Math/Physics Circuit Problem from CAP contest involving Resistance

Question

Consider a circuit made of a wire with uniform resistance in a shape of a circle as shown in the picture. The circle is connected diagonally from point A to point B with the same type of wire. If the current passing through the circuit is i0, what is the current passing through the wire AB as a function of angle θ?

Answer 1

Excuse the clumsy 'artwork':

Without loss of generality we can take the resistance per unit length to be one and we can take the radius to be one.

One approach is to use Norton's theorem. (a) shows the general circuit and (b) shows how the current splits between two resistors.

(c) Shows how the Norton equivalent resistance is computed and (d) shows how the Norton short circuit current is computed (with the aid of (b)).

(e) Shows the Norton equivalent circuit and from this (again with the aid of (b)) shows that $i_D = {1 \over 1 + {4 \over \pi}}(1-{2 \theta \over \pi}) i_0$.

Another approach would be to perform a $\Delta-Y$ conversion on the three rightmost resistors in (a).