If we start with,

d = r₁t₁ d = r₂t₂

then,

r₁ = d/ t₁ r₂ = d/ t₂

and

t₁ = d/ r₁ t₂ = d/ r₂

Total distance is 2d because both equations travel over the same distance.

Therefore,

2d = r_{avg. rate} (t₁+ t₂)

then,

2d = r_{avg. rate} (d/ r₁ + d/ r₂)

distance can be removed and the equation is

2 = r_{avg. rate} (1/ r₁ + 1/ r₂)

r_{avg. rate} = 2/(1/ r₁ + 1/ r₂) = (2 r₁ r₂)/ ( r₁ + r₂) =

(r₁ r₂)/ ( r₁/2 + r₂/2) = (GM)² /AM