then 6. ... AE > A'B', AE > AB which equals A'B'. Subst. ... minor AFE> AFB, so that E does not lie on AFB. 7. .. OC cuts AE, as at G, and OD OG. ... OD < OC, ... OG <OC. 8. Th. 4 I, th. 20 Ax. 8 Step 4 Theorem 8. In the same circle or in equal circles, chords that are equidistant from the center are equal; and of two chords unequally distant, the one nearer the center is the greater. Proof. If c, c' are two chords of the same circle or of equal circles, and d, d' are the respective perpendiculars from the center upon them; then from th. 7, Hence the converses are true by the Law of Converse. COROLLARY. The diameter is the greatest chord in a circle. For its distance from the center is zero. EXERCISES. 300. AB is a fixed chord of a circle, and XY is any other chord having its mid-point P on AB. What is the greatest and what is the least length that XY can have ? 301. What is the locus of the mid-points of equal chords of a circle ? Theorem 9. Of all lines passing through a point on a circumference, the perpendicular to the radius drawn to that point is the only one that does not meet the circumference again. Given point P on the circumference of a with center O, and AB, PC, respectively perpendicular and oblique to OP at P. that AB does not meet the circumference again, but that To prove PC does. Proof. 1. Let OM 2. Then M -B PC, and OX be any oblique to AB. OM < OP. I, th. 20 3... M is within the O, and PC cuts the circumference again. Def. O, cors. 4, 5 Why? 5. .. X, any point except P on AB, is without the O. Def. O, cor. 4 6... the perpendicular does not meet the circumference again, but an oblique does. DEFINITIONS. Steps 3, 5 The unlimited straight line which meets. the circumference of a circle in but one point is said to touch, or be tangent to, the circle at that point. The point is called the point of contact, or point of tangency, and the line is called a tangent. A tangent from a point to a circle is to be understood as the segment of the tangent between the point and the circle. If the two points in which a secant cuts a circumference continually approach, the secant approaches the condition of tangency. Hence the tangent is sometimes spoken of as a secant at its limiting position. COROLLARIES. 1. One, and only one, tangent can be drawn to a circle at a given point on the circumference. For the tangent is perpendicular to the radius at that point, and there is only one such perpendicular. (Has this been proved ?) 2. Any tangent is perpendicular to the radius drawn to the point of contact. (Why?) 3. A line perpendicular to a radius at its extremity on the circumference is tangent to the circle. (Why?) 4. The center of a circle lies on the perpendicular to any tangent at the point of contact. For the radius to that point is perpendicular to the tangent, and as there is only one such perpendicular at that point (prel. th. 2), that perpendicular must be the radius. 5. The perpendicular from the center to a tangent meets it at the point of contact. For the radius to that point is perpendicular to the tangent, and there is only one perpendicular from the center to the tangent. Theorem 10. An unlimited straight line cuts a circumference, touches the circle, or does not meet the circle, according as its distance from the center of the circle is less than, equal to, or greater than the radius. 4. And N cannot meet O M because it cannot cross T. COROLLARY. The converses are true. I, th. 16, cor. 3 Let the student state this corollary in full, and show that the Law of Converse applies. Section 3.- Angles formed by Chords, Secants, and Tangents. DEFINITIONS. A segment of a circle is either of the two portions into which the circle is cut by a chord. If a segment is not a semicircle, it is called a major or a minor segment according as its arc is a major or minor arc. E.g. DBC is a minor segment, and BDE is a major segment. The fact that the word segment is used to mean a part of a line, and also a part of a circle, will not present any difficulty, since the latter use is rare, and the sense in which the word is used is always evident. It means 66 a part cut off," and is therefore applicable to both cases. E C B The angle, not reflex, formed by two chords which meet on the circumference is called an inscribed angle, and is said to stand upon, or be subtended by, the arc which lies within the angle and is cut off by the arms. It is also called an angle inscribed in, or simply an angle in, the segment whose arc is the conjugate of the arc on which it stands. ADB is an inscribed angle, standing on AB; it is also an angle in the segment BCDEA. Points lying on the same circumference are called concyclic. 302. If from the extremities of any chord perpendiculars to that chord are drawn, they will cut off equal segments measured from the extremities of any diameter. (Draw a perpendicular from the center to the chord.) 303. If a tangent from a point B on a circumference meets two tangents from A, C, on the circumference, in points X, Y; and if the lines joining the center to A, X, Y, C, are a, x, y, c, respectively, then, 2xy=ax + yc, and XY = AX + YC. Theorem 11. An inscribed angle equals half the central angle standing on the same arc. Given FIG. 1. FIG. 2. AVB an inscribed angle, and AOB the central angle Proof. 1. Suppose VO drawn through center O, and produced. NOTE. to meet the circumference at X. 2. Then 3. And 4. 5. XVB/VBO. <XOB=/XVB+ZVBO. ../XVB=XOB. AVX=AOX. I, th. 3 Why? Step 2 Ax. 7 6. Similarly AVX AOX. (Each zero in Fig. 2.) 7. .. ZAVB ZAOB. = Ax. 2 The proof holds for all three figures, point A having moved to X (Fig. 2), and then through X (Fig. 3); another illustration of the Principle of Continuity. COROLLARIES. 1. Angles in the same segment, or in equal segments, of a circle are equal. (Why?) 2. If from a point on the same side of a chord as a given segment, lines are drawn to the ends of that chord, the angle included by those lines is greater than, equal to, or less than an angle in that segment, according as the point is within, on the arc of, or without the segment. This follows from cor. 1 and from I, th. 9. Draw the figure and prove. |
