Computation of a circle radius knowing a certain angle and a certain distance

Question

Hi guys I'm having trouble determining the equation to find an unknown radius of a circle that is tangent to 2 lines, and the minimum distance from the intersection of the 2 lines to the circumference of the circle is known (Intersection of the bisector line to the circumference of the circle).

I attached a picture to show the information i have for the problem, the basic problem is, R=??? based on D and A?: Diagram of Problem

Answer 1

Let $I$ be the point of intersection of the tangents.

Let T be the point of tangency of one of these tangents.

Let $O$ be the center of the circle.

Let $B:=\tfrac{A}{2}$.

(I am indebted to @amd who has "short circuited" a first unduly long proof):

In right triangle OTI,

$$\sin(B)=\dfrac{OT}{OI} \ \ \ \iff \ \ \ \sin(B)=\dfrac{R}{R+D}$$

which we can invert under the following form:

$$R=D \dfrac{\sin(B)}{1-\sin(B)} \ \ \ \iff \ \ \ R=D \dfrac{\sin(A/2)}{1-\sin(A/2)}$$

There is also a proof using the power of a point (here $I$) but it is more complicated.(https://en.wikipedia.org/wiki/Power_of_a_point));

Remark: I think that this question arose from a metrology problem. I am right ?