Given side lengths find largest angle

Question

It is given that the three side lengths of a triangle are m, n, and sqrt(m^2+mn+n^2), determine the longest side and find the largest angle.

Answer 1

If side m = side n, then the angle between them, Q is the largest angle and angle M=angle N.

If m ≠ n then Q is still the largest angle because the square root of either m² or n² plus a little more is still the largest side, and m+n < (m² +mn +n²) , i.e. sum of 2 sides have to be less than the 3rd side.