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\(D\) is the midpoint of side \(BC\) and \(AE \perp BC\). If \(BC = a\), \(AC = b\), \(AB = c\), \(ED = x\), \(AD = p\) and \(AE = h\), prove that
(i) \(b^2 = p^2 + ax + \frac{a^2}{4}\)
(ii) \(c^2 = p^2 - ax + \frac{a^2}{4}\)
(iii) \(b^2 + c^2 = 2p^2 + \frac{a^2}{2}\)
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