Question

In a cyclic quadrilateral ABCD, prove that tan^2B/2 = ((s - a)(s - b)/(s - c)(s - d)) a, b, c and d being the lengths of sides AB, BC, CD and DA, respectively, and ‘s’ is semi-perimeter of quadrilateral.

Answer 1

See Fig. In DABC

AC² = a² + b² − 2ab cos B  ....(1)

In  ΔADC

AC² = c² + d² − 2cd cos D

= c² + d² − 2cd cos (180° − B)

= c² + d² + 2cd cos B     ......(2)

From Eqs. (1) and (2), we get