We can show that Chris's solution is the only one as follows:
Treat this as a quadratic in a: a2 - (b+c)a +b2+c2-bc = 0
Solve for a using the quadratic formula
a = (b + c ± [(b+c)2 - 4(b2 + c2 - bc)]1/2)/2
a = (b+c ± [b2 + c2 + 2bc - 4b2 - 4c2 +4bc]1/2)/2
a = (b+c ± [6bc - 3b2 - 3c2]1/2)/2
a = (b+c ± [-3(b - c)2]1/2)/2
a = (b+c ± i*(b-c)*√3)/2
This can only be real (presumably the triangle has real sides!) if b = c so that the imaginary part vanishes.
When b = c then a = (b + c)/2 = (b + b)/2 = b
So all the sides are the same length.