In the construction of a triangle given the length of one side, the length of the median to that side and the length of the radius of the circumcircle, one starts with the side of the triangle and locates the circumcenter by drawing circles with radius equal to that of the circumcircle and centers at the endpoints of the triangle side. Once the circumcenter is located as the intersection of these circles, the circumcircle can be drawn. The third vertex of the triangle must be on the circumcircle.
a) How can the third vertex be located?
Bisect the given side and using this midpoint as center draw the circle with radius equal to the length of the median. The intersections of this circle with the circumcircle are the points which can be used as the third vertex.
b) How many solutions are there if the radius of the circumcircle is equal to half of the length of the given side? (Hint: there are two possibilities depending on the length of the median.)
If the median length equals the radius of the circumcircle then the two circles coincide and there are an infinite number of solutions. For any other length of the median, the circles will not intersect, so there will be no solutions.