Plane trigonometry, by S.L. Loney.

228

TRIGONOMETRY.

[Exs. XXXV.]

17. The lengths of two sides of a triangle are one foot and ^/2 feet
respectively and the angle opposite the shorter side is 30~; prove that
there are two triangles satisfying these conditions, find their angles, and
shew that their areas are in the ratio,3+1: J3-1.
18. Find by the aid of the tables the area of the larger of the two
triangles given by the data
A=31~ 15', a=5 ins. and b=7ins.
199. On the circles connected with a given
triangle.
The circle which passes through the angular points of
a triangle ABC is called its circumscribing circle or, more
briefly, its circumcircle. The centre of this circle is
found by the construction of Euc. IV. 5. Its radius is
always called R.
The circle which can be inscribed within the triangle
so as to touch each of the sides is called its inscribed
circle or, more briefly, its incircle. The centre of this
circle is found by the construction of Euc. IV. 4. Its radius
will be denoted by r.
The circle which touches the side BC and the two
sides AB and AC produced is called the escribed circle
opposite the angle A. Its radius will be denoted by rl.
Similarly r2 denotes the radius of the circle which
touches the side CA and the two sides BC and BA
produced. Also r3 denotes the radius of the circle touching AB and the two sides CA and CB produced.
200. To find the magnitude of R, the radius of the
circumcircle of any triangle ABC.
Bisect the two sides BC and CA in D and E respectively, and draw DO and EO perpendicular to BC and CA.