Use this method, or that of Example (20), to find a by the aid of the Tables, when b=347, c=293, A=39° 42′. (22) Prove that in any triangle [See Art. 260.] (23) If D, E, F are the middle points of BC, CA, AB, then (24) If x=II1, y=II2, z=II ̧, d=2R, prove that xyz+d(x2+ y2+22)=4d3. (25) The circle escribed to the side BC of the triangle ABC will be the inscribed circle of the triangle whose sides are (26) If a point Q be taken within ABC such that the angles BQC, CQA, AQB are each 120o, prove that (27) If sin A and cos A are both given, prove that in general n different values and no more can be assigned to sin A n (28) If from a point on a circular arc (of radius r) which subtends an angle a at the centre, perpendiculars are drawn to its bounding radii, then the distance between the feet of the perpendiculars is r sin a. (29) If two circles, radii a and b, touch externally, the angle between their common tangents = 2 tan-1√ b (30) If the inscribed circle touches the sides in DEF, prove that EF: FD: DE=cos B C A : COS : COS • 2 2 (31) If the bisectors of the angles A, B, C meet the opposite 2bc A sides in D, E, F then (i) AD= COS (ii) CD= (iii) The area of DEF b+c B a sin B sin B+sin C C sin sin en 2 2 B-C C-A A-B COS COS COS 2 2 2 (32) If a, ß, y, d are the angles of a quadrilateral inscribed in a circle, then cos (a+B). cos (3+ y). cos (y+8). cos (8+ a) = (1 - cos2 a - cos2 B)2. (33) If 01 and 62 be two values of found from the equation a cos 0+b sin 0=c, then 100+ sin +1.00 -0. prove that cos 20+ cos 20+ cos 2x + 4 cos 0. cos.cosx+1=0, hence [see Ex. (20), p. 143], prove that (0±±x)=(2n+1) π. (35) A circle is drawn touching the inscribed circle and the sides AB, AC (not produced) of the triangle ABC. Another circle is drawn touching the circle and the same side. Prove that the radius of the nth circle thus drawn is r 1-sin 1 + sin that the sum of the radii of all the circles, when n=∞, is and (36) If AO, BO, CO pass through the centre of the circumscribing circle and meet the opposite sides on DEF, then (37) If h and k be the lengths of the diagonals of a quadrilateral and the angle between them, prove that the area of the quadrilateral ishk sin 0. (39) Two regular polygons of n and 2n sides are described such that the circle inscribed in the first circumscribes the second; also the radius of the circle inscribed in the second is to that of the circle circumscribing the first as 3+√√3 is to 4√2. Prove that n=6. A N (40) Prove by the aid of a figure like that of Art. 282 that I102=R2+2Rr1. (41) If yz + zx+xy=1, prove by Trigonometry that |