502. The bisector of an angle of a triangle divides the opposite side into segments that are proportional to the adjacent sides of the angle. Α Let BD be the bisector of B of the ▲ ABC. CONVERSELY. A line drawn through the vertex of an angle of a triangle, dividing the opposite side into segments pro portional to the adjacent sides of the angle, bisects the 503. EXERCISE. = Z1=22. (?) and The triangle ABC has AB = 8 in., BC = 6 in., AC 12 in. BD bisects B. What are the lengths of the segments into which it divides AC? 504. EXERCISE. BD is the bisector of B in the triangle ABC. The segments of AC are AD = 5 in. and DC = 2 in. The sum of the sides AB and BC is 14 in. Find the lengths of AB and BC. 505. EXERCISE. Construct a triangle having given two sides and one of the two segments into which the third side is divided by the bisector of the opposite angle. (Two constructions.) 506. DEFINITION. A point C, taken on the line AB between the points A and B, is said to divide the line AB internally into two segments, CA and CB. A point c', taken on AB pro duced, is said to divide AB C externally into two segments, C'A and C'B. In each case, the segments are the distances from C' (or c') to the extremities of AB. PROPOSITION XX. THEOREM 507. The bisector of an exterior angle of a triangle divides the opposite side externally into two segments that are proportional to the adjacent sides of the angle. F Α Let BD bisect the exterior / CBF of the ▲ ABC. 508. EXERCISE. The lengths of the sides of a triangle are 4, 5, and 6 yards, respectively. Find the lengths of the segments into which the bisector of the angle exterior to the largest angle of the triangle divides the opposite side externally. CONVERSELY. A line drawn through the vertex of an angle of a triangle dividing the opposite side externally into segments proportional to the adjacent sides of the angle, bisects the exterior angle. it is divided internally and externally in the same ratio. 510. EXERCISE. The bisector of an angle of a triangle and the bisector of its adjacent exterior angle divide the opposite side harmonically. (§§ 502, 507.) 511. EXERCISE. To divide a line internally and externally so that its segments shall have a given ratio, i.e. to divide a line harmonically. Let AB be the given line, and m and n lines in the given ratio. Required to divide AB internally and externally into segments having Draw EC and prolong it until it meets AB prolonged at some point F. By means of similar triangles, show 512. DEFINITION. If the line AB is divided harmonically at C and D, and the four points A, B, C, and D are connected with any other point 0, the resulting figure is called a harmonic pencil. The point o is called the vertex of the pen cil, and the four lines OA, OC, OB, and OD are called rays. A C B 513. EXERCISE. O-ACBD is a harmonic pencil. through C parallel to OD, and limited EF is drawn Multiply (1), (2), and (3) together member by member. Q.E.D. |