Determining distance to an object based on the distance from it to two objects on a perpendicular line and the angle between them

Question

Is possible to determine length $d$, given I only know lengths $a$ and $b$ and $\Theta$ ($\angle$ ACB )? More importantly, how?

Answer 1

Yes, it is possible. Use properties of similar triangles to get $$\dfrac{a}{d}=\dfrac{d}{b}\\ \implies d=\sqrt{ab}$$ There are many other such relations.

Answer 2

Let $\alpha$ be the angle at $C$ opposite $a$, and let $\beta$ be the angle at $C$ opposite $b$. Then $\alpha+\beta=\Theta$.

Note that $\tan \alpha=\frac{a}{d}$ and $\tan \beta=\frac{b}{d}$. Using the Addition Law for $\tan(\alpha+\beta)$, we get $$\frac{\frac{a}{d}+\frac{b}{d}}{1-\frac{ab}{d^2}}=\tan\Theta.$$ This equation can be manipulated to a quadratic in $d$. Solve.

Remark: There is a minor technical hitch if $\Theta=\frac{\pi}{2}$, but then the problem is trivial. Or else, sloppily, we can say that it corresponds to the denominator $1-\frac{ab}{d^2}$ being $0$.