Distance to circle inside triangle

Question

I apologise for the lack of a precise term or title, math isn't my strong suit.

I'm trying to calculate the length of L, given angle A and radius of circle D, so that lines b and c tangents with circle D.

Can anyone help me?

Answer 1

Hint: Using the definition of sine and the orthogonality of the tangent one has $$\sin\left(\alpha/2\right)=\frac{\text{radius}}{\text{distance from center}}$$

Answer 2

L = R / Sin(A/2)

where R is the radius of the circle and A is the angle

Answer 3

Supplementary to b00n heT's answer:

Let's call the intersection of $b$ with the circunference $R$ and $r=d(R,D)$, the radius of the circumference.

Considering the right-angle triangle [ARD], we have that $\sin(RÂD)=\frac{r}{L}$, from the definition of the $\sin$ of an angle.

Then, if $A=2RÂD$, then: $$\sin\left(\frac{A}{2}\right)=\frac{r}{L} \Leftrightarrow L=\frac{r}{\sin\left(\frac{A}{2}\right)}$$