There are $12$ bars which comprise three groups of $6$, $3$ and $3$: each group has identical bars but every group has a distinct length of bars.
- Group 1 may consist of six bars each of $10$ units length,
- Group 2 may consist of three bars each of $3$ units length,
- Group 3 may consist of three bars each of $2$ units length.
By using these bars, you are forming a 12-sided convex polygon (Dodecagon) by randomly putting bars next to each other.
Interestingly, you notice that all of the vertices of this Dodecagon are on a circle with an integer-valued radius and more interestingly all bars have integer-valued lengths as well.
In this case,
What is the minimum value for the radius that satisfies the above condition?
Hint: Hexagon in a circle try to solve this first.