Observe the setup illustrated in the figure. The work performed in...

Points $A$, $B$, and $C$ lie on a circular arc with the point charge $q$ at its center. Therefore, the distances from the charge $q$ to points $A$, $B$, and $C$ are equal to the radius $r$.

The electric potential $V$ at a distance $r$ from a point charge $q$ is $V = \dfrac{1}{4\pi\epsilon_0}\dfrac{q}{r}$.

Since the distances are equal, the electric potentials at points $A$, $B$, and $C$ are the same:

$V_A = V_B = V_C$

The work done $W$ in moving a test charge $Q$ from point $P$ to a point $X$ is $W_X = Q(V_X - V_P)$.

Because $V_A = V_B = V_C$, the work done is equal for all three paths:

$W_A = W_B = W_C$