at the circumference C is measured by half the arc BD: therefore the halves of the arcs AC, BD, and consequently the arcs themselves, are also equal. Q. E. D. THEOREM LVII. When a tangent and chord are parallel to each other, they intercept equal arcs. Let the tangent ABC be parallel to the chord DF; then are the arcs BD, BF, equal; that is, BD = BF. Draw the chord BD. Then, because the lines AB, DF, are parallel, the alternate angles D and B are equal (th. But the angle B, formed by a tangent and chord, is measured by half the arc BD (th. 48); and the other angle 12). at the circumference D is measured by half the arc BF (th. 49); therefore the arcs BD, BF, are equal. Q. F. D. THEOREM LVIII. A Let the two chords AB, CD, intersect at the point E: then the angle AEC, or DEB, is measured by half the sum of the two arcs AC, DB. The angle formed within a circle, by the intersection of two chords, is measured by half the sum of the two intercepted arcs. B C Draw the chord AF parallel to CD. Then, because the lines AF, CD, are parallel, and AB cuts them, the angles on the same side A and DEB are equal (th. 14). But the angle at the circumference A is measured by half the arc BF, or of the sum of FD and DB (th. 49); therefore, the angle E is also measured by half the sum of FD and DB. Again, because the chords AF, CD, are parallel, the arcs AC, FD, are equal (th. 56); therefore, the sum of the two arcs AC, DB, is equal to the sum of the two FD, DB; and consequently the angle E, which is measured by half the latter sum, is also measured by half the former. Q. E. D. 1 12 THEOREM LIX. The angle formed out of a circle, by two secants, is measured by half the dif ference of the intercepted arcs. E Let the angle E be formed by two secants EAB and ECD; this angle is measured by half the difference of the two arcs AC, DB, intercepted by the two secants. Draw the chord AF parallel to CD. Then, because the lines AF, CD, are parallel, and AB cuts them, the angles on the same side A and BED are equal (th. 14). But the angle A, at the circumference, is measured by half the arc BF (th. 49), or of the difference of DF and DB: therefore, the equal angle E is also measured by half the difference of DF, DB. B Again, because the chords AF, CD are parallel, the arcs AC, FD, are equal (th. 56); therefore, the difference of the two arcs AC, DB, is equal to the dif ference of the two DF, DB. Consequently, the angle E, which is measured by half the latter difference, is also measured by half the former. Q. E D. BB THEOREM LX. The angle formed by two tangents, is measured by half the difference of the two intercepted arcs. Let EB, ED, be two tangents to a circle at the points A, C; then the angle E is measured by half the difference of the two arcs CFA, CGA. Draw the chord AF parallel to ED. Then, because the lines AF, ED, are parallel, and EB meets them, the angles on the same side A and E are equal (th. 14). But the angle A, formed by the chord AF and tangent AB, is measured by half the arc AF (th. 48); therefore, the equal angle E is also measured by half the same arc AF, or half the difference of the arcs CFA and CF, or CGA (th. 57.) Corol. In like manner it is proved, that the angle E, formed by a tangent ECD, and a secant EAB, is measured by half the difference of the two intercepted arcs CA and CFB. THEOREM LXI. Let the two lines AB, CD, meet each other in E; then the rectangle of AE, EB, will be equal to the rectangle of CE, ED. Or, AE. EB = CE. ED. When two lines, meeting a circle each in two points, cut one another, either within it or without it; the rectangle of the parts of the one, is equal to the rectangle of the parts of the other; the parts of each being measured from the point of meeting to the two intersections with the circumference. For, through the point E draw the diameter FG; also, from the centre H draw the radius DH, and draw HI perpendicular to CD. E Then, since DEH is a triangle, and the perp. HI bisects the chord CD (th. 41), the line CE is equal to the difference of the segments DI, EI, the sum of them being DE. 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. B H But the rectangle of the sum and difference of the two sides of a triangle is equal to the rectangle of the sum and difference of the segments of the base (th. 35); therefore the rectangle of FE, EG, is equal to the rectangle of (E, 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 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. 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. 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 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. THEOREM LXII. In equiangular triangles, the rectangles of the corresponding or like sides, taken alternately, are equal. 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. E D Let CD be the perpendicular, and CE the diameter of the circle about the triangle ABC; then the rectangle CA. CB is the rectangle CD.CE. E 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. 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 to the rectangle DE. AC. Q. E. D. A 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. For, join BE: then in the two triangles ACD, ECB, the angles A and E are equal, standing on the same arc BC (th. 50); also the right angle D is equal to the angle B, which is also a right angle, being in a semicircle (th. 52): 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. 62). B 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 CD be produced to meet the circumscribing circle at E, and join AE. D 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 (cor. 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 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. P. B 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. A 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, 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). D B 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. 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 ACBC + DC. OF RATIOS AND PROPORTIONS. 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 2 divided by 6, or gor: 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 ; &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 quantities A (2), B (4), C (8), where ==, 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 (4), B (2), C (10), D (5), where 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. The proportionality of quantities may also be expressed very generally by the A с equality of fractions, as at pa. 121. Thus, if = D' B then A B C : D, also B: A:: D: C, A: C:: B : D. and C: A :: D : B. : 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. |