THEOREM. sides of the other, each to each, they are therefore equal, and the angle ADB opposite to the side AB is equal to the angle DBC opposite to the side CD; consequently the side AD is parallel to BC (67). For a similar reason AB is parallel to CD; therefore the quadrilateral ABCD is a parallelogram. ?) 87. If two opposite sides AB, CD (fig. 44), of a quadrilateral Fig. 44. are equal and parallel, the two other sides will also be equal and parallel, and the figure ABCD will be a parallelogram. Demonstration. Let the diagonal BD be drawn; since AB is parallel to CD, the alternate angles ABD, BDC, are equal (67). Besides, the side AB= CD, and the side DB is common, there. fore the triangle ABD is equal to the triangle DBC (36), and the side AD= BC, the angle ADB = DBC, and consequently AD is parallel to BC; therefore the figure ABCD is a parallelogram. 32 88. The two diagonals AC, DB (fig. 45), of a parallelogram Fig. 45. mutually bisect each other. Demonstration. If we compare the triangle ADO with the triangle COB, we find the side AD= CB, and the angle ADO = CBO (67); also the angle DAO = OCB; therefore these two triangles are equal (38), and consequently A0, the side opposite to the angle ADO, is equal to OC, the side opposite to the angle OBC; DO likewise is equal to OB. 89. Scholium. In the case of the rhombus, the sides AB, BC, being equal, the triangles AOB, OBC, have the three sides of the one equal to the three sides of the other, each to each, and are consequently equal; whence it follows, that the angle AOB=BOC, and that thus the two diagonals of a rhombus cut each other mutually at right angles. THEOREM. Fig. 46. 90. The circumference of a circle is a curved line all the points of which are equally distant from a point within called the centre. The circle is the space terminated by this curved line*. 91. Every straight line CA, CE, CD (fig. 46), &c. drawn from the centre to the circumference is called a radius or semidiameter, and every straight line, as AB, which passes through the centre and is terminated each way by the circumference, is called a diameter. By the definition of a circle the radii are all equal, and all the diameters also are equal and double of the radius. 92. An arc of a circle is any portion of its circumference, as FHG. The chord or subtense of an arc is the straight line FG, which joins its extremities**. 93. A segment of a circle is the portion comprehended between an arc and its chord. 94. A sector is the part of a circle comprehended between an arc DE and the two radii CD, CE, drawn to the extremities of this arc. 95. A straight line is said to be inscribed in a circle, when its Fig. 47. extremities are in the circumference of the circle, as AB (fig. 47). An inscribed angle is one whose vertex is in the circumference, and which is formed by two chords, as BAC. An inscribed triangle is a triangle whose three angles have their vertices in the circumference of the circle, as BAC. * In common discourse the circle is sometimes confounded with its circumference; but it will always be easy to preserve the exactness of these expressions by recollecting that the circle is a surface which has length and breadth, while the circumference is only a line. ** The same chord, as FG, corresponds to two arcs, and consequently to two segments; but, in speaking of these, the smaller is always to be understood, when the contrary is not expressed: And in general an inscribed figure is one, all whose angles have their vertices in the circumference of the circle. In this case, the circle is said to be circumscribed about the figure. 96. A secant is a line, which meets the circumference in two points, as AB (fig. 48). Fig. 48. 97. A tangent is a line which has only one point in common with the circumference, as CD. The common point M is called the point of contuct. Also two circumferences are tangents to each other (fig. 59, 60), Fig. 59, when they have only one point common. A polygon is said to be circumscribed about a circle, when all its sides are tangents to the circumference; and in this case the eircle is said to be inscribed in the polygon. 60. 98. Every diameter AB (fig. 49) bisects the circle and its cir- Fig. 49. cumference. Demonstration. If the figure AEB be applied to AFB, so that the base AB may be common to both, the curved line AEB must fall exactly upon the curved line AFB; otherwise, there would be points in the one or the other unequally distant from the centre, which is contrary to the definition of a circle. -2 THEOREM. 99. Every chord is less than the diameter. Demonstration. If the radii CA, CD (fig. 49), be drawn from Fig. 49. the centre to the extremities of the chord AD, we shall have the straight line AD< AC + CD, that is, AD < AB (91). 100. Corollary. Hence the greatest straight line that can be inscribed in a circle is equal to its diameter. THEOREM. 101. A straight line cannot meet the circumference of a circle in more than two points. Demonstration. If it could meet it in three, these three points being equally distant from the centre, there might be three equal straight lines drawn from a given point to the same straight line; which is impossible (54). 102. In the same circle, or in equal circles, equal arcs are subtended by equal chords, and conversely, equal chords subtend equal arcs. Fig. 50. Demonstration. The radius AC (fig. 50) being equal to the radius EO, and the arc AMD equal to the arc ENG; the chord AD will be equal to the chord EG. For, the diameter AB being equal to the diameter EF, the semicircle AMDB may be applied exactly to the semicircle ENGF, and then the curved line AMDB will coincide entirely with the curved line ENGF; but the portion AMD being supposed equal to the portion ENG, the point D will fall upon G; therefore the chord AD is equal to the chord EG. Conversely, AC being supposed equal to EO, if the chord AD= EG, the arc AMD will be equal to the arc ENG. For, if the radii CD, OG, be drawn, the two triangles ACD, EOG, will have the three sides of the one equal to the three sides of the other, each to each, namely, AC= EO, CD = OG and AD= EG; therefore these triangles are equal (43); hence the angle ACD=EOG. Now, if the semicircle ADB be placed upon EGF, because the angle ACD = EOG, it is evident, that the radius CD will fall upon the radius OG, and the point D upon G, therefore the arc AMD is equal to the arc ENG. THEOREM. 103. In the same circle, or in equal circles, if the arc be less than half a circumference, the greater arc is subtended by the greater chord ; and, conversely, the greater chord is subtended by the greater arc. Fig. 50. Demonstration. Let the arc AH (fig. 50) be greater than AD, and let the chords AD and AH, and the radii CD, CH, be drawn. The two sides, AC, CH, of the triangle ACH, are equal to the two sides AC, CD, of the triangle ACD, and the angle ACH is greater than ACD; hence the third side AH is greater than the third side AD (42), therefore the greater arc is subtended by the greater chord. Conversely, if the chord AH be greater than AD, it may be inferred from the same triangles that the angle ACH is greater than ACD, and that thus the arc AH is greater than AD. 104. Scholium. The arcs of which we have been speaking, are supposed to be less than a semicircumference; if they were greater, the contrary would be true; in this case, as the arc increases, the chord would diminish, and the reverse; thus, the arc AKBD being greater than AKBH, the chord AD of the first is less than the chord AH of the second. >YY THEOREM. 105. The radius CG (fig. 51), perpendicular to a chord AB, Fig. 31. bisects this chord and the arc subtended by it AGB. Demonstration. Draw the radii CA, CB; these radii are, with respect to the perpendicular CD, two equal oblique lines, there fore they are equally distant from the perpendicular (52), and AD= DB. Again, since AD = BD, and CG is a perpendicular erected upon the middle of AB, each point in CG is at equal distances from A and B (55). The point G is one of these points; therefore AG=GB. But, if the chord AG is equal to the chord GB, the arc AG will be equal to the arc GB (102); therefore the radius CG, perpendicular to the chord AB, bisects the arc suba tended by this chord in the point G. 106. Scholium. The centre C, the middle D of the chord AB, and the middle G of the arc subtended by this chord, are three points situated in the same straight line perpendicular to the chord. Now, two points in a straight line are sufficient to determine its position; therefore a straight line which pa ses through any two of these points must necessarily pass through the third ; and must be perpendicular to the chord, It follows also, that a perpendicular erected upon the middle of a chord passes through the centre, and the middle of the arc subtended by that chord. For this perpendicular is the same as that let fall from the centre upon the same chord, since they both pass through the middle of the chord (51). THEOREM. 107. The circumference of a circle may be made to pass through any three points, A, B, C (fig. 52), which are not in the same Fig. 52. Geom. 4 |