The four distinct points \(P_i\) (\(i=1\), \(2\), \(3\), \(4\)) are the
vertices, labelled anticlockwise, of a cyclic quadrilateral. The
lines \(P_1P_3\) and \(P_2P_4\) intersect at \(Q\).
By considering the
triangles \(P_1QP_4\) and \(P_2QP_3\) show that
\((P_1Q)( QP_3) = (P_2Q) (QP_4)\,\).
Let \(\+p_i\) be the position vector of the point \(P_i\)
(\(i=1\), \(2\), \(3\), \(4\)). Show that there
exist numbers \(a_i\), not all zero, such
that
\begin{equation}
\sum\limits_{i=1}^4 a_i =0 \qquad\text{and}\qquad
\sum\limits_{i=1}^4 a_i \+p_i ={\bf 0} \,. \tag{\(*\)}
\end{equation}
Let \(a_i\) (\(i=1\),~\(2\), \(3\),~\(4\)) be any numbers, not all zero,
that satisfy~\((*)\). Show that \(a_1+a_3\ne 0\) and that the lines
\(P_1P_3\) and \(P_2P_4\) intersect at the point with position vector
\[
\frac{a_1 \+p_1 + a_3 \+p_3}{a_1+a_3} \,.
\]
Deduce that \(a_1a_3 (P_1P_3)^2 = a_2a_4 (P_2P_4)^2\,\).