7．Triangle \(A B C\) ，with \(B C = a , A C = b\) and \(A B = c\) is inscribed in a circle．Given that \(A B\) is a diameter of the circle and that \(a ^ { 2 } , b ^ { 2 }\) and \(c ^ { 2 }\) are three consecutive terms of an arithmetic progression（arithmetic series），
（a）express \(b\) and \(c\) in terms of \(a\) ，
（b）verify that \(\cot A , \cot B\) and \(\cot C\) are consecutive terms of an arithmetic progression． In an acute－angled triangle \(P Q R\) the sides \(Q R , P R\) and \(P Q\) have lengths \(p , q\) and \(r\) respectively．
（c）Prove that $$\frac { p } { \sin P } = \frac { q } { \sin Q } = \frac { r } { \sin R }$$ Given now that triangle \(P Q R\) is such that \(p ^ { 2 } , q ^ { 2 }\) and \(r ^ { 2 }\) are three consecutive terms of an arithmetic progression，
（d）use the cosine rule to prove that \(\frac { 2 \cos Q } { q } = \frac { \cos P } { p } + \frac { \cos R } { r }\) ．
（6）
（e）Using the results given in parts（c）and（d），prove that \(\cot P , \cot Q\) and \(\cot R\) are consecutive terms in an arithmetic progression． Marks for style，clarity and presentation： 7