or, and AD BD $ 278 (the homologous sides of similar S are proportional). ..BDXCE=AD X B C. Again, in the A ABE and B C D, LABE= DBC, Cons. Z BAE=2BDC, $ 203 (each being measured by t of the arc BC). .. A ABE and B C D are similar, (two A are similar when two & of the one are equal respectively to two & of the other). : BD, the longest side of the other, § 278 A B A E BD - CD ..BD X A E= A B XC D. Adding these two equalities, BD (A E + C E) = A B XCD + A D X BC, or BD X A C= AB XCD + AD 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. ON CONSTRUCTIONS. PROPOSITION XXI. PROBLEM. 302. To divide a given straight line into equal parts. B Let A B be the given straight line. Take any convenient length, and apply it to A O 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 Il to C B. These lines divide A B into equal parts, $ 274 (if a series of Ils 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. н к в As 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 the indefinite line A X. CE=n, EF=0. Draw FB. From E and C draw E K and C H Il to FB. K and H are the division points required. (A K AHHK KB For § 275 of (E) = A C = C E = EF'. (a line drawn through two sides of a A ll to the third side divides those sides proportionally). ..AH : HK : KB :: AC : CE : EF. Substitute m, n, and o for their equals A C, C E, and E F. Then AH : H K : KB :: m :n : 0. Q. E. F. PROPOSITION XXIII. PROBLEM. 304. To find a fourth proportional to three given straight lines. AS ---- B ----- 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. · 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. Join BC. CE is a third proportional to A B and AC. § 251 AB AC $ 275 BD - CE' (a line drawn through two sides of a All to the third side divides those sides proportionally). Substitute, in the above equality, A C for its equal BD; AB:AC :: AC : CE. or, Q. E. F. |