Given an arbitrary quadrilateral with known side lengths and one input angle (the single degree of freedom), I am trying to pick the input angle that minimizes the difference between the resulting other 3 angles.
I want each of the calculated angles to be as close to each other as possible: for example some configurations result in one angle being 180° while another is 90°. My goal in this case would be to choose an input angle such that these two angles are more "evenly spaced", like 135° and 135° for example. I'm not sure what relations or equations I would need to figure this out, but any direction would be helpful!
Let the quadrilateral have sides $a, b, c, d$ where all sides have known lengths and fixed order. Let $\theta$ be the input angle between $a$ and $b$. Let $\phi$ be the angle between $b$ and $c$, and $\gamma$ be the angle between $c$ and $d$. I would like to find a value for $\theta$ such that $\phi - \gamma$ is minimal.