Given an equilateral triangle $\triangle ABC$ and a right-angled triangle $\triangle ABD$ where $\angle ADB$ is the right angle and, therefore, the hypotenuse $AB$ is shared with the $\triangle ABC$. Let's say that the angle $\angle DAB$ is $\theta$ and the length of $AB$ is $d$. The problem:
How do you define angle $\angle ADC$ in terms of $\theta$? And also how do you define length of $DC$ in terms of $\theta$ and $d$? Basically I need to figure out their functions.
Here's essentially what it looks like 
Can't seem to figure out how to derive the solution myself, and I don't even know how to describe this problem so that I could Google it.
In the real world, I got inspired by a tripod that was standing in the corner of my room with two of it's legs touching the walls. It got me thinking: if I rotated the tripod around the corner so that the two legs would keep touching the walls what trajectory would the third leg follow? And so the problem just stuck in my head for half a day.
If it's a known generic problem I would like to know what it's called, if not then what steps are necessary to derive the solution? Thanks.