result.