...they are in proportion by subtraction. This is sometimes called proportion by division. THEOREM 124. In a series of equal ratios the sum of the antecedents...consequents as any antecedent is to its consequent. THEOREM 98. If three parallels cut two transversals, the segments on one transversal have the same...

...similar to the given polygon. 288. Two regular polygons of the same number of sides are similar. 289. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as ... what? *2Q0. The perimeters of two similar polygons are to each other as any two corresponding sides....

...terms is to the second term as the difference between the last two terms is to the fourth term. 8. In a series of equal ratios the sum of the antecedents...consequents as any antecedent is to its consequent. 9. If a:b = c:d, then a8 : 6s = c3 : cf . 10. If a:6 = 6:c, then a:c = a2:62. 11. If a : 6 = 6 : c,...

...bd " bd ' • r •„ , ' — *- I Jo ( \ III, § 144] GENERAL THEOREMS 145 Theorem H. In a scries of equal ratios the sum of the antecedents is to the...consequents as any antecedent is to its consequent. Given a/b = c/d = e/f = •••, to prove that a + c + e + ••• _?_£._£_ b+d+f+- -b=d=f= '••'...

...transformation, divide the proportion resulting in (3) by that resulting in (4). 22. Theorem. — (9) In a series of equal ratios, the sum of the antecedents...consequents, as any antecedent is to its consequent. /,. ac ,ea Glven_ . ~ =?= | to prove CL (* P Proof: Since -, ~, -, etc., are all equal, denote the...

...equations have the same denominators; then divide the first by the second. Proposition IX. Theorem 197. In a series of equal ratios, the sum of the antecedents...consequents as any antecedent is to its consequent. „ „ . ace Hyp.: Grant -r— = = -,- . Det.: To prove that (a + c + e) : (b + d + f) = a : b. Proof:...

...Prove that if any number of quantities are in proportion any antecedent will be to its consequent as the sum of the antecedents is to the sum of the consequents. 9. The area of a circle is equal to the circumference multiplied by one-half the radius. Demonstrate....

...number of magnitudes are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents (see Euclid [E]). In modern symbols, if sn = X/«=i ai, then ai : o2 = (sn - an) . (sn - ai)Hence,...

...d = a : c. 3. d : b = c : a. 7. c : a = d : b. 4. d : c = b : a. 8. c : d = a : b. 361. Theorem IV. In a series of equal ratios the sum of the antecedents...consequents as any antecedent is to its consequent. aceo Proof. If T — i — "7.— 7 ? let # represent the value of each bdfh of the ratios. Then a...

...the terms D, E, F are proportional. Since A : B = B : C, by composition A + B:B = B + C:G; therefore the sum of the antecedents is to the sum of the consequents in the same ratio, that is, A + 2B + C:B + C = B + C:C. Now D = A + 2B + C, E = B + C, and F = C ;...