we have .. AF-AE, BD=BF, CD=CE; AF+(BD+CD) = half the sum of the sides; .. AF+a=s. .. AF=8-a=AE. Similarly, BD=BF=s-b, CD=CE=s-c. B 2 C Similarly, r=(s-b) tan, r=(s-c) tan. Again, AF AE, BF1=BD1, CE1=CD1; .. 2AF AF1+AE1=(AB+BD1)+(AC+CD1) = EXAMPLES. XVIII. a. 1. Two sides of a triangle are 300 ft. and 120 ft., and the included angle is 150°; find the area. 2. Find the area of the triangle whose sides are 171, 204, 195. 3. Find the sine of the greatest angle of a triangle whose sides are 70, 147, and 119. 4. If the sides of a triangle are 39, 40, 25, find the lengths of the three perpendiculars from the angular points on the opposite sides. 5. One side of a triangle is 30 ft. and the adjacent angles are 22 and 112°, find the area. 6. Find the area of a parallelogram two of whose adjacent sides are 42 and 32 ft., and include an angle of 30°. 7. The area of a rhombus is 648 sq. yds. and one of the angles is 150°: find the length of each side. 8. In a triangle if a=13, b=14, c=15, find r and R. 9. Find 71, 72, 73 in the case of a triangle whose sides are 17, 10, 21. 10. If the area of a triangle is 96, and the radii of the escribed circles are 8, 12, 24, find the sides. 31. If the perpendiculars from A, B, C to the opposite sides are P1, P2, P3 respectively, prove that 34. 4A (cot A+cot B+cot C)=a2+b2+c2. 36. a2b2c2 (sin 24+ sin 2B+ sin 2C)=32A3. 37. a cos A+bcos B+c cos C-4R sin A sin B sin C. 38. a cot A+b cot B+c cot C=2(R+r). A B 39. (b+c) tan+(c+a) tan+(a+b) tan C =4R (cos A+cos B+cos C). 40. r (sin A+ sin B+sin C)=2R sin A sin B sin C. Inscribed and circumscribed Polygons. 214. To find the perimeter and area of a regular polygon of n sides inscribed in a circle. Let r be the radius of the circle, and AB a side of the polygon. Join OA, OB, and draw OD bisecting LAOB; then AB is bisected at right angles in D. Perimeter of polygon=nAB=2nAD=2n0A sin AOD Area of polygon=n (area of triangle AOB) 215. To find the perimeter and area of a regular polygon of n sides circumscribed about a given circle. Letr be the radius of the circle, and AB a side of the polygon. Let AB touch the circle at D. Join OA, OB, OD; then OD bisects AB at right angles, and also bisects LAOB. Area of polygon=n (area of triangle AOB) B 216. There is no need to burden the memory with the formulæ of the last two articles, as in any particular instance they are very readily obtained. Example 1. The side of a regular dodecagon is 2 ft., find the radius of the circumscribed circle. Let r be the required radius. In the adjoining figure we have Example 2. A regular pentagon and a regular decagon have the same perimeter, prove that their areas are as 2 to √5. Let AB be one of the n sides of a regular polygon, O the centre of the circumscribed circle, OD perpendicular to AB. Each side of the decagon is c, and its area is c2 cot π 10 |