Fig. 46. SECTION SECOND. Of the Circle and the Measure of Angles. DEFINITIONS. 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, In this And in general an inscribed figure is one, all whose angles have their vertices in the circumference of the circle. 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). 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 contact. Fig. 48. 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 circle 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. 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). Fig. 50. Fig. 50. THEOREM. 102. In the same circle, or in equal circles, equal arcs are subtended by equal chords, and conversely, equal chords subtend equal arcs. 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. 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. THEOREM. 105. The radius CG (fig. 51), perpendicular to a chord AB, Fig. 51. 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, therefore 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 subtended 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 passes 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 straight line, but the circumference of only one circle may be made to pass through the same points. Demonstration. Join AB, BC, and bisect these two straight lines by the perpendiculars DE, FG; these perpendiculars will meet in a point 0. For the lines DE, FG, will necessarily cut each other, if they are not parallel. Let us suppose that they are parallel; the line AB perpendicular to DE will be perpendicular to FG (65), and the angle K will be a right angle; but BK, which is BD produced, is different from BF, since the three points A, B, C, are not in the same straight line; there are then two perpendiculars BF, BK, let fall from the same point upon the same straight line, which is impossible (50); therefore the perpendiculars DE, FG, will always cut each other in some point 0. Now the point O, considered with reference to the perpendicular DE, is at equal distances from the two points A and B (55); also this same point O, considered with reference to the perpendicular FG, is at equal distances from the two points B and C, hence the three distances OA, OB, OC, are equal; therefore the circumference, described from the centre O with the radius OB, will pass through the three points A, B, C. It is thus proved, that the circumference of a circle may be made to pass through any three given points, which are not in the same straight line; it remains to show, that there is only one circle, which can be so described. If there were another circle, the circumference of which passed through the three given points A, B, C, its centre could not be without the line DE (55), since, in this case, it would be at unequal distances from A and B; neither can it be without the line FG, for a similar reason; it will then be in both of these lines at the same time. But two lines can cut each other in only one point (32); there is therefore only one circle, whose circumference can pass through three given points. 108. Corollary. Two circumferences can meet each other only in two points; for, if they had three points common, they would have the same centre, and would make one and the same circumference. |