This last form of the proportion appears very frequently. In general form it is stated as follows: 374. COROLLARY 1. If a line is drawn through two sides of a triangle, parallel to the third side, A B one of those sides is to either part cut off by the parallel line as the other side is to the corresponding part. 375. COROLLARY 2. If two tines are cut by any number of parallels, the corresponding segments are proportional. Then But Draw BK and DL parallel to AG. m' = m, p" = p' = p, and r' = r. Why? § 372 PROPOSITION II. PROBLEM 376. To construct the fourth proportional to three given lines. Required to construct the fourth proportional to m, n, and p. Construction. 1. Draw any angle GAH. 2. On AG take AB = m, BC = n; on AH take AD = p. 3. Draw BD. Through C draw a line parallel to BD, meeting AH in E. Then DE is the required fourth proportional. Proof. (To be completed.) PROPOSITION III. PROBLEM 377. To divide a given line into segments proportional to any number of given lines. Given the lines AB, m, n, and p. Required to divide AB into segments proportional to m, n, and p. Construction. 1. Draw AC, making any convenient angle with AB. (Construction and proof to be completed.) EXERCISES 1. In how many ways can the construction of § 376 be made? Will the result be the same in each case? (Check by measurement.) 2. Find a third proportional to two given lines m and n. 3. If a, b, c, d, and e are five given lines, how would you construct 4. If a, b, c, and d are four given lines, how would you construct 5. Two lines, a and b, are the dimensions of a rectangle. Construct the altitude of an equal rectangle whose base is equal to a third given line c, without actually constructing the first rectangle. 6. Construct the altitude of a rectangle of base b, which is equal to a square of side c, without actually constructing the square. 7. Divide a given square into two rectangles which are to each other as two given lines mand n. 8. If m, n, p, q, and rare the sides of a polygon, and AG is a given line, show how Proposition III may be applied to construct the sides of a polygon similar to the given polygon, with AG as a side homologous to m. PROPOSITION IV. THEOREM 378. If a line divides two sides of a triangle proportionally, it is parallel to the third side. Given the triangle ABC, and the line DE drawn so that or Proof. 1. From the given proportion, by addition, AD + DB: AD = AE + EC : AE, AB: AD = AC : AE. § 364 2. Through D draw a line DG || to BC, and suppose that it Discussion. Why is the addition form of the proportion used in the above proof? (This theorem is the converse of Proposition I, § 372.) SIMILAR TRIANGLES PROPOSITION V. THEOREM 379. If two triangles have the angles of one equal respectively to the angles of the other, the triangles are similar. Given the triangles ABC and A'B'C', having the angles A, B, C equal to the angles A', B', C' respectively. Proof. 1. On AB lay off AD = A'B', and on AC lay off 5. In like manner, by laying off the corresponding sides of AA'B'C' from B on BA and BC it can be shown that AB: A'B' = BC : B'C'. That is, the homologous sides of AABC and A'B'C' are 380. COROLLARY 1. If two triangles have two angles of one equal respectively to two angles of the other, the triangles are 381. COROLLARY 2. If two right triangles have an acute angle of one equal to an acute angle of the other, the right triangles are similar. similar. 382. COROLLARY 3. If two isosceles triangles have the angle at the vertex, or a base angle of one equal to the corresponding angle of the other, they are similar. 383. COROLLARY 4. Two equilateral triangles are similar. 384. COROLLARY 5. If two triangles have their sides respectively parallel, they are similar. Produce one or both of two nonparallel sides until they meet. It can then be shown that the triangles have their angles respectively equal. 385. COROLLARY 6. If two triangles have their sides respectively perpendicular, they are similar. The proof is similar to that of Corollary 5. REMARK. It can be shown that in Corollary 5 and Corollary 6 the parallel sides and the perpendicular sides respectively are homologous sides. EXERCISES 1. The diagonals of any trapezoid divide each other in the same ratio. 2. A line parallel to one side of a triangle, cutting the other two sides, produced if necessary, forms a tri angle similar to the given triangle. 3. In the accompanying figure ABCD is a parallelogram. AB is produced to E. Find all the similar triangles. D A B E F C 4. ABCD is any quadrilateral inscribed in a circle. The diago nals intersect at O. Prove that AOB~△COD. |