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104. Escribed Circle. To find the radii of the escribed circles of a triangle.

A circle, which touches one side of a triangle and the other two sides produced, is called an escribed circle of the triangle.

Let O be the centre of the escribed circle which touches the side BC and the other sides produced, at the points D, E, and F, respectively, and let the radius of this circle be r1.

We then have from the figure

E

= { r1(b + c − a) =rı(s—a). (Art. 99)

Similarly it may be proved that if r1⁄2, are the radii of the circles touching AC and AB respectively,

therefore the arc BD is equal to the arc DC, and DOH bisects BC at right angles.

Draw IM perpendicular to AC. Then

1. The sides of a triangle are 18, 24, 30; find the radii of its inscribed, escribed, and circumscribed circles.

Ans. 6, 12, 18, 36, 15.

2. Prove that the area of the triangle ABC is

3. Find the area of the triangle ABC when

(1) a=4, b= 10 ft., C = 30°.

(2) b=5, c = 20 inches, A = 60°.

14, c = 15 chains.

(3) a = 13, b =

Ans. 10 sq. ft.

43.3 sq. in. 84 sq. chains.

6. Prove that the area of the triangle ABC is represented by each of the three expressions:

2 R2 sin A sin B sin C,

rs, and

Rr (sin A+ sin B + sin C).

7. If A = 60°, a=√3, b=√2, prove that the area =4(3+√3).

8. Prove R(sin A+ sin B + sin C) = s.

9. Prove that the bisectors of the angles A, B, C, of a triangle are, respectively, equal to

Let ABCD be the quadrilateral, and a, b, c, and d its sides.

Join BD.

S

ad sin A+ be sin C

= (ad + bc) sin A

=

(1)

Now in ▲ ABD, BD2=a2+d2 - 2 ad cos A,

B

(2 ad+2bc)2- (a2 — b2 — c2 + d2)2

2 (ad + bc)

*See Geometry, Art. 251.

√[(a + d)2 − (b − e)2] [ (b + c)2 - (a — d)2]

2(ad+bc)

√(a+d+b−c)(a+d−b+c)(b+c+a−d)(b+c−a+d)

The more important formulæ proved in this chapter are summed up as follows:

1

= √2b2c2+2 c2a2 + 2 a2b2 — aa — ba — c1.

2 bc

8. Area of A= √s (s—a) (s—b) (s—c).

In a right triangle ABC, in which C is the right angle,

9. (sin Asin B)2+ (cos A+ cos B)2 = 2.

In any triangle ABC, prove the following statements: