8. In a spherical triangle if A=B = 2C shew that

9. If the equal sides of an isosceles triangle ABC be bisected by an arc DE, and BC be the base, shew that

10. If c,, c, be the two values of the third side when A, a, b are given and the triangle is ambiguous, prove that

VII.

CIRCUMSCRIBED AND INSCRIBED CIRCLES.

89. To find the angular radius of the small circle inscribed in a given triangle.

Let ABC be the triangle; bisect the angles A and B by arcs meeting in P; from P draw PD, PE, PF perpendicular to the sides. Then it may be shewn that PD, PE, PF are all equal; also that AE = AF, BF=BD, CD=CE. Hence BC + AF = half

the sum of the sides = s; therefore AF=8· -α. Let PF=r.

The value of tan r may be expressed in various forms; thus from Art. 45, we obtain

tan r=

√cos S cos (S – A) cos (S– B) cos (S′ – C')}

2 cos A cos B cos C

4 cos 4 cos Bcos C= cos S+cos(S-A) + cos(S-B) + cos (S — C');

90. To find the angular radius of the small circle described so as to touch one side of a given triangle, and the other sides produced.

B

Let ABC be the triangle; and suppose we require the radius of the small circle which touches BC, and AB and AC produced. Produce AB and AC to meet in A'; then we require the radius of the small circle inscribed in A'BC, and the sides of A'BC are a, π-b, π-c respectively. Hence if r, be the required radius, and s denote as usual § (a + b + c), we have from Art. 89,

A

tan r
= tan

sin s

2

(1).

From this result we may derive other equivalent forms as in the preceding article; or we may make use of those forms immediately, observing that the angles of the triangle ABC are A, -B, π-C respectively. Hences being (a+b+c) and S being (A+B+C) we shall obtain

These results may also be found independently by bisecting two of the angles of the triangle A'BC, so as to determine the pole of the small circle, and proceeding as in Art. 89.

91. A circle which touches one side of a triangle and the other sides produced is called an escribed circle; thus there are three escribed circles belonging to a given triangle. We may denote the radii of the escribed circles which touch CA and AB respectively by r, and r,, and values of tan r, and tan r, may be found from what has been already given with respect to tan r, by appropriate changes in the letters which denote the sides and angles.

In the preceding article a triangle ABC was formed by producing AB and AC to meet again in A'; similarly another triangle may be formed by producing BC and BA to meet again, and another by producing CA and CB to meet again. The original triangle ABC and the three formed from it have been called associated triangles, ABC being the fundamental triangle. Thus the inscribed and escribed circles of a given triangle are the same as the circles inscribed in the system of associated triangles of which the given triangle is the fundamental triangle.

92. To find the angular radius of the small circle described about a given triangle.

Let ABC be the given triangle; bisect the sides CB, CA in D and E respectively, and draw from D and E arcs perpendicular to CB and CA respectively, and let P be the intersection of these

arcs.

Then P will be the pole of the small circle described about ABC. For draw PA, PB, PC; then from the right-angled triangles PCD and PBD it follows that PB = PC; and from the right-angled triangles PCE and PAE it follows that PA PC; hence PA = PB = PC. Also the angle PAB Also the angle PAB = the angle PBA, the angle PBC= the angle PCB, and the angle PCA = the angle PAC; therefore PCB + A = 1⁄2 (4+ B+ C'), and PCB = S – A.

The value of tan R may be expressed in various forms; thus

α

if we substitute for tan from Art. 49, we obtain

tan R =

cos (S—A) cos (S – B) cos (S – C)

Again cos (S- A) = cos {} (B + C) − 1⁄2 4}

= cos (B+C) cos § A + sin † (B + C) sin & A sin cos A

{cos (b+c) + cos (b- c)}, (Art. 54,)

cos bcosc;

cos a

sin A

cosa

Substitute in the last expression the value of sin A from