wlog, assume the first duck to be at 0 degrees.

wlog, assume the second duck to be between 0 and 180 degrees.

The second duck is in a tiny slice between Θ and Θ+dΘ.

The likelihood the duck has landed there is dΘ/π

The third duck can land anywhere from 0 to 2π.  All 3 ducks being on the same side of a diameter means that duck 3 cannot land between π and π+Θ.  There is a (2π-Θ)/2π probability of success.

So integrate over Θ from 0 to π, ((2π-Θ)/2π) dΘ/π

or  (1/(2*π^2) integral 0 to π of (2π-Θ) dΘ

1/(2*π^2) (2π*Θ - (Θ^2)/2)  eval at π minus eval at zero

1/(2*π^2) * (2*π^2 - π^2/2)

(1/(2*π^2))*(3*π^2/2) = 3/4