First, we have to assume that the triangle is uniformly dense!
The centroid is at the point where all the medians of the triangle meet.
Geometrically this is easy to find, simply bisect any 2 sides and connect the point of bisection to the opposite vertex.
The centroid divides each median in a ratio of 2:1 with the larger end toward the vertex.
Since you have all the sides of the triangle, you can compute all the angles and with a little more trig you can figure the length of the medians and the distance from a vertex to the centoid.
Does that fix it for you?
But if you want to find the Cartesian co-ordinates of the centroid without finding the co-ordinates of the vertices, I think the answer is probably no...though there may be a formula which I have never seen. You should check in a textbook devoted entirely to plane analytic geometry.
