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CIRCLES AND TANGENTS The basic principle is the definition:
A circle is a plane figure bounded by a curved line, all points of which are equally distant from a point within called the center.
157. Define Secant; also Circle; Radius; Arc.
160. a. Only one circle can be drawn with a given center and a given radius.
b. All radii of the same or equal circles are equal.
164. Define Central Angle. 165. Define Circumscribed. 166. Define Inscribed. *167. Through three points not in a straight line, one circle and only one can be drawn.
168. Through three (or more) points in a straight line, a circle cannot be drawn. *169. PROBLEM: Circumscribe a circle about a triangle. *170. A straight line perpendicular to a radius at its end is a tangent to the circle.
*171. A tangent to a circle is perpendicular to the radius drawn to the point of tangency.
*172. Two tangents to a circle from an external point are equal. *173. PROBLEM: Inscribe a circle in a triangle. *174. A circle can be circumscribed about or inscribed in any regular polygon.
175. State two methods of proving a line tangent to a circle.
State one additional method of proving lines perpendicular.
EXERCISES 1. Draw a tangent to a circle at a point on the circumference.
2. A perpendicular to a tangent at the point of tangency passes through the center.
3. If two circles are tangent, the two radii to the point of tangency are in a straight line.
4. If a line is drawn from the center of a circle to the intersection of two tangents, this line: a. Bisects the angle between the tangents; b. Is the perpendicular bisector of the chord between the points of tangency.
5. If two circles are tangent externally, the tangents to them from any point in the common internal tangent are equal.
6. The sum of two opposite sides of a circumscribed quadrilateral is equal to the sum of the other two sides.
7. If two common external tangents are drawn to two circles, the parts intercepted between the points of tangency are equal.
8. Find the locus of the centers of circles, which have a given radius and touch a given line.
9. Find the locus of the centers of circles, which touch a given line at a given point.
10. Find the locus of the centers of circles, which touch each of two parallel lines.
11. Find the locus of the centers of circles, which touch each of two non-parallel lines.
12. Find the locus of points at a given distance from a given point.
13. Find a point at a given distance from a given point, and equidistant from two given points.
14. Find a point at a given distance from a given point, and equidistant from two non-parallel lines.
15. Find a point at a given distance from a given point, and equidistant from two parallel lines.
16. Find a point at a given distance from a given point, and at a given distance from a given line.
EQUALITY OF ARCS AND CHORDS
The two basic methods are:
177. In the same or equal circles, equal central angles intercept equal arcs.
178. In the same or equal circles, equal arcs are intercepted by equal central angles.
*179. In the same or equal circles, equal chords subtend equal arcs.
*180. In the same or equal circles, equal arcs are subtended by equal chords.
*181. A radius perpendicular to a chord of a circle bisects the chord and its arc.
*182. Parallel lines intercept equal arcs on a circumference. (Three cases.)
*183. a. In the same or equal circles, equal chords are equally distant from the center.
b. In the same or equal circles, chords equally distant from the center are equal.
184. PROBLEM: Given an arc, find the center of the circle. 185. PROBLEM: Bisect a given arc.
SUMMARY 186. State four methods of proving arcs equal.
State four additional methods of proving straight lines equal.
EXERCISES 1. Prove Art. 160f. by Art. 177.
2. There are four ideas in Art. 181: (1) Radius; (2) perpendicular; (3) bisects the chord; (4) bisects the arc. If any two of these are
given, the other two can be proved. In Art. 181 (1) and (2) were given, and (3) and (4) were proved.
Let (1) and (3) be given; prove (2).
Make three other combinations and prove them. 3. If two circles intersect, the line of centers is the perpendicular bisector of the common chord.
4. If A,B, D, and Care four points in order on a circumference such that BC equals AD, then chord AB equals chord CD.
5. If A,B, D, and Care four points in order on a circumference, such that AB equals CD, then chord BC equals chord AD.
6. The opposite arcs intercepted by two diameters are equal.
7. Two chords drawn perpendicular to a third chord at its ends are equal.
8. Two parallel chords drawn through the ends of a diameter are equal.
9. If two unequal circles have the same center (concentric), any two chords of the larger circle tangent to the smaller circle are equal.
10. If from a point in a circumference two chords are drawn making equal angles with the radius to that point, these chords are equal.
11. If two intersecting chords make equal angles with the radius through the point of intersection, these chords are equal.
12. The diameter of the circle inscribed in a right triangle equals the difference between the sum of the short sides and the hypotenuse.
13. Find the locus of the middle points of chords of a given length in a given circle.
14. Find the locus of the middle points of chords drawn from a given point in a given circumference.
15. Find the locus of the middle points of straight lines drawn from a given external point to a given circumference.
16. Find the locus of the end points of straight lines of a given length, touching a given circumference at one end and parallel to a given line.
17. A ladder, leaning against a perpendicular wall, slips down until it is flat upon the ground. Find the locus of its middle point.
18. Divide a circle into four equal arcs; six; three. 19. A trapezoid inscribed in a circle is isosceles.
20. If two circles are tangent to two parallel lines, any other parallel line, cutting the circle, forms two equal chords.
INEQUALITY OF ARCS AND CHORDS
The basic method is:
188. In the same or equal circles, the greater of two central angles intercepts the greater arc.
189. In the same or equal circles, the greater of two arcs is intercepted by the greater central angle.
*190. In the same or equal circles, the greater of two chords subtends the greater arc.
191. In the same or equal circles, the greater of two arcs is subtended by the greater chord.
*192. In the same or equal circles, the greater of two chords is nearer the center.
*193. In the same or equal circles, if two chords are unequally distant from the center, the chord nearer the center is the greater.
SUMMARY 194. State two methods of proving arcs unequal.
195. State two additional methods of proving straight lines unequal. 196.
EXERCISES 1. The diameter of a circle is greater than any other chord. (Two ways.)
2. Prove Art. 193 by the Indirect Method.
3. A chord drawn perpendicular to a diameter is smaller than any other chord through the point of intersection.
4. If AB and CD are two intersecting chords of a circle, and AB is greater than CD, and arcs ACB and DAC are each less than a semicircumference, BC is greater than AD.
5. In the same figure, if BC is greater than AD, prove AB greater than CD.