5. Circles are said to touch mutually, if they meet, but do not cut each other. 6. The point where a straight line touches a circle, or one circle touches another, is called the point of contact. 7. A straight line is said to be inflected from a point, when it terminates in another straight line, or at the circumference of a circle. PROP. I. THEOR. A circle is bisected oy its diameter. The circle ADBE is divided into two equal portions, by the diameter AB. For let the portion ADB be reversed and applied to AEB, the straight line AB and its middle point, or the centre C, remaining the same. And since the radii of the circle are all equal, or the distance of C from any point in the boundary ADB is equal to its distance from any point of the opposite boundary AEB, every point D A B C E D of the former must meet with a corresponding point of the latter, and consequently the two portions ADB and AEB will entirely coincide. Cor. The portion ADB limited by a diameter, is thus a semicircle, and the arc ADB is a semicircumference. PROP. II. THEOR. A straight line cuts the circumference of a circle only in two points. If the straight line AB cut the circumference of a circle in D, it can only meet it again in another point E. For join D and the centre C; and because from the point C on AD C EB : ly two equal straight lines, such as CD and CE, can be drawn to AB (I. 17. cor.) the circle described from C through the point D will cross AB again only at E. PROP. III. THEOR. The chord of an arc lies wholly within the cir cle. The straight line AB which joins two points A, B in t circumference of a circle, lies wholly within the figure. For, from the centre C, draw CD to any point in AB, and join CA and Св. C A D B Because CDA is the exterior angle of the triangle CDB, it is greater (I. 8.) than the interior CBD or CBA; but CBA, being (I. 10.) equal to CAB or CAD, the angle CDA is consequently greater than CAD, and its opposite side CA (I. 13.) greater than CD, or CD is less than CA, and therefore the point D must lie within the circle. Cor. Hence a circle is concave towards its centre. PROP. IV. THEOR. A straight line drawn from the centre of a circle at right angles to a chord, likewise bisects it; and, conversely, the straight line which joins the centre with the middle of a chord, is perpendicular to it. The perpendicular let fall from the centre C upon the chord AB, cuts it into two equal parts AD, DB. For join CA, CB: And, in the triangles ACD, BCD, the side AC is equal to CB, CD is common to both, and the right angle ADC is equal to BDC; these trian gles having thus their corresponding angles at A and B both acute, are equal (I. 21.) and consequently the side AD is equal to BD. C D B Again, let AD be equal to BD; the bisecting line CD is at right angles to AB. For join CA, CB. The triangles ACD and BCD, having the sides AC, AD equal to CB, BD, and the remaining side CD common to both, are equal (I. 2.), and consequently the angle CDA is equal to CDB, and each of them a right angle. Cor. Hence a straight line cutting two concentric circles has equal portions intercepted by their circumferences. PROP. V. THEOR. A straight line which bisects a chord at right angles, passes through the centre of the circle. If the perpendicular FE bisect a chord AB, it will pass through G the centre of the circle. For in FE take any point D, and join DA and DB. The triangles ADC and BDC, having the side AC equal to BC, CD common, and E D G A C B F the right angle ACD equal to BCD, are equal (I. 3.), and consequently the base AD is equal to BD. The point D is, therefore, the centre of a circle described through A and B; and thus the centres of the circles that can pass through A and B are all found in the straight line EF. The centre G of the circle AEBF must hence occur in that perpendicular. Cor. The centre of a circle may hence be found by bisecting the chord AB by the diameter EF (I. 7.), and bisecting this again in G. PROP. VI. THEOR. The diameter is the greatest line that can be inflected within a circle. or to the whole diameter AB. Wherefore AB is greater than DE. PROP. VII. THEOR. If from any eccentric point, two straight lines be drawn to the circumference of a circle; the one which passes nearer the centre, is greater than that which lies more remote. |