SECOND BOOK. THE CIRCLE, AND ITS COMBINATION WITH THE STRAIGHT LINE. DEFINITIONS. I. The circumference of a circle is a curved line, all the points of which are equally distant from a point within called the centre. circle is the space bounded by the circumference. The II. Any straight line drawn from the centre to the circumference is called a radius. Hence, all radii of the same circle are equal. A line passing through the centre and terminating in both directions by the circumference, is called a diameter. Hence, all diameters of the same circle are equal, each being made up of two radii. III. Any portion of the circumference of a circle is called an arc. One fourth of the entire circumference is called a quadrant. The straight line joining the extremities of an arc is called a chord. The chord is said to subtend the arc. Every chord corresponds always to two arcs, which together make up the entire circumference. It is the smaller arc which is referred to as the subtended arc, unless otherwise expressed. The portion of a circle included by an arc and its chord is called a segment. The portion included between two radii and the intercepted arc is called a sector. IV. When a straight line cuts the circumference of a circle it is called a secant. When a straight line touches the circumference in only one point it is called a tangent; and the common point of the line and circumference is called the point of contact. Two circumferences are tangent to each other when they have only one point in common. Two circumferences are concentric when they have the same centre. V. A line is inscribed in a circle when its extremities are in the circumference. An angle is inscribed in a circle when its sides are inscribed. A polygon is inscribed in a circle when its sides are inscribed; and under the same circumstances, the circle is said to circumscribe the polygon. A circle is inscribed in a polygon when its circumference touches each side, and the polygon is said to be circumscribed about the circle. By an angle in a segment of a circle, is to be understood an angle whose vertex is in the arc, and whose sides intercept the chord of said arc; and by an angle at the centre, is meant one whose vertex is at the centre. In both cases the angles are said to be subtended by the chords or arcs which their sides include. VI. Any polygonal figure is said to be equilateral when all its sides are equal; and it is equiangular when all its angles are equal. Two polygons are said to be mutually equilateral when their corresponding sides, taken in the same order, are equal. When this is the case with the corresponding angles, the polygons are said to be mutually equiangular. A regular polygon has all its sides equal, and all its angles equal. If all the sides are not equal, or all the angles are not equal, the polygon is irregular. A regular polygon may have any number of sides not less than three. The equilateral triangle is a regular polygon of three sides. The square is also a regular polygon of four sides. OF CHORDS, SECANTS, AND TANGENTS. THEOREM I. Every diameter divides the circle and its circumference into two equal parts. A C B Revolve the portion ACB about the diameter AB as a hinge, until it returns to its primitive plane, on the opposite side of AB; then will the portion of the circumference ACB wholly coincide with ADB. For, if not, there would be points in the circumference unequally distant from the centre, which is impossible (D. II.). Hence, the diameter divides the circle and its circumference into two equal parts. D THEOREM II. A straight line cannot meet the circumference in more than two points. For if it could, drawing radii to these points, we should have more than two equal lines drawn from the same point to a straight line, which is impossible (B. I., T. XII., C. II.).* * When a reference is made from one Book to another Book, the Book referred to will be given as above; but when the Book is not given, the reference is confined to the Book in which the proposition occurs. THEOREM III. D C The diameter of a circle is greater than any chord. The diameter AB is greater than any chord, as CD. For, drawing the radii EC and ED, we have EC+ED> CD (B. I., A T. VII.). But the diameter AB (D. II.); hence, AB > CD. EC+ ED B E THEOREM IV. The radius drawn perpendicular to a chord, bisects the chord, and also bisects its subtended arc. Let the radius CD be drawn perpendicular to the chord AB. A F E B Produce DC to F, and apply the figure DAF to DBF by revolving it about DF. Then the arc DAF will coincide with DBF (T. I.). And since the angles DEA and DEB are right, the line EA will coincide with EB. Hence we have AE=EB, arc AD=arc DB. Also we have arc AF: =arc FB. D Scholium. The straight line CD fulfils four different conditions: 1st, it passes through the centre; 2d, it passes through the middle of the chord; 3d, it passes through the middle of the subtended arc; 4th, it is perpendicular to the chord. Any two of these conditions are sufficient to determine the direction of the line. Hence we also have the following propositions: A radius drawn bisecting a chord is perpendicular to it, and bisects the subtended arc. A radius drawn bisecting an arc, will also bisect its chord perpendicularly. A line drawn bisecting a chord and its subtended arc, will pass through the centre, and be perpendicular to the chord. A perpendicular bisecting a chord, will also bisect its subtended arc, and pass through the centre. A line drawn from the middle of an arc perpendicular to its chord, will bisect the chord, and pass through the centre. THEOREM V. A line perpendicular to a radius at its extremity, is tangent to the circumference. If ED is drawn perpendicular to the radius CA, at its extremity, it will be tangent to the circumference; that is, it will have only the point A in common with the circumference. For any other point, as F, being joined with the centre, gives an ob A F E D lique line greater than the radius (B. I., T. XII.). Hence, A is the only point common to the straight line and the circumference; consequently, this line is tangent to the circumference (D. IV.). Cor. I. If a straight line is tangent to the circumference of a circle, it will be perpendicular to the radius drawn to the point of contact. For all other points of the tangent line, except that of contact, are situated without the circumference, and therefore at a greater distance from the centre than the radius. Hence the radius, being the shortest line which can be drawn from the centre to the tangent, is perpendicular to it (B. I., T. XII.). Cor. II. Only one tangent can be drawn through the same point of the circumference. Cor. III. From a point within the circumference no tangent can be drawn. Cor. IV. The perpendiculars drawn at the extremities of a diameter will be parallel tangents, and conversely two parallel tangents will have their points of contact situated at the extremities of the same diameter. THEOREM VI. Parallel secants or tangents intercept equal arcs of the circumference. First. When the parallels are both secants. Draw the radius EF perpendicular to AB, and it will also be perpendicular to its parallel CD, and we shall have (T. IV.) arc GF = arc KF and arc HF: arc LF. Consequently, arc GF — arc HF F H A LB C G D K E |