(the homologous sides of similar ▲ are proportional). .. BDX CE = ADX BC. Again, in the A ABE and B C D, $278 (two are similar when two of the one are equal respectively to two of the other). (the homologous sides of similar are proportional). § 278 But Adding these two equalities, or BD (AE+CE) = ABX CD + ADX BC, BDXACABX CD + ADX 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. ON CONSTRUCTIONS. PROPOSITION XXI. PROBLEM. 302. To divide a given straight line into equal parts. A 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 O. Take any convenient length, and apply it to AO as many times as the line AB is to be divided into parts. From the last point thus found on A O, as C, draw C B. Through the several points of division on A O draw lines Il 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. I. When the common tangent is exterior. II. When the common tangent is interior. 303. To divide a given straight line into parts proportional to any number of given lines. Let AB, 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. (a line drawn through two sides of a ▲l to the third side divides those sides proportionally). .. AH: HK:KB:: ACCE: EF. Substitute m, n, and o for their equals A C, C E, and E F. Then AH: HK: KB:: m n o. Q. E. F. PROPOSITION XXIII. PROBLEM. 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. Draw the indefinite line A R, making any convenient with A B. On A R take A C = m, and C S = o. Draw CB. From S draw SF to CB, to meet A B produced at F. BF is the fourth proportional required. For, : AC AB CS: BF, § 275 (a line drawn through two sides of a ▲ | 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:o : BF. Q. E. F. lines. PROPOSITION XXIV. PROBLEM. 305. To find a third proportional to two given straight Let A B and A C 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 B D = A C. Join BC. Through D draw DE to BC to meet A C produced at E. CE is a third proportional to A B and A C. § 251 $275 (a line drawn through two sides of a ▲l to the third side divides those sides proportionally). Substitute, in the above equality, A C for its equal B D ; |