235. The following exercise consists of miscellaneous questions involving the properties of triangles. EXAMPLES. XVIII. e. 1. If the sides of a triangle are 242, 1212, 1450 yards, shew that the area is 6 acres. 2. One of the sides of a triangle is 200 yards and the adjacent angles are 22.5° and 67.5°: find the area. 3. If r1=212=2r3, shew that 3a=4b. 4. If a, b, c are in A. P., shew that r1, 72, 73 are in H. P. 5. Find the area of a triangle whose sides are 6. If sin A sin C=sin (A – B) : sin (B–C), shew that a2, b2, c2 are in A. P. 9. (r2+r3)(~3+r1) (r1+r2)=4R ( ̃1⁄2o3+~3~1+~1~2). 13. The perimeter of a right-angled triangle is 70, and the in-radius is 6: find the sides. 14. If f, g, h are the perpendiculars from the circum-centre on the sides, prove that 15. An equilateral triangle and a regular hexagon have the same perimeter: shew that the areas of their inscribed circles are as 4 to 9. *16. Shew that the perimeter of the pedal triangle is equal to abc/2R2. *17. Shew that the area of the ex-central triangle is equal to abc (a+b+c)/4A. 18. In the ambiguous case, if A, a, b are the given parts, and c1, c2 the two values of the third side, shew that the distance between the circum-centres of the two triangles is C1 ~c2 2 sin A' *19. If ẞ be the angle between the diagonals of a cyclic quadrilateral, shew that sin B= 2S ac+bd *20. Shew that g3. II. II. II3=IA2. IB2. IC2. *21. Shew that the sum of the squares of the sides of the ex-central triangle is equal to 8R (4R+r). *22. If circles can be inscribed in and circumscribed about a quadrilateral, and if ß be the angle between the diagonals, shew that 23. If l, m, n are the lengths of the medians of a triangle, prove that (1) 4 (12+m2+n2)=3(a2+b2+c2); (2) (b2-c2) 12+(c2 — a2) m2+(a2 — b2) n2=0; (3) 16 (74+m2+n1) = 9 (a1+b2+c1). 24. Shew that the radii of the escribed circles are the roots of the equation x3- (4R+r) x2+s2x − s2r=0. 25. If A1, A2, A, be the areas of the triangles cut off by tangents to the in-circle parallel to the sides of a triangle, prove that *26. The triangle LMN is formed by joining the points of contact of the in-circle; shew that it is similar to the ex-central triangle, and that their areas are as r2 to 4R2. 27. In the triangle PQR formed by drawing tangents at A, B, C to the circum-circle, prove that the angles and sides are 180°-24, 180°-2B, 180° - 2C′; 28. If p, q, r be the lengths of the bisectors of the angles of a triangle, prove that 29. If the perpendiculars AG, BH, CK are produced to meet the circum-circle in L, M, N, prove that (1) area of triangle LMN=8A cos A cos B cos C; (2) AL sin A+BM sin B+CN sin C=8R sin A sin B sin C. 30. If ra, b, re be the radii of the circles inscribed between the in-circle and the sides containing the angles A, B, C respectively, shew that π-A (1) ra=rtan2 ; (2) √rore+√rera+√rarb=r. *31. Lines drawn through the angular points of a triangle ABC parallel to the sides of the pedal triangle form a triangle XYZ shew that the perimeter and area of XYZ are respectively 2R tan A tan B tan C and R2 tan A tan B tan C. *32. A straight line cuts three concentric circles in A, B, C and passes at a distance p from their centre: shew that the area of the triangle formed by the tangents at A, B, C is BC.CA.AB MISCELLANEOUS EXAMPLES. F. 1. If a+B+y+8=180°, shew that cos a cos B+cos y cos d=sin a sin ẞ+sin y sin d. 2. Prove that cos (15° – A) sec 15° – sin (15° – A) cosec 15° = 4 sin A. 3. Shew that in a triangle cot A+ sin A cosec B cosec C retains the same value if any two of the angles A, B, C are interchanged. 4. If a=2, b=√8, A=30°, solve the triangle. 5. Shew that (1) cot 18°=√5 cot 36° ; (2) 16 sin 36° sin 72° sin 108° sin 144°=5. 6. Find the number of ciphers before the first significant digit in ('0396), given log 2=30103, log 3=47712, log 11=104139. 7. An observer finds that the angle subtended by the line joining two points A and B on the horizontal plane is 30°. On walking 50 yards directly towards A the angle increases to 75° : find his distance from B at each observation. 8. Prove that cos2 a+cos2 B+cos2 y+cos2 (a+B+y) 9. Shew that =2+2 cos (B+y) cos (y+a) cos (a+B). (1) tan 40°+cot 40°=2 sec 10°; (2) tan 70°+tan 20° = 2 cosec 40°. 10. Prove that (1) 2 sin 4a-sin 10a+sin 2a=16 sin a cos a cos 2a sin2 3a; 11. If B=30°, b=3√2-√6, c=6-2/3, solve the triangle. 12. From a ship which is sailing N.E., the bearing of a rock is N.N.W. After the ship has sailed 10 miles the rock bears due W. find the distance of the ship from the rock at each observation. 14. If cos (0-a), cose, cos (0+a) are in harmonical progression, shew that' 15. If sin ß be the geometric mean between sin a and cos a, prove that are cos 28=2 cos2 (+a). 16. Shew that the distances of the orthocentre from the sides 2R cos B cos C, 2R cos Ccos A, 2R cos A cos B. 18. If the sides of a right-angled triangle are 2 (1+sin 6)+cos and 2 (1 + cos 0) + sin 0, prove that the hypotenuse is 3+2 (cos +sin @). *19. Prove that the distances of the in-centre of the excentral triangle III, from its ex-centres are *20. Prove that the distances between the ex-centres of the ex-central triangle III, are 3 |