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 co2+ the rectangle AD. DB is the rectangle AC

. CB.

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

B

E

Then the two triangles ACE, ECD, 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 CD+the rectangle CD. DE (th. 30); therefore also the former rectangle AC. CB is also CD2 + CD. DE, or equal to CD2 + 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 rectan

BC.

gle AB. DC + the rectangle AD

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 ▲ 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 BC; 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.

the rectangles AC. BE+AC. DE is

But the former sum of

the rectangle AC. BD
.BD is equal to the

(th. 30): therefore the same rectangle Ac
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 relation which one magnitude bears to another magnitude of the same kind with respect to quantity.

The quantity or measure of a ratio is expressed by divid ing the leading quantity or antecedent by the following or consequent. Thus the ratio of 6 to 2 is 3, the ratio of 20 to 8 is 202, the ratio of 2 to 6 is 3, and the ratio of to 20 is.

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 quantities ▲ (2), B (4), c (8), where 21=1, 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), p where ==, both the same ratio,

Note.

Note. To denote that four quantities, A, B, C, D, are proportional, they are usually stated or placed thus, A: BC: D; and read thus, A is to в as c is to D. But when three quanti. ties 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 1:23:6; 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 1:32: 6.

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

Corol. Hence, like parts of quantities have the same ratio as the wholes; because the wholes are equimultiples of the like parts, or ▲ and в are like parts of ma and me.

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 Con

If Four Quantities be Proportional; they will be in Propor tion by Inversion, or Inversely.

LET A: B: Ma: mв; then will в : A :: MB : ma.

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

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 ma+mB 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.

LET A BMA : mв; also, let PA and pma be any equimultiples of the two antecedents, and qв and qmв any equimultiples of the two consequents; then will

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: B ±na :: ma : MB ±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 A B : MA MB NA : nв, &c; then will A: BA+MA+NA :: B+MB+NB, &c.