BOOK III. THE CIRCLE, AND THE MEASURE OF ANGLES. Definitions. 1. A circle is a plane figure bounded by a line, every point of which is equally distant from a point within, called the center. This bounding line is called the circumference of the circle. 2. A radius of a circle is a straight line drawn from the center to the circumference. A diameter of a circle is a straight line. passing through the center, and terminated both ways by the circumference. Cor. All the radii of a circle are equal; all the diameters are equal also, and each double of the radius. 3. An arc of a circle is any part of the circumference. The chord of an arc is the straight line which joins its two extremities. 4. A segment of a circle is the figure included between an arc and its chord. 5. A sector of a circle is the figure included between an arc, and the two radii drawn to the extremities of the arc. 6. A straight line is said to be inscribed in a circle, when its extremities are on the circumference. An inscribed angle is one whose sides are inscribed. 7. A polygon is said to be inscribed in a circle, when all its sides are inscribed. The circle is then said to be described about the polygon. 8. A secant is a line which cuts the circumference, and lies partly within and partly without the circle. 9. A straight line is said to touch a circle, when it meets the circumference, and, being produced, does not cut it. Such a line is called a tangent, and the point in which it meets the circumference, is called the point of contact. 10. Two circumferences touch each other when they meet, but do not cut one another. 11. A polygon is described about a circle, when each side of the polygon touches the circumference of the circle. In the same case, the circle is said to be inscribed in the polygon. PROPOSITION I. THEOREM. Every diameter divides the circle and its circumference into two equal parts. A B Let ACBD be a circle, and AB its diameter. The line AB divides the circle and its circumference into two equal parts. For, if the figure ADB be applied to the figure ACB, while the line AB remains common to both, the curve line ACB must coincide exactly with the curve line ADB. For, if any part of the curve ACB were to fall either within or without the curve ADB, there would be points in one or the other unequally distant from the center, which is contrary to the definition of a circle. Therefore, every diameter, &c. D PROPOSITION II. THEOREM. A straight line can not meet the circumference of a circle in more than two points. For, if it is possible, let the straight line ADB meet the circumference CDE in three points, C, D, E. Take F, the center of the circle, and join FC, FD, FE. Then, because F is the center of the circle, the three straight lines FC, FD, FE are all equal to each other; hence, three equal straight lines have Α F B D been drawn from the same point to the same straight line, which is impossible (Prop. XVII., Cor. 2, Book I.). Therefore, a straight line, &c. In equal circles, equal arcs are subtended by equal chords; and, conversely, equal chords subtend equal arcs. being equal to the diameter EF, the semicircle ADB may be applied exactly to the semicircle EHF, and the curve line AIDB will coincide entirely with the curve line EMHF (Prop. I.). But the arc AID is, by hypothesis, equal to the arc EMH; hence the point D will fall on the point H, and therefore the chord AD is equal to the chord EH (Axiom 11, B. I.). Conversely, if the chord AD is equal to the chord EH, then the arc AID will be equal to the arc EMH. For, if the radii CD, GH are drawn, the two triangles ACD, EGH will have their three sides equal, each to each, viz.: AC to EG, CD to GH, and AD equal to EH; the triangles are consequently equal (Prop. XV., B. I.), and the angle ACD is equal to the angle EGH. Let, now, the semicircle ADB be applied to the semicircle EHF, so that AC may coincide with EG; then, since the angle ACD is equal to the angle EGH, the radius CD will coincide with the radius GH, and the point D with the point H. Therefore, the arc AID must coincide with the arc EMH, and be equal to it. Hence, in equal circles, &c. PROPOSITION IV. THEOREM. In equal circles, equal angles at the center, are subtended by equal arcs; and, conversely, equal arcs subtend equal angles at the center. Let AGB, DHE be two equal circles, and let ACB, DFE be equal angles at their centers; then will the arc AB be equal to the arc DE. Join AB, DE; and, because the cir cles AGB, DHE are equal, their radii are equal. Therefore, the two sides CA, CB are equal to the two sides FD, FE; also, the angle at C is equal to the angle at F; therefore, the base AB is equal to the base DE (Prop. VI., B.I.). And, because the chord AB A G BD H is equal to the chord DE, the arc AB must be equal to the arc DE (Prop. III.). Conversely, if the arc AB is equal to the arc DE, the angle ACB will be equal to the angle DFE. For, if these angles are not equal, one of them is the greater. Let ACB be the greater, and take ACI equal to DFE; then, because equal angles at the center are subtended by equal arcs, the arc AI is equal to the arc DE. But the arc AB is equal to the arc DE; therefore, the arc AI is equal to the arc AB, the less to the greater, which is impossible. Hence the angle ACB is not unequal to the angle DFE, that is, it is equal to it. Therefore, in equal circles, &c. PROPOSITION V. THEOREM. In the same circle, or in equal circles, a greater arc is subtended by a greater chord; and, conversely, the greater chord subtends the greater arc. In the circle AEB, let the arc AE be greater than the arc AD; then will the chord AE be greater than the chord AD. A D E B Draw the radii CA, CD, CE. Now, if the arc AE were equal to the arc AD, the angle ACE would be equal to the angle ACD (Prop. IV.); hence it is clear that if the arc AE be greater than the arc AD, the angle ACE must be greater than the angle ACD. But the two sides AC, CE of the triangle ACE are equal to the two AC, CD of the triangle ACD, and the angle ACE is greater than the angle ACD; therefore, the third side AE is greater than the third side AD (Prop. XIII., B. I.); hence the chord which subtends the greater arc is the greater. Conversely, if the chord AE is greater than the chord AD, the arc AE is greater than the arc AD. For, because the two triangles ACE, ACD have two sides of the one equal to two sides of the other, each to each, but the base AE of the one is greater than the base AD of the other, therefore the angle ACE is greater than the angle ACD (Prop. XIV. B. I.); and hence the arc AE is greater than the arc AD (Prop. IV.). Therefore, in the same circle, &c. Scholium. The arcs here treated of are supposed to be less than a semicircumference. If they were greater, the opposite property would hold true, that is, the greater the arc the smaller the chord. PROPOSITION VI. THEOREM. The radius which is perpendicular to a chord, bisects the chord, and also the arc which it subtends. Let ABG be a circle, of which AB is a chord, and CE a radius perpendicular to` it; the chord AB will be bisected in D, and the arc AEB will be bisected in E. Draw the radii CA, CB. The two rightangled triangles CDA, CDB have the side AC equal to CB, and CD common; there- A fore the triangles are equal, and the base AD is equal to the base DB (Prop. XIX., B. I.). Secondly, since ACB is an isosceles triangle, and the line CD bisects the base at right angles, it bisects also the vertical angle ACB (Prop. X., Cor. 1, B. I.). And, since the angle ACE is equal to the angle BCE, the arc AE must be equal to the arc BE (Prop. IV.); hence the radius CE, perpendicular to the chord AB, divides the arc subtended by this chord, into two equal parts in the point E. Therefore, the radius, &c. Scholium. The center C, the middle point D of the chord AB, and the middle point E of the arc subtended by this chord, are three points situated in a straight line perpendicular to the chord. Now two points are sufficient to determine the position of a straight line; therefore any straight line which passes through two of these points, will necessarily pass through the third, and be perpendicular to the chord. Also, the perpendicular at the middle of a chord passes through the center of the circle, and through the middle of the arc subtended by the chord. |