In the figure, \(AB\) is parallel to \(DC\), and \(AC\) intersects \(PQ\) at \(O\). Show that \(OA \times CQ = OC \times AP\).
Given \(AB||DC\).
\(AC\) and \(PQ\) intersect each other at the point \(O\).
In \(\Delta AOP\) and \(\Delta COQ\),
\(\angle AOP\ = \) (vertically opposite angles)
\(\angle APO =\) (alternate angles)
Thus, \(\Delta AOP \sim \) (by similarity criterion).
Hence, corresponding sides are proportional.
\(\frac{OA}{OC}\ = \)
\(OA \cdot CQ = OC \cdot AP\)
Hence proved.