Plane trigonometry, by S.L. Loney.

AREA OF A QUADRILATERAL.

251

Ce" +_ b2 - cf- _ d2
220. Since cos B = a2 b2 _ 2 -2 (ab + cd)
we have A = a2 + b2 2ab cos B
a2 + b2 - C2-d2
= a2 + b2 - ab
ab + cd
(a2 + b2) cd + ab (c2 + d2)
ab + cd
(ac + bd) (ad + be)
ab + cd
Similarly it could be proved that
BD =_ (ab + cd) (ac + bd)
ad + be
It follows by multiplication that
A C". BD2 = (ac + bd)2,
i.e. AC. BD = AB. CD +BC. AD.
This is Euc. vi. Prop. D.
221. If we have any quadrilateral, not necessarily
inscribable in a circle, we can express its area in terms of
its sides and the sum of any two opposite angles.
For let the sum of the two angles B and D be denoted
by 2a, and denote the area of the
quadrilateral by A. A d
Then / D
A =area of ABC + area of ACD
= ab sinB n B + cd sin D,
soB th
so that b

4A = 2ab sin B + 2cd sin D... (1).