or, § 278 A D BD CE BO' ... BD X CE A D X BC. Again, in the A ABE and B C D, LABE= 2 DBC, Cons. § 203 and ZBA E= BDC, $ 280 (two A are similar when two ts of the one are equal respectively to two 5 of the other). Whence A B, the longest side of the one, : BD, the longest side of the other, or, § 278 А В A E BD CD BD (A E + C E) = A B X C D + A D X BC, or BD X AC=AB XCD + A D X BC. Q. E. D. Ex. If two circles are tangent internally, show that chords of the greater, drawn from the point of tangency, are divided proportionally by the circumference of the less. 302. To divide a given straight line into equal parts. А B 0 Let A B be the given straight line. It is required to divide A B into equal parts. From A draw the indefinite line A 0. Take any convenient length, and apply it to A 0 as many times as the line A B is to be divided into parts. From the last point thus found on A 0, as C, draw C B. Through the several points of division on A O draw lines ll to C B. These lines divide A B into equal parts, § 274 (if a series of lls intersecting any two straight lines, intercept equal parts on one of these lines, they intercept equal parts on the other also). Q. E. F. Ex. To draw a common tangent to two given circles. PROPOSITION XXII. PROBLEM. 303. To divide a given straight line into parts proportional to any number of given lines. Let A B, m, n, and o be given straight lines. It is required to divide A B into parts proportional to the given lines m, n, and o. Draw FB. From E and C draw E K and CH || to FB. K and H are the division points required. (AE) $ 275 А К АН HK KB CE (a line drawn through two sides of a A ll to the third side divides those sides proportionally). .:. A H : HK : KB : :AC : CE : EF. Substitute m, n, and o for their equals AC, CE, and EF. Then AH : HK : KB : : m :n : 0. Q. E. F. 304. To find a fourth proportional to three given straight lines. Let the three given lines be m, n, and o. It is required to find a fourth proportional to m, n, and o. Take A B equal to n. Draw the indefinite line A R, making any convenient 2 with A B. On A R take A C=m, and CS=0. Draw C B. From S draw SF || to C B, to meet A B produced at F. B F is the fourth proportional required. For, AC : AB ::CS : BF, $ 275 (a line drawn through two sides of a A ll to the third side divides those sides proportionally). Substitute m, n, and o for their equals A C, A B, and C S. Then m :n ::0 : BF. Q. E. F. PROPOSITION XXIV. PROBLEM. 305. To find a third proportional to two given straight lines. Let A B and AC be the two given straight lines. It is required to find a third proportional to A B and A C. Place A B and A C so as to contain any convenient Z. Produce A B to D, making BD= AC. Join BC. Through D draw D E ll to B C to meet A C produced at E. C E is a third proportional to A B and A C. § 251 А С For, AB $ 275 CE' (a line drawn through two sides of a A ll to the third side divides those sides proportionally). Substitute, in the above equality, A C for its equal B D; Then АВ AC or, AB: AC :: AC : CE. Q. E. F. |