Tangent Line Length At Top Of Inside Circle

Question

Question In Picture Form

From the picture, r is known and θ is known. I am trying to find the length of line X. The circle is tangent to the two lines of the triangle and the triangle is isosceles. Is there any other information I need to solve this problem?

Thank you in advance for any assistance. Everything I found during research is for a fully inscribed circle.

Answer 1

Let $|AB|=x$, $|AC|=|BC|=a$ and $S$ is the area of $\triangle ABC$.

Then the circle is the excircle of $\triangle ABC$, for which it is known that its radius \begin{align} r_c&=\frac{2S}{|AC|+|BC|-|AB|} =\frac{2S}{2a-x} ,\\ S&=\tfrac12a^2\sin\theta ,\\ a&=\frac{x}{2\sin\tfrac\theta2} ,\\ \end{align}

and after simplifications, we can find $x$ as \begin{align} x&=\frac{2r_c(1-\sin\tfrac\theta2)}{\cos\tfrac\theta2} . \end{align}

Edit

Or, we can get the same result just from

\begin{align} r_c= |CO_t|\sin\tfrac\theta2 &= (|CC_t|+r_c)\sin\tfrac\theta2 \\ &= (\tfrac{x}2\cot\tfrac\theta2+r_c)\sin\tfrac\theta2 . \end{align}

Answer 2

Assuming that $BB' \perp OC$

we have

$$ \phi = \angle{AOC} = \frac{\pi}{2}-\frac{\theta}{2} $$

hence

$$ x = 2 r \tan\left(\frac{\phi}{2}\right) $$