Also, because H is the centre of the circle, and the radii DH, FH, GH, are all equal, the line EG is equal to the sum of the sides DH, HE, and EF is equal to their difference.

E

G

H

But the rectangle of the sum and difference of the two sides of a triangle is equal to the De rectangle of the sum and difference of the segments of the base (th. 35, B. I.); therefore the rectangle of FE, EG is equal to the rectangle of CE, ED. In like manner it is proved that the same rectangle of FE, EG is equal to the rectangle of AE, EB. Consequently, the rectangle of AE, EB is also equal to the rectangle of CE, ED (ax. 1). Q. E. D.

Corol. 1. When one of the lines in the second case, as DE, by revolving about the point E, comes into the position of the tangent EC or ED, the two points C and D running into D one, then the rectangle of CE, ED becomes the square of CE, because CE and DE are then equal. Consequently, the rectangle of the parts of the secant AE. EB is equal to the square of the tangent CE2.

E

Corol. 2. Hence both the tangents EC, EF, drawn from the same point E, are equal, since the square of each is equal to the same rectangle or quantity AE. EB.

THEOREM XXII.

In equiangular triangles, the rectangles of the corresponding or like sides, taken alternately, are equal.

Let ABC, DEF 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 the like sides G 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.

H

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 GH.

Then the angle G is equal to the angle C on the same arc BH, and the angle H equal to the angle B on the same arc CG (th. 10); also the opposite angles at A are equal (th. 7, B. I.); 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, B. I.), 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. 21); consequently the rectangle DF. AB is equal to the rectangle DE. AC. Q. E. D.

THEOREM XXIII.

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 CE the diameter of the circle about the triangle ABC; then the rectangle CA. CB is the rectangle A CD.CE.

B

For, join BE; then, in the two triangles ACD, ECB, the angles A and E are equal, standing on the same arc BC (th. 10); also the right angle D is equal to the angle B, which is also a right angle, being in a semicircle (th. 12); therefore these two triangles have also their 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 CE. CD of the other two (th. 22).

THEOREM XXIV.

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 CD be produced to meet the circumscribing circle at E, and join AE.

A

D

B

Then the two triangles AEC, 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. 10); consequently the third angles at A and D are equal (Cor. 1, th. 18, B. I.); 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. 22). But the latter rectangle CD. CE is=CD2+the rectangle CD. DE (th. 30, B. I.); therefore the former rectangle AC. CB is also=CD2+ CD. DE, or equal to CD2+AD. DB, since CD. DE is= AD.DB (th. 21). Q. E. D.

THEOREM XXV.

The rectangle of the two diagonal 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.

D

A

E

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 CD; 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. 22).

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. 22).

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, B. I.); therefore the same rectangle AC. BD is equal to the latter sum, the rectangle AD. BC+the rectangle AB. DC (ax. 1). Q. E. D.

Corol. Hence, if ABD be an equilateral triangle, and C any point in the arc BCD of the circumscribing circle, we have AC BC+DC. For AC. BD being AD. BC+ AB. DC; dividing by BD=AB=AD, there results AC= BC+DC.

BOOK III.

OF RATIOS AND PROPORTIONS.

DEFINITIONS.

DEF. 78. RATIO is the proportion or relation which one magnitude bears to another magnitude of the same kind with respect to quantity.

ter.

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 latSo the ratio of a quantity expressed by the number 2 to a like quantity expressed by the number 6, is denoted by 2 divided by 6, oror: 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; the ratio of 4 to 6 is or; that of 6 to 4 is & or 3, &c.

79. Proportion is an equality of ratios. Thus,

80. 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 A (2), B (4), C (8), where ==, both the same ratio.

81. 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 (4), B (2), C (10), D (5), where 4-10-2, both the same ratio.

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 B 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.

C