I need to find the distance from the barycenter of an equilateral triangle to the edge in a given angle.
Here's a little sketch:
Given the outer radius of the triangle, the angle and the rotation (assuming the rotation in the picture would be $0$), I need to find the distance from the point on the edge (marked as red in the sketch) to the center.
This expression works for your particular choice of the angle:
where $R$ is the radius of circumscribed circle (I assume this was meant by "outer radius of the triangle").