247. DEFINITIONS. BOOK II A circle is a portion of a plane bounded by a curved line, all the points of which are equally distant from a point within called the center. The bounding line is called the circumference. A straight line from the center to any point in the circumference is a radius. It follows from the definition of circle that all radii of the same circle are equal. Arc Chord Diameter Radius A straight line passing through the center and limited by the circumference is a diameter. Every diameter is composed of two radii; therefore all diameters of the same circle are equal. An arc is any portion of a circumference. A chord is a straight line joining the extremities of an arc. A chord is said to subtend the arc whose extremities it joins, and the arc is said to be subtended by the chord. Every chord subtends two different arcs; A thus the chord AB subtends the arc ANB, and also the arc AMB. Unless the contrary is specially stated, we shall assume the chord to belong to the smaller arc. An inscribed polygon is a polygon whose vertices are in the circumference and whose sides are chords. [The polygon ABCD is inscribed in the circle; the circle is also said to be circum- A scribed about the polygon.] N B B M PROPOSITION I. PROBLEM 248. To find the center of a given circle. Join any two points on the circumference, as A and B, by the line AB. Bisect AB by the perpendicular DC. Bisect DC. Then is o the center of the circle. By definition, the center of the circle is equally distant from A and B. By § 48 the center is on DC. By definition the center of the circle is equally distant from D and C. Since the center is on DC, and is also equally distant from D and C, it must be at the middle point of DC, that is, at 0. Therefore, is the center of the circle xyz. Q.E.F. 249. COROLLARY. A line that is perpendicular to a chord and bisects it, passes through the center of the circle. NOTE. It follows from § 249 that the only chords in a circle that can bisect each other are diameters. 250. EXERCISE. Describe a circumference passing through two given points. How many different circumferences can be described passing through two given points ? 251. EXERCISE. Describe a circumference, with a given radius, and passing through two given points. How many circumferences can be described in this case? PROPOSITION II. THEOREM 252. A diameter divides a circle and also its circumference into two equal parts. B Let AB be a diameter of the circle whose center is 0. To Prove that AB divides the circle and also its circumference into two equal parts. Proof. Place ACB upon ADB so that AB is common. Then will the curves ACB and ADB coincide, for if they do not there would be points in the two arcs unequally distant from the center, which contradicts the definition of circle. Therefore AB divides the circle and also its circumference into two equal parts. Q.E.D. 253. EXERCISE. Through a given point draw a line bisecting a given circle. When can an infinite number of such lines be drawn? SANDERS' GEOM.- -6 PROPOSITION III. THEOREM 254. A diameter of a circle is greater than any other chord. B Let AB be a diameter of the O whose center is 0, and CD be any other chord. Apply § 168 to A OCD, recollecting that AB OC + OD. Q.E.D. 255. EXERCISE. Prove this Proposition (§ 254), using a figure in which the given chord CD intersects the diameter AB. 256. EXERCISE. Through a point within a circle draw the longest possible chord. 257. EXERCISE. The side AC of an inscribed triangle ABC is a diameter of the circle. Compare the angle B with angles A and C. 258. EXERCISE. AB is perpendicular to the chord CD, and bisects it. 259. EXERCISE. The diameter AB and E the chord CD are prolonged until they meet at E. Prove and EA EC EB> ED. B PROPOSITION IV. THEOREM 260. A straight line cannot intersect a circumference in more than two points. A PE B R Let CDR be a circumference and AB a line intersecting it at C and D. To Prove that AB cannot intersect the circumference at any other point. Proof. Suppose that AB did intersect the circumference in a third point E. Draw the radii to the three points. Now we have three equal lines (why equal?) drawn from the point o to the line AB, which contradicts (?). Therefore the supposition that AB could intersect the circumference in more than two points is false. 261. EXERCISE. Q.E.D. Show by §§ 249 and 92 that AB cannot intersect the circumference in three points (C, D, and E). 262. DEFINITION. A secant is a straight line that cuts a circumference. B |