Shortest path between two points around an obstacle? I'm trying to figure out a problem that goes like this:

A particle originally placed at the origin tries to reach the point $(12,16)$ whilst covering the shortest distance possible. But there is a circle of radius $3$, centered at the point $(6,8)$, and the point cannot go through the circle. (Click on image to view larger picture.)


My original thought was to travel in a straight line until reaching the circle, and then travel along the circumference until we reach the point on the circumference that is the shortest distance to $(12,16)$. However I feel like this path should be longer than a path along a curve that is tangent to the circle and passes through both the origin and the given point. Now I'm just stuck on how to find this specific curve.
Since the curve must be tangent to the circle at some point I can equate the derivative at some point, but what point exactly? 

Hint: (Developing a little more the comment by Harshal Gajjar) 
You have two rectangular triangles, and you could know the position of the vertex, where the angle A is, since we  know that the sides of this triangle are $3,\sqrt 91$ and $10$.
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My calculations give me that this vertex has coordinates $\left(6+3\frac{9-\sqrt 91}{50},8+3\frac{\sqrt{2328}}{50}\right)$.
Since the positions $(0,0)$ and $(12,16)$ are symmetrically allocated with respect to the center of the circle, the calculations for the triangle based on the origin is similar. 
The final step is the calculation of the little arc on the circle between the points of tangency. 
