Page 8 - MATINF Nr. 6

8 E.A. Jos´e Garc´ıa

Fig. 2: A bicentric quadrilateral ABCD.

Proof. Since a + c = b + d in a bicentric quadrilateral, the formula (3) reduces to

α ad
cos 2 = .
2 ad + bc

Similarly we can get sin 2 α = bc . Now, following the same steps as in Brahmagupta’s
2 ad+bc
formula
◦
ad sin α bc sin (180 − α)
∆ 2 = +
2 2
ad sin α bc sin α
= +
2 2
α α
= sin cos (ad + bc)
2 2
Ê Ê
bc ad
= (ad + bc)
ad + bc ad + bc
√
= abcd.

The following theorem generalizes Theorem 1 for a general convex quadrilateral.

Theorem 5. Let a, b, c, d be the sides of a general convex quadrilateral, s is the semiperimeter,
and α and γ are opposite angles, then

α γ
ad sin 2 + bc cos 2 = (s − a)(s − d) (5)
2 2

and
γ α
bc sin 2 + ad cos 2 = (s − b)(s − c). (6)
2 2

Proof. By the Law of Cosines,

2
2
2
2
a + d − 2ad cos α = b + c − 2bc cos γ.