Let E1(r), E2(r) and E3(r) be the respective electric fields at a distance r from a point charge Q, an infinitely long wire with constant linear charge density λ, and an infinite plane with uniform surface charge density σ. If E1(r0) = E2(r0) = E3(r0) at a given distance r0, then:

Q/(4πϵ₀r₀²) = λ/(2πϵ₀r₀) = σ/(2ϵ₀)
E1(r0/2) = Q/(4πϵ₀(r₀/2)²) = Q/(πϵ₀r₀²)
E2(r0/2) = λ/(2πϵ₀(r₀/2)) = λ/(πϵ₀r₀)
E3(r0/2) = σ/(2ϵ₀)
Since E1(r0) = E2(r0) = E3(r0), we have:
Q/(4πϵ₀r₀²) = λ/(2πϵ₀r₀) = σ/(2ϵ₀)
From the above equations, we get:
Q = 2πr₀²σ and λ = σr₀/π
Now, let's find the ratio of E1(r0/2) and E2(r0/2):
E1(r0/2) / E2(r0/2) = [Q/(πϵ₀r₀²)] / [λ/(πϵ₀r₀)] = Q/(r₀λ) = (2πr₀²σ)/(r₀(r₀σ/π)) = 2π²
This is not equal to 2. Let's consider the relationship between E1(r0/2) and E2(r0/2) and E3(r0/2):
E1(r0/2) = Q/(πϵ₀r₀²) = 2σr₀²/ϵ₀r₀² = 2σ/ϵ₀ = 4E3(r₀/2)
E2(r0/2) = λ/(πϵ₀r₀) = (r₀σ/π)/(πϵ₀r₀) = σ/(π²ϵ₀)
E3(r0/2) = σ/(2ϵ₀)
E1(r0/2) / E2(r0/2) = 4π² ≠ 2
E2(r0/2) / E3(r0/2) = σ/(π²ϵ₀) / (σ/2ϵ₀) = 2/π² ≠ 4
From Q/(4πε₀r₀²) = σ/(2ε₀), we have Q = 2πr₀²σ. Also, from λ/(2πε₀r₀) = σ/(2ε₀), we have λ = r₀σ/π. Therefore,
Q = 4πr₀²σ is incorrect, r₀ = λ/(2πσ) is incorrect, and E2(r0/2)=4E3(r0/2) is also incorrect.
The correct relationship is E1(r0/2) = 2E2(r0/2).