Let ABC, DEP, be two equiangular triangles, having the angle A = the angle D. the angle B= the angle E, and the angle c= the angle F; also G B the like sides AB, DE, and ac, DF, being those opposite the equal angles : then will the rectangle of AB, DF, be equal to the rectangle of AC, DE. In BA produced take ag equal to df; and through the three points B, C, G, conceive a circle bcgh to be described, meeting ca produced at h, and join G. Then the angle G is equal to the angle c' on the same arc BH, and the angle u equal to the angle B on the same arc cg (th. 50); also the opposite angles at A are equal (th. 7) : therefore the triangle agh is equiangular to the triangle ACB, and consequently to the triangle DFE also. But the two like sides AG. DF, are also equal by supposition; consequently the two triangles AGH, DFE, are identical (th. 2), having the two sides AG, AH, equal to the two DF, DE, each to each. But the rectangle GA. AB is equal to the rectangle HA. AC (th. 61): consequently the rectangle DF. AB is equal the rectangle de Q. ED. AC. THEOREM LXIII. The Rectangle of the two Sides of any Triangle, is Equal to the Rectangle of the Perpendicular on the third Side and the Diameter of the Circumscribing Circle. Let cd be the perpendicular, and OE the diameter of the circle about the triangle ABC ; then the rectangle cĄ . ce is = the А. B yectangle CD. CE E For, join Be: then in the two triangles ACD, ECB, the angles A and I are equal, standing on the same arc Bc (th. 50): also the right angle a is equal to the angle B, which is also a right angle, being in a semicircle (th. 52): therefore these two triangles have also thcir third angles equal, and are equiangular. Hence, AC, CE, and CD, CB, being like sides, subtending the equal angles, the rectangle ac . CB, of the first and last of them, is equal to the rectangle çE.CD, of the other two (th 62). THEOREM THEOREM LXIV. The Square of a line bisecting any Angle of a Triangle, together with the Rectangle of the Two Segments of the opposite Side, is Equal to the Rectangle of the two other Sides inciuding the Bisected Angle. LET CD bisect the angle c of the triangle ABC ; !hen the square cp + the rectangle AD . DB is the rectangle ac .CB. For, let cd be produced to meet the cir. cumscribing circle at E, and juin AE. Then the two triangles ACE, BCD, are equiangular: for the angles at c are equal by supposition, and the angles B and E are equal, standing on the same arc ac (th. 50); consequently the third angles at A and D are equal (corol. I, th. 17): also AC. cp, and CE, CB, are like or corresponding sides, being opposite to equal angles: therefore the rectangle ac the rectangle cd. ce (th. 62). But the latter rectangle cd · çe is = CD2 + the rectangle cd . DE (th. 30); therefore also the former rectangle ac · CB is also = CD2 + CD. DE, or cqual to co? + AD. DB, since cd. de is = AD. DB (th. 61). Q. E. D. CB is THEOREM LXV. The Rectangle of the two Diagonals of any Quadranglc Inscribed in a Circle, is equal to the sum of the two Rect: angles of the Opposite Sides, LET ABCD be any quadrilateral inscribed in a circle, and AC, BD, its two diagonals: then the rectangle Ac.BD is the rectangle AB · DC + the rectangle AD.Bc. For, let ce be drawn, making the angle BCE equal to the angle dCA. Then the two triangles AcD, BCE, are equiangular ; for the angles A and B are equal, standing on the same arc dc ; and the angles DCA, BCR, are equal by supposition; consequently the third angles Adc, bec are also equal : also, ac, rc, and AD, BE, are like or corresponding sides, being opposite to the equal anyles: therefore the rectangle ac. Be is = the rectangle AD, BC (th. 62), Again, the two triangles ABC, DEC, are equiangular: for the angles BAC, BDC, are equal, standing on the same arc Bc; and the angle oce is equal to the angle BCA, by adding the common angle ace to the two equal angles DCA, BCE ; therefore the third angles E and abc are also equal : but ac, DC, and AB, DE, are the like sides : therefore the rectangle ac. DE is the rectangle AB . DC (th. 62). Hence, by equal additions, the sum of the rectangles ac . BE + AC. DE is = AD . BC + AB. DC. But the former sum of the rectangles ac , BE + AC . De is = the rectangle ac. BD (th. 30): therefore the same rectangle ac BD is equal to the latter sum, the rect. AD . BC + the rect. AB . DC (ax 1). Q. E, D. OF RATIOS AND PROPORTIONS. DEFINITIONS. Def. 76. Ratio is the proportion or relation which one magnitude bears to another inagnitude of the same kind with respect to quantity. Note. The measure, or quantity, of a ratio, is conceived, by considering what part or parts the leading quantity, called the Antecedent, is of the other, called the Consequent; or what part or parts the number expressing the quantity of the former, is of the number denoting in like manner the luiter. So, the ratio of a quantity expressed by the number 2, to a like quantity expressed by the number 6, is denoted by 6 divided by 2, or or 3 : the number 2 being 3 times contained in 6, or the third part of it. In like manner, the ratio of the quantity 3 to 6, is measured by çor 2; the ratio of 4 to 6 isor 1; that of 6 to 4 is or ; &c. 77. Proportion is an equality of ratios. Thus, 78. Three quantities are said to be Proportional, when the ratio of the first to the second is equal to the ratio of the second to the third. As of the three quantiiies A (2), 3 (4), c (8), where = 2, both the same ratio 79. Four quantities are said to be Proportional, when the ratio of the first to the second, is the same as the ratio of the third to the fourth. As of the four, A (2), B (4), c (5), D (10), where = "= 2, both the same ratio. Note. Note. To denote that four quantities, A, B, C, D, are proa portional, they are usually stated or placed thus, A:B::cid; and read thus, a is to e as c is io D But when three quantities are proportional, the middle one is repeated, and they are written thus, A ; B::B:C. 80. Of three proportional quantities, the middle one is said to be a Mean Proportional between the other two ; and the last, a Third Proportional to the first and second. 81. Of four proportional quantities, the last is said to be a Fourth Proportional to the other thrce, taken in order. 82. Quantities are said to be Continually Proportional, or in Continued Proportion, when the ratio is the same between every two adjacent terms, viz. when the first is to the second, as the second to the third, as the third to the fourth, as the fourth to the fifth, and so on, all in the same common ratio. As in the quantities 1, 2, 4, 8, 16, &c; where the common ratio is equal 102. 83. Of any number of quantities, A, B, C, D, the ratio of the first, A, to the last D, is said to be Compounded of the ratios of the first to the second, of the second to the third, and so on to the last. 84. Inverse ratio is, when the antecedent is made the conscquent, and the consequent the antecedent.-Thus, if 1:2::3:6; then inversely, 2:1::6:3, 85. Alternate proportion is, when antecedent is compared with antecedent, and consequent with consequent.--As, if 1:2:: 3:6; then, by alternation, or permutation, it will be 1:3:: 2 : 6. 86. Compounded ratio is, when the sum of the antecericnt and consequent is compareci, cither with the consequent, or with the antecedent.-Thus, if i : 2::3:6, then by composition, I +2:1::3 +6:3, and 1 + 2: 2::3 +6: 6. 37. Divided ratio, is when the difference of the antecedent and consequent is compared, either with the antecedent or with the consequent. Thus, if l: 2 :: 3 : 6, then, by division, 2-1:1::6-3: 3, and 2 1:2;:6-3: 6. Note. The term Divided, or Division, here means subtracting, or parting ; being used in the sense opposed to compounding, or adding, ia dof. 86. TIÆOREL THEOREM LXVI. Cquimultiples of any two Quantities have the same Ratio as the Quantities themselves. Let A and B be any two quantities, and ma, mb, any equimultiples of them. m being any number whatever : then will ma and mb have the same ratio as A and B, or A i B :: ZA : MB MB B For the same ratio. MA Corol. Hence, like parts of quantities have the same ratio as the wholes; because the wholes are equimultiples of the like parts, or A and B are like parts of mA and MB. THEOREM LXVI. If Four Quantities, of the Same Kind, be Proportionals; they will be in Proportion by Alternation or Permutation, If Four Quantities be Proportional ; they will be in Pro. portion by Inversion, or Inversely. MA both the same ratio. A THEOREM LXIX. portion by Composition and Division. and BŁA:B:: MB + ma : MB. |