BOOK IV PROPORTIONAL LINES AND SIMILAR FIGURES Proposition I 289. 1. Draw a line parallel to the base of a triangle through the middle point of one side and cutting the other side. How do the segments of the other side compare in length? 2. Draw a line parallel to the base one fourth, one sixth, or any part of the distance from the extremity of the base to the vertex. How do the segments of the other side compare? 3. How does the ratio of one of these sides to either of its segments compare with the ratio of the other to its corresponding segment? Theorem. A line which is parallel to one side of a triangle and meets the other two sides divides those sides proportionally. Data: Any triangle, as ABC, and any line parallel to AB, as DE, meeting AC and BC in D and E, respectively. D E To prove CD: DA=CE: EB. A M B Proof. Case I. When CD and DA are commensurable. Suppose that M is a unit of measure common to CD and DA, and that M is contained in CD 3 times and in DA 2 times. Divide CD and DA into parts each equal to the common measure M, and from each point of division draw lines parallel to AB. § 157, these lines divide CE into 3 and EB into 2 equal parts; and, Ax. 1, CE: EB CD: DA 3:2, Case II. When CD and DA are incommensurable. Since CD and DA are incommensurable, suppose that CD and DF are commensur able and that FA < M. If M is indefinitely diminished, the ratios CD: DF and CE: EG remain equal, and indefinitely approach their limiting ratios CD: DA and CE: EB, respectively. Hence, § 222, CD: DACE: EB. Therefore, etc. Q.E.D. 290. Cor. A line which is parallel to one side of a triangle and meets the other two sides divides them so that one side is to either of its segments as the other side is to its corresponding segment. Proposition II 291. Draw a line dividing each of two sides of a triangle into halves, or into other proportional parts. What is the direction of this line with reference to the third side? Theorem. A line which divides two sides of a triangle proportionally is parallel to the third side. (Converse of Prop. I.) Proof. If DE is not parallel to AB, some other line drawn through D will be parallel to AB. Suppose that DF is that line. But this is impossible unless F coincides with E; that is, Ax. 11, unless DF coincides with DE. Therefore, the hypothesis, that some line other than DE drawn through D is parallel to AB, is untenable. Hence, Therefore, etc. DE AB. Proposition III Q.E.D. 292. Draw a triangle whose sides are 6", 5", and 3", or any other dimensions; bisect any one of its angles and produce the bisector to meet the opposite side. How does the ratio of the segments of this side made by the bisector compare with the ratio of the sides of the triangle adjacent to these segments? Theorem. The bisector of an angle of a triangle divides the opposite side into segments which are proportional to the adjacent sides. Data: Any triangle, as ABC, and CD the bisector of one of its angles, ACB. To prove AD: DB AC: CB. A D Proof. From B draw a line parallel to CD and meeting AC produced in E. Substituting CB in the proportion for its equal CE, Therefore, etc. AD: DB AC: CB. Why? Why? Why? Q.E.D. Proposition IV 293. Draw a triangle whose sides are 6", 5", and 3", or any other dimensions; bisect an exterior angle at any vertex and produce the bisector to meet the opposite side produced. How does the ratio of the distances from the point of meeting to each extremity of the opposite side compare with the ratio of the other sides of the triangle? Theorem. The bisector of an exterior angle of a triangle meets the opposite side produced at a point the distances of which from the extremities of this side are proportional to the other two sides. Substituting BC in the proportion for its equal FC, Therefore, etc. AE: BE AC: BC. Why? Why? Why? Q.E.D. 294. Sch. This proposition is not true, if the triangle is isosceles or equilateral. Why? Ex. 416. The base of a triangle is 10 ft. and the other sides 8 ft. and 12 ft. Find the segments of the base made by the bisector of the vertical angle. Ex. 417. The sides AC and BC of the triangle ABC are 5 ft. and 8 ft. respectively. If a line drawn parallel to the base divides AC into segments of 2 ft. and 3 ft., what are the segments into which it divides BC? 295. If a straight line is divided at a point between its extremities, it is said to be divided internally. The line is equal to the sum of the internal segments. If a straight line is produced and divided at a point on the part produced, it is said to be divided externally. The line is equal to the difference between the external segments. In Fig. 2, AB is produced and divided externally at E, and -B 296. If a line is divided at a given point so that one segment is a mean proportional between the whole line and the other segment, it is said to be divided in extreme and mean ratio. AB is divided internally in extreme and mean ratio at the point C. In Fig. 1, if AB is divided externally in extreme and mean ratio at the point E. 297. If a line is divided internally and externally into segments which have the same ratio, it is said to be divided harmonically. and AB is divided harmonically at the points C and E. Ex. 418. A line drawn parallel to the base of triangle ABC divides AC into segments of 3dm and 8dm respectively, and the segment of BC, corresponding to 3dm, is 5dm, What is the length of BC? |