We are going to prove that the electric field strength is zero at the so-called incentre, the centre of the triangle’s inscribed circle (which has radius r in the figure)

Let us consider a small length of rod at position P on one of the sides of the triangle; let it subtend an angle Δϕ at the incentre (see figure). Its distance from the incentre is r/cos φ φ . Its small length Δ x can be found by noting that P is a distance x = r tan φ φ along the rod from the fixed point Q and so

Δ x = (r Δϕ ) / (cos 2 ϕ ). Consequently the charge it carries is

Δ q =

where λ is the linear charge density on the rods. The magnitude of the elementary contribution of this

small piece to the electric field at the incentre is

Δ E =

It can be seen from this result that the same electric field (in both magnitude and direction) would be produced by an arc of the inscribed circle that subtends Δϕ at the circle’s centre and carries the same linear charge density λ as the rod.

Summing up the contributions of the small arc pieces corresponding to all three sides of the triangle,

we will, because of the circular symmetry, obtain zero net field. It follows that the electric field strength produced by the charged sides of the triangle is also zero at the incentre.