ILL 1972 Problem 35

$(a)$ Prove that for $a, b, c, d \in\mathbb{R}, m \in [1,+\infty)$ with $am + b =-cm + d = m$ , $$(i)\sqrt{a^2 + b^2}+\sqrt{c^2 + d^2}+\sqrt{(a-c)^2 + (b-d)^2}\ge \frac{4m^2}{1+m^2},\text{ and}$$ $$(ii) 2 \le \frac{4m^2}{1+m^2} < 4.$$ $(b)$ Express $a, b, c, d$ as functions of $m$ so that there is equality in $(i).$