The geometric mean is
√(ab)
The arithmetic mean is
(a+b)/2
These are equal when
(a+b)/2 = √ab
a + b = 2√ab
(a +b)² = 4ab . . . . square both sides
a² +2ab +b² = 4ab . . . . expand the square of the binomial
a² -2ab +b² = 0 . . . . . . subtract the right side
(a -b)² = 0 . . . . . . . . . . . write as a square
a -b = 0 . . . . . . . . . . . . . take the square root
a = b . . . . . . . . . . . . . . . add b
The arithmetic and geometric means of a and b are equal when ...
a = b
