Consecutively use the inequalities of means:
$$\begin{aligned} & \left(\frac{a+b}{2c}\right)^2 + \left(\frac{b+c}{2d}\right)^2 + \left(\frac{c+d}{2a}\right)^2 + \left(\frac{d+a}{2b}\right)^2 \ge \frac{ab}{c^2} + \frac{bc}{d^2} + \frac{cd}{a^2} + \frac{da}{b^2} \ge \\ & \ge 2\sqrt{ab} \sqrt{bc} \frac{\sqrt{ab}}{c} \frac{\sqrt{bc}}{d} + 2\sqrt{cd} \sqrt{da} \frac{\sqrt{ab}}{a} \frac{\sqrt{bc}}{b} \ge 4\sqrt{\frac{ab}{c} \frac{\sqrt{bc}}{d} \frac{\sqrt{cd}}{a} \frac{\sqrt{da}}{b}} = 4. \end{aligned}$$