Given circle O with

PERSPECTIVE ON HISTORY Circumference of the Earth By traveling around the earth at the equator, one would traverse the circumference of the earth. Early mathematicians attempted to discover the numerical circumference of the earth. But the best approximation of the circumference was due to the work of the Greek mathematician Eratosthenes 276–194 B . C .. In his day, Eratosthenes held the highly regarded post as the head of the museum at the university in Alexandria. What Eratosthenes did to calculate the earth’s circumference was based upon several assumptions. With the sun at a great distance from the earth, its rays would be parallel as they struck the earth. Because of parallel lines, the alternate interior angles shown in the diagram would have the same measure indicated by the Greek letter ␣. In Eratosthenes’ plan, an angle measurement in Alexandria would be determined when the sun was directly over the city of Syene. While the angle suggested at the center of the earth could not be measured, the angle in Alexandria formed by the vertical and the related shadow could be measured; in fact, the measure was . Eratosthenes’ solution to the problem was based upon this fact: The ratio comparing angle measures is equivalent to the ratio comparing land distances. The distance between Syene and Alexandria was approximately 5,000 stadia ␣ L 7.2° 1 stadium 516.73 ft. Where C is the circumference of the earth in stadia, this leads to the proportion Solving the proportion and converting to miles, Eratosthenes approximation of the earth’s circumference was about 24,662 mi, which is about 245 mi less than the actual circumference. Eratosthenes, a tireless student and teacher, lost his sight late in life. Unable to bear his loss of sight and lack of productivity, Eratosthenes committed suicide by refusing to eat. ␣ 360° = 5000 C or 7.2 360 = 5000 C L PERSPECTIVE ON APPLICATIONS Sum of Interior Angles of a Polygon Suppose that we had studied the circle before studying polygons. Our methods of proof and justifications would be greatly affected. In particular, suppose that you do not know the sum of interior angles of a triangle but that you do know these facts: 1. The sum of the arc measures of a circle is 360°. 2. The measure of an inscribed angle of a circle is the measure of its intercepted arc. Using these facts, we prove “The sum of the interior angles of a triangle is 180°.” Proof: In , , , and . Then 1 2 mBC ¬ + mAC ¬ + mAB ¬ = 1 2 360° = 180°. m ∠A + m∠B + m∠C = m ∠C = 1 2 m AB ¬ m ∠B = 1 2 mAC ¬ m ∠A = 1 2 mBC ¬ 䉭ABC 1 2 Alexandria Syene Figure 6.62 Using known facts 1 and 2, we can also show that “The sum of the interior angles of a quadrilateral is 360°.” However, we would complete our proof by utilizing a cyclic quadrilateral. The strategic ordering and association of terms leads to the desired result. Proof: For quadrilateral HJKL in Figure 6.64, or 1 2 360° + 1 2 360° 1 2 m LKJ ២ + m LHJ ២ + 1 2 mHLK ២ + m HJK ២ = 1 2 m LKJ ២ + 1 2 mHLK ២ + 1 2 m LHJ ២ + 1 2 m HJK ២ m ∠H + m∠J + m∠K + m∠L = B A C Figure 6.63 In turn, we see that 180° + 180° or 360° m ∠H + m∠J + m∠K + m∠L = L H J K Figure 6.64 We could continue in this manner to show that the sum of the five interior angles of a pentagon using a cyclic pentagon is 540° and that the sum of the n interior angles of a cyclic polygon of n sides is . n - 2180° Summary A LOOK BACK AT CHAPTER 6 One goal in this chapter has been to classify angles in- side, on, and outside the circle. Formulas for finding the measures of these angles were developed. Line and line segments related to a circle were defined, and some ways of finding the measures of these segments were de- scribed. Theorems involving inequalities in a circle were proved. A LOOK AHEAD TO CHAPTER 7 One goal of Chapter 7 is the study of loci plural of locus, which has to do with point location. In fact, a locus of points is often nothing more than the description of some well-known geometric figure. Knowledge of locus leads to the determination of whether certain lines must be con- current meet at a common point. Finally, we will extend the notion of concurrence to develop further properties and terminology for regular polygons. KEY CONCEPTS

6.1

Circle • Congruent Circles • Concentric Circles • Center of the Circle • Radius • Diameter • Chord • Semicircle • Arc • Major Arc • Minor Arc • Intercepted Arc • Congruent Arcs • Central Angle • Inscribed Angle

6.2

Tangent • Point of Tangency • Secant • Polygon Inscribed in a Circle • Cyclic Polygon • Circumscribed Circle • Polygon Circumscribed about a Circle • Inscribed Circle • Interior and Exterior of a Circle 6.3 Tangent Circles • Internally Tangent Circles • Externally Tangent Circles • Line of Centers • Common Tangent • Common External Tangents • Common Internal Tangents

6.4

Constructions of Tangents to a Circle • Inequalities in the Circle