Construct segment c (with endpoints A and B). Draw the circle centered at A with radius length of b. Let L be a point on this circle and construct the segment from L to B with M as its midpoint. Let K be the midpoint of c.
Construct the circle centered at A with radius of the length of the median of a and the circle centered at K and passing through M. The intersection of these two circles (point N) is the midpoint of side a.
We can find vertex C by constructing the circle centered at N and passing through B and finding its intersection with the first circle constructed centered at A. All vertices have all been found and the triangle can be constructed.
Click here for a GSP sketch.