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 including the Bisected Angle.

LET CD bisect the angle c of the triangle ABC; then the square cD2 + the rectangle AD. DB is the rectangle Ac. CB.

For, let co be produced to meet the circumscribing circle at E, and join aɛ.

E

B

Then the two triangles ACE, BCD, are equiangular for the angles at c are equal by supposition, and the angles в and E are equal, standing on the same arc AC (th. 50); consequently the third angles at A and D are equal (corol. 1, th. 17): also AC, CD, and CE, CB, are like or corresponding sides, being opposite to equal angles: therefore the rectangle Ac CB is the rectangle CD. CE (th. 62). But the latter rectangle CD CE is CD2 + the rectangle CD. DE (th. 30); therefore also the former rectangle AC .CB is also = CD2 + CD. DE, or equal to CD + AD. DB, since CD. DE is = AD. DB (th. 61).

Q. E. D.

THEOREM LXV.

The Rectangle of the two Diagonals of any Quadrangle Inscribed in a Circle, is equal to the sum of the two Rectangles 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.

B

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 в are equal, standing on the same arc DC; and the angles DCA, BCE, are equal by supposition; consequently the third angles ADC, BEC are also equal: also, AC, BC, and AD, BE, are like or corresponding sides, being opposite to the equal angles: 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 Ec; and the angle DCE 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. 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).

DE is

OF RATIOS AND PROPORTIONS.

Q. E. D.

DEFINITIONS.

DEF. 76. RATIo is the proportion or relation which one magnitude bears to another magnitude 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 latter. 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 or 2; the ratio of

4 to 6 is or 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 quantitics a (2), B (4), c (8), where = 3 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), в (4), c (5), D (10), where 4 10 2, both the same ratio.

5

Note.

Note. To denote that four quantities, A, B, C, D, are proportional, they are usually stated or placed thus, A : B ::C: D; and read thus, A is to в as c is to 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 three, 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 to 2.

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 consequent, 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::5:6; then, by alternation, or permutation, it will be 1:3::26.

86. Compounded ratio is, when the sum of the antecedent and consequent is compared, either with the consequent, ar with the antecedent. Thus, if 1:23: 6, then by composition, 2:1:: 36: 3, and 1 + 2: 2 :: 36: 6.

87. Divided ratio, is when the difference of the antecedent and consequent is compared, either with the antecedent or with the consequent. Thus, if 1:2:: 3 : 6, then, by division, 2-1:1::63: 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, in def. 86.

THEOREM

THEOREM LXVI.

Equimultiples of any two Quantities have the same Ratio as the Quantities themselves.

LET A and B be any two quantities, and mA, mв, any equimultiples of them, m being any number whatever: then will ma and mв have the same ratio as a and b, or a : B ::

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 mв.

THEOREM LXVII.

If Four Quantities, of the Same Kind, be Proportionals; they will be in Proportion by Alternation or Permutation, or the Antecedents will have the Same Ratio as the Consequents.

LET A B: ma: mB; then will A mA :: B: MB.

If Four Quantities be Proportional; they will be in Pro, portion by Inversion, or Inversely.

LET A: B :: ma: me; then will B: A :: MB : Ma.

If Four Quantities be Proportional; they will be in Proportion by Composition and Division.

LET A B :: MA: MB;

Then will B±A: A :: MB ± ma : ma, and BA : B :: MB ± MA : MB.

Corol. It appears from hence, that the Sum of the Greatest and Least of four proportional quantities, of the same kind, exceeds the Sum of the Two Means. For, since

A: A+B :: ma : ma + mв, where A is the least, and mamB the greatest; then m +1.A+ mB, the sum of the greatest and least exceeds m +1. A + B the sum of

the two means.

THEOREM LXX.

If, of Four Proportional Quantities, there be taken any Equimultiples whatever of the two Antecedents, and any Equimultiples whatever of the two Consequents; the quantities resulting will still be proportional.

LETA B ma mв; also, let a and pmA be any equimultiples of the two antecedents, and qв and qmв any cquimultiples of the two consequents; then will ha qв fima : qmB.,

If there be Four Proportional Quantities, and the two Consequents be either Augmented or Diminished by Quantities that have the Same Ratio as the respective Antecedents; the Results and the Antecedents will still be Proportionals.

LET A B: ma: mB, and na and nmA any two quantities having the same ratio as the two antecedents; then will A: BNA :: ma: mв ±nma.

If any Number of Quantities be Proportional, then any one of the Antecedents will be to its Consequent, as the Sum of all the Antecedents is to the Sum of all the Consequents.

LET AB: ma: MB :: na nв, &c; then willA B A+ ma + na :: B+ MB + NB, &c.