M and N are the mid-points of the sides QR and PQ respectively of a triangle PQR, right
angled at Q. Prove that:
\text { (i) } \quad \mathrm{PM}^{2}+\mathrm{RN}^{2}=5 \mathrm{MN}^{2}
\text { (ii) } 4 \mathrm{PM}^{2}=4 \mathrm{PQ}^{2}+\mathrm{QR}^{2}
\text { (iii) } \quad 4 \mathrm{RN}^{2}=\mathrm{PQ}^{2}+4 \mathrm{QR}^{2}
\text { (iv) } 4\left(\mathrm{PM}^{2}+\mathrm{RN}^{2}\right)=5 \mathrm{PR}^{2}