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In Fig. \( \quad \mathrm{AD} \) is a median of a triangle \( \mathrm{ABC} \) and
\( \mathrm{AM} \perp \mathrm{BC} \). Prove that :
(i) \( \mathrm{AC}^{2}=\mathrm{AD}^{2}+\mathrm{BC} \cdot \mathrm{DM}+\left(\frac{\mathrm{BC}}{2}\right)^{2} \)
\( { }_{-}^{-} \)(ii) \( \mathrm{AB}^{2}=\mathrm{AD}^{2}-\mathrm{BC} \cdot \mathrm{DM}+\left(\frac{\mathrm{BC}}{2}\right)^{2} \)
- (iii) \( \mathrm{AC}^{2}+\mathrm{AB}^{2}=2 \mathrm{AD}^{2}+\frac{1}{2} \mathrm{BC}^{2} \)
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