THEOREM L. All angles in the same segment of a circle, or standing on the same arc, are equal to each other. LET C and D be two angles in the same segment ACDB. or, which is the same thing, standing on the supplemental arc AEB; then will the angle C be equal to the angle D. For each of these angles is measured by half the arc AEB; and thus, having equal measures, they are equal to each other (ax. 11). B THEOREM LI. An angle at the centre of a circle is double the angle at the circumference, when both stand on the same arc. LET C be an angle at the centre C, and D an angle at the circumference, both standing on the same arc or same chord AB: then will the angle C be double of the angle D, or the angle D equal to balf the angle C. For, the angle at the centre C is measured by the whole arc AEB (def. 60), and the angle at the circumference D is measured by half the same arc AEB (th. 49); therefore the angle D is only half the angle C, or the angle C double the angle D. B E THEOREM LII. An angle in a semicircle, is a right angle. IF ABC or ADC be a semicircle; then any angle D in that semicircle, is a right angle. For, the angle D, at the circumference, is measured by half the arc ABC (th. 49), that is, by a quadrant of the circumference: and a quadrant is the measure of a right angle (cor. 4, th. 6; or cor. 2, th, 48). Therefore the angle D is a right angle. THEOREM LIII. The angle formed by a tangent to a circle, and a chord drawn from the point of contact, is equal to the angle in the alternate segment. IF AB be a tangent, and AC a chord, and D any angle in the alternate segment ADC; then will the angle D be equal to the angle BAC made by the tangent and chord of the arc AEC. For the angle D, at the circumference, is measured by half the arc AEC (th. 49); and the angle BAC, made by the tangent and chord, is also measured by the same half arc AEC (th. 48); therefore, these two angles are equal (ax. 11). D B THEOREM LIV. The sum of any two opposite angles of a quadrangle inscribed in a circle, is equal to two right angles. LET ABCD be any quadrilateral inscribed in a circle; then shall the sum of the two opposite angles A and C, or B and D, be equal to two right angles. For the angle A is measured by half the arc DCB, which it stands upon, and the angle C by half the arc DAB, (th. 49); therefore the sum of the two angles A and C is measured by half the sum of these two arcs, that is, by half the circumference. But half the circumference is the measure of two right angles (cor. 4, th. 6); therefore the sum of the two opposite angles A and C is equal to two right angles. In like manner it is shown, that the sum of the other two opposite angles, D and B, is equal to two right angles. THEOREM LV. If any side of a quadrangle, inscribed in a circle, be produced out, the outward angle will be equal to the inward opposite angle. If the side AB, of the quadrilateral ABCD, inscribed in a circle, be produced to E; the outward angle DAE will be equal to the inward opposite angle C. B For, the sum of the two adjacent angles DAE and DAB is equal to two right angles (th. 4); and the sum of the two opposite angles C and DAB is also equal to two right angles (th. 54); therefore the former sum, of the two angles DAE and DAB, is equal to the latter sum, of the two C and DAB (ax. 1). From each of these equals taking away the common angle DAB, there remains the angle DAE equal the angle C. THEOREM LVI. Any two parallel chords intercept equal arcs. LET the two chords AB, CD, be parallel: then will the arcs AC, BD, be equal; or AC = BD. Draw the line BC. Then, because the lines AE, CD, are parallel, the alternate angles B and C are equal (th. 12). But the angle at the circumference B, is measured by half the arc AC (th. 49); and the other equal angle at the cir cumference C is measured by half the arc BD; therefore the halves of the arcs AC, BD, and consequently the arcs themselves, are also equal. 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. 12), But the angle B, formed by a tangent and chord, is measured by half the arc BD (th. 48); and the other angle at D B c the circumference D is measured by half the arc BF (th. 49); therefore the arcs BD, BF, are equal. THEOREM LVIII. The angle formed, within a circle, by the intersection of two chords, is measured by half the sum of the two intercepted arcs. 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. A C KE D B 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. THEOREM LIX. The angle formed, out of a circle, by two secants, is measured by half the difference of the intercepted arcs. 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. 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 difference of the two DF, BD; and consequently the angle E, which is measured by half the latter difference, is also measured by half the former. 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 E G B 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). Cor. 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. 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. 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. For, through the point E draw the diameter FG; also, from the centre H draw the radius DH, and draw HI perpendicular to CD. 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: and 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. D IXE 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 CE, ED. In like manner it is proved, that the same rectangle of FE, EG, is equal to the rectangle of AE, EB: and consequently, the rectangle of AE, EB, is also equal to the rectangle of CE, ED (ax. 1). Cor. 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 CE2, the square of the tangent. Cor. 2. Hence both the tangents EC, EF, drawn from the D C E same point E, are equal; since the square of each is equal to the same rectangle or quantity AE . EB. THEOREM LXII. 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 equal to the angle D, the angle B to the angle E, and the angle C to 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. 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. G H D 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 GA. AB = HA. AC (th. 61): consequently, DF. AB = DE. 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. A D B LET CD be the perpendicular, and CE the diameter of the circle about the triangle ABC; then CA . CB = CD . CE. 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 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). E 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 we shall have CD + AD . DB = AC. CB. For, let CD be produced to meet the circumscribing circle at E, and join 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); |