Plane trigonometry, by S.L. Loney.

254

TRIGONOMETRY.

[Exs. XXXVIII.]

4. Prove that the area of any quadrilateral is one-half the product of
the two diagonals and the sine of the angle between them.
5. If a quadrilateral can be inscribed in one circle and circumscribed
about another circle, prove that its area is V/abcd and that the radius of
the latter circle is
2 V'abcd
a+b+c+d'
6. A quadrilateral ABCD is described about a circle; prove that
A B 0. D
AB sin - sin - = CD sin - sin.
2 2 '2 2
7. a, b, c, and d are the sides of a quadrilateral taken in order, and a
is the angle between the diagonals opposite to b or d; prove that the area
of the quadrilateral is
1 2
(a2- b+ c2 - d2) tan a.
8. If a, b, c, d be the sides and x and y the diagonals of a quadrilateral, prove that its area is
[4x22 - (b2 + d2 - a2 - c2)2?]2
4
9. If a quadrilateral can be inscribed in a circle, prove that the angle
between its diagonals is
sin-1 [2/(s - a) (s - b) (s - c) (s - d) -(ac + bd)].
If the same quadrilateral can also be circumscribed about a circle, prove
that this angle is then
ac - bd
COS-1 --
ac + bd
10. The sides of a quadrilateral are divided in order in the ratio
m: n, and a new quadrilateral is formed by joining the points of division;
prove that its area is to the area of the original figure as mP1+n2 to
(m + n)2.
11. If a quadrilateral can be inscribed in a circle, prove that the
radius of the circle is

1 /(ab + cd) (ac + bd) (ad + bc)
4 (s- a) (s -b) (s- c) (s- d) '