Modern geometry . fig. 16. A 1;CA + A IjAB - A liBC = A ^ow A liCA = ^IjYi. CA A liAB = |riC,A liBC = ^r^a; h + o — a 2 = A. But 6 + c — a = (a + 6 + c)-2ffl= 2s - 2a;.*. ri (s - a) = A;A Similarly r, =-^^, r,= ?^^. THE TRIANGLE 27 1 SR Ex. 48. Prove that in an equilateral triangle r=^R, r\=ri=ri=-^. Ex. 48. If the ex-centres be joined, the triangle so formed is similar tothe triangle XYZ. Ex. SO. Prove that the circle on llj as diameter passes through B and construct a triangle, having given BC, z B, and the length iii, Ex. 51. AZi + AYi = AB-|-AC + BC. Ex. 62. AYi = AZi = s. Ex.
Modern geometry . fig. 16. A 1;CA + A IjAB - A liBC = A ^ow A liCA = ^IjYi. CA A liAB = |riC,A liBC = ^r^a; h + o — a 2 = A. But 6 + c — a = (a + 6 + c)-2ffl= 2s - 2a;.*. ri (s - a) = A;A Similarly r, =-^^, r,= ?^^. THE TRIANGLE 27 1 SR Ex. 48. Prove that in an equilateral triangle r=^R, r\=ri=ri=-^. Ex. 48. If the ex-centres be joined, the triangle so formed is similar tothe triangle XYZ. Ex. SO. Prove that the circle on llj as diameter passes through B and construct a triangle, having given BC, z B, and the length iii, Ex. 51. AZi + AYi = AB-|-AC + BC. Ex. 62. AYi = AZi = s. AZ + AY = AB + AC-BO. AY = AZ=s-a. Ex. 55. ZZi = YYi=a. Ex. 56. BXi = CX=s-c. BX = CX, = s-6. Ex. 68. XXx=o ~ 6. Theorem 13. (i) AYi = AZi = s. (ii) AY = AZ = s - a. (iii) YYi = ZZi = a. (iv) BXj = CX = s-o. (v) XXi = c~6. 28. fig. 17. (i) AYi + AZi = AC + CYi + AB + BZ, = AC + CXi + AB + BXi (since tangents to a circle= AC + AB + BC from a point are equal) = 2«. But AYi = AZi,/. AYi = AZi = S. ao (iii) AY + AZ = AC - CY + AB - BZ= AC - CX + AB - BX= AC + AB-BC= 2« - 2a. But AY = AZ, /. AY = AZ = s - a. YYi = AYi - AY = s — {s — a) Similarly ZZj = a. THE TRIANGLE 29 (iv) BX, = BZi = AZi - AB = CX = « — c, by proof similar to (ii). (v) XXj = BC - CX - BXi = a - 2 (s — c) = a- (a + b + c) + 2o = c-b. If the figure were drawn with 6 > c, it would be found thatXXi =b-c. Ex. 69. Find the lengths of the segments into which- the point ofcontact of the in-cirole divides the hypotenuse of a right-angled trianglewhose sides are 6 and 8 feet. Ex. 60. The distance between X and the mid-point of BC is ^{b ~ c). Ex. 61. The in-radius of a right-angled triangle is equal to half thedifference between the sum of the sides and the hypotenuse. Ex. 62. If the diagonals of a quadrilateral ABCD intersect a
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