17. Corollary II. The middle points of any number of parallel chords all lie in the same diameter perpendicular to the chords. In other words, the locus of the middle points of a system of parallel chords is the diameter perpendicular to these chords. PROPOSITION VIII.-THEOREM. 18. In the same circle, or in equal circles, equal chords are equally distant from the centre; and of two unequal chords, the less is at the greater distance from the centre. 1st. Let AB, CD, be equal chords; OE, OF, the perpendiculars which measure their distances from the centre 0; then, OE= OF. For, since the perpendiculars bisect the chords (15), AECF; hence (I. 83), the right triangles AOE and COF are equal, and OE OF = A B E H G D 2d. Let CG, AB, be unequal chords; OE, OH, their distances from the centre; and let CG be less than AB; then, OH> OE. For, since chord AB > chord CG, we have arc AB > arc CG; so that if from C we draw the chord CD: AB, its subtended arc CD, being equal to the arc AB, will be greater than the arc CG. Therefore the perpendicular OH will intersect the chord CD in some point I. Drawing the perpendicular OF to CD, we have, by the first part of the demonstration, OF OE. But OH > OI, and OI> OF (I. 28); still more, then, is OH> OF, or OH> OE. If the chords be taken in two equal circles, the demonstration is the same. = 19. Corollary I. The converse of the proposition is also evidently true, namely: in the same circle, or in equal circles, chords equally distant from the centre are equal; and of two chords unequally distant from the centre, that is the greater whose distance from the centre is the less. 20. Corollary II. The least chord that can be drawn in a circle through a given point P is the chord, AB, perpendicular to the line OP joining the given point and the centre. For, if CD is any other chord drawn through P, the perpendicular OQ to this chord is less than OP; therefore, by the preceding corollary, CD is greater than AB. B PROPOSITION IX.-THEOREM. 21. Through any three points, not in the same straight line, a circumference can be made to pass, and but one. Let A, B, C, be any three points not in the same straight line. B D 1st. A circumference can be made to pass through these points. For, since they are not in the same straight line, the lines AB, BC, AC, joining them two and two, form a triangle, and the three perpendiculars DE, FG, HK, erected at the middle points of the sides, meet in a point O which is equally distant from the three points A, B, C, (I. 131). Therefore a circumference described from O as a centre and a radius equal to any one of the three equal distances OA, OB, OC, will pass through the three given points. 2d. Only one circumference can be made to pass through these points. For the centre of any circumference passing through the three points must be at once in two perpendiculars, as DE, FG, and therefore at their intersection; but two straight lines intersect in only one point, and hence O is the centre of the only circumference that can pass through the three points. 22. Corollary. Two circumferences can intersect in but two points; for, they could not have a third point in common without having the same centre and becoming in fact but one circumference. TANGENTS AND SECANTS. F 23. Definitions. A tangent is an indefinite straight line which has but one point in common with the circumference; as ACB. The common point, C, is called the point of contact, or the point of tangency. The circumference is also said to be tangent to the line AB at the point C. A secant is a straight line which meets the circumference in two points; as EF. E 24. Definition. A rectilinear figure is said to be circumscribed about a circle when all its sides are tangents to the circumference. In the same case, the circle is said to be inscribed in the figure. C PROPOSITION X.-THEOREM. 25. A straight line oblique to a radius at its extremity cuts the circumference. Let AB be oblique to the radius OC at its extremity C; then, AB cuts the circumference at C, and also in a second point D. For, let OE be the perpendicular from O upon AB; then OE OC, and the point E is within the circumference. Therefore AB cuts the circumference in C, and must evidently cut it in a second point D. C E B PROPOSITION XI.-THEOREM. 26. A straight line perpendicular to a radius at its extremity is a tangent to the circle. Let AB be perpendicular to the radius OC at its extremity C; then, AB is a tangent to the circle at the point C. For, from the centre O draw the oblique line OD to any point of AB except C. Then, OD > OC, and D is a point without the circumference. Therefore AB having all its B points except C without the circumference, has but the point C in common with it, and is a tangent at that point (23). 27. Corollary. Conversely, a tangent AB at any point C is perpendicular to the radius OC drawn to that point. For, if it were not perpendicular to the radius it would cut the circumference (25), and would not be a tangent. E FI B 28. Scholium. If a secant EF, passing through a point C of the circumference, be supposed to revolve upon this point, as upon a pivot, its second point of intersection, D, will move along the circumference and approach nearer and nearer to C. the second point comes into coincidence with C, the revolving line ceases to be strictly a secant, and becomes the tangent AB; but, continuing the revolution, When DI E' the revolving line again becomes a secant, as E'F', and the second point of intersection reappears on the other side of C, as at D'. If, then, our revolving line be required to be a secant in the strict sense imposed by our definition, that is a line meeting the circumference in two points, this condition can be satisfied only by keeping the second point of intersection, D, distinct from the first point, C, however near these points may be brought to each other; and, therefore, under this condition, the tangent is often called the limit of the secants drawn through the point of contact; that is to say, a limit toward which the secant continually approaches, as the second point of intersection (on either side of the first) continually approaches the first, but a limit which is never reached by the secant as such. On the other hand, as the tangent is but one of the positions of our revolving line, it has properties in common with the secant; and in order to exhibit such common properties in the most striking manner, it is often expedient to regard the tangent as a secant whose two points of intersection are coincident. But it is to be observed that we then no longer consider the secant as a cutting line, but simply as a line drawn through two points of the curve; and we include the tangent as that special case of such a line in which the two points are coincident. In this, we generalize in the same way as in algebra, when we say that the expression x = ab signifies that x is the difference of a and b, even when a = b, and there is really no difference between a and b. PROPOSITION XII.-THEOREM. 29. Two parallels intercept equal arcs on a circumference. We may have three cases: 1st. When the parallels AB, CD, are both secants; then, the intercepted arcs AC and BD are equal. For, let OM be the radius drawn perpendicular to the parallels. By Prop. VII. the point M is at once the middle of the arc AMB and of the arc CMD, and hence we have AM BM and CM= DM, whence, by subtraction, E M C G that is, AM-CM-BM-DM; AC: = BD. 2d. When one of the parallels is a secant, as AB, and the other is a tangent, as EF at M, then, the intercepted arcs AM and BM are equal. For, the radius OM drawn to the point of contact is perpendicular to the tangent (27), and consequently perpendicular also to its parallel AB; therefore, by Prop. VII., AM = BM. 3d. When both the parallels are tangents, as EF at M, and GH |