162. Theorem. In the same circle or in equal circles, two chords unequally distant from the center are unequal, the chord at the less distance being the greater. Hypothesis. In circle with center O, AB and CD are chords, OM⊥ AB, ON I CD, OM < ON. Conclusion. AB > CD. Suggestions. Draw AE = CD, OP | AE, and draw PM. Proceed with a direct proof. A short indirect proof may be given also, based upon § 159 and § 161. Write out a complete proof. EXERCISES 1. A diameter is the longest chord of a circle. 2. The chord drawn through a given point within a circle, perpendicular to the diameter through that point, is the shortest chord which can be drawn through that point. 3. If two chords which intersect within the circle make equal angles with the radius drawn through the point of intersection, the chords are equal. 4. If the vertices of a polygon lie on a circle and the sides are equidistant from the center, it is equilateral. 5. If two equal chords intersect, the segments of one are equal to the corresponding segments C B A D 0 of the other. That is, if AB = DC, prove that BE = CE and AE = DE. 6. In this circle, a large number of equal chords are drawn. Explain why they appear to enclose a smaller circle with the same center. 7. Why is a board sawed from a circular log narrower than another board sawed from the same log nearer the center? 8. If PA and PC are two secants which make equal angles with the line connecting P to the center O, chords AB and CD which the circle cuts from the secants are equal. 9. The locus of the middle points of all of the equal chords of a circle is a concentric circle. 163. Theorem. - A straight line perpendicular to a radius at its outer extremity is tangent to the circle. Hypothesis. OC is a radius of circle with center O, and ABLOC at C. Conclusion. AB is tangent to the circle. Proof. 1. Let D be any point of AB except C. Draw OD. 2. Then OD > OC. § 145 3. ... Dis outside of the circle. § 151, (8) 4. ... since is the only point of AB that is on the circle, Def. tangent AB is tangent to the circle. 164. Theorem. A tangent to a circle is perpendicular to the radius drawn to the point of contact. Suggestions. Let AB be tangent at Cand let OC be a radius. Take D any point of AB except C. Show that OD > OC by § 151, (8), etc. Write the proof in full. EXERCISES 1. The perpendicular to a tangent at the point of contact passes through the center of the circle. SUGGESTION. - Use indirect proof. Draw the radius to the point of contact. How many perpendiculars can be drawn to a given line at a given point? 2. A perpendicular from the center of a circle to a tangent passes through the point of contact. SUGGESTION. - Draw the radius to the point of contact. 3. The tangents to a circle at the extremities of any diameter are parallel. 4. The diameter of a circle bisects all chords parallel to the tangent at the extremity of the diameter. 5. The locus of the centers of circles tangent to a straight line at a given point is a straight line perpendicular to the line at that point. 6. If two circles are concentric, a chord of the larger which is tangent to the smaller is bisected at the point of contact. 7. If two circles are concentric, chords of the larger which are tangent to the smaller are equal. A C ! 8. All equal chords of a circle are tangent to a concentric circle. 9. If the vertices of an equilateral hexagon are on a circle, a concentric circle can be drawn to which all sides of the hexagon are tangent. SUGGESTION. - The exercise may be proved if it can be proved that perpendiculars from the center to the sides of the hexagon are equal. By what theorem may the sides then be proved tangents? B Hypothesis. AB and AC are tangents to circle with center O at Band C, respectively. AB = AC. Conclusion. Suggestions. It may be proved that AB = AC if it can be shown that AB and AC are corresponding sides of congruent triangles. It may be proved that A AOC≃△AOB if it is first proved that ∠C and ∠ B are right angles and that OC = OB. Hence, begin by drawing OA, OB, and ос, proving that ∠C and ∠ B are right angles, etc. 166. Circumscribed and inscribed polygons. — A polygon all of whose sides are tangent to a circle is said to be circumscribed about the circle, and the circle is said to be inscribed in the polygon. A CIRCUMSCRIBED POLYGON AN INSCRIBED POLYGON AND A polygon all of whose vertices are on a circle is said to be inscribed in the circle, and the circle is said to be circumscribed about the polygon. EXERCISES 1. If two tangents are drawn from a point to a circle, the line joining the point to the center of the circle bisects the angle between the tangents. 2. In the figure of § 165, A O bisects BC. 3. In the figure of § 165, AO is the perpendicular bisector of the chord joining the points of contact of the tangents. 4. The instrument called a center square is used for locating the cen ters of circular objects. It consists of a and AD. When the center of a circular N A C D B are tangent to it. Prove that when this is done AB passes over the center of the object. 5. The prongs AC and AD of the center square in Ex. 4 are equal. Prove that if the circular object is so large that when the instrument is applied to it the ends of the prongs, C and D, rest against the object, A C and AD becoming secants, then AB passes over the center of the object. 6. The bisectors of the angles of a circumscribed polygon pass through the center of the circle. 7. The locus of the centers of all circles tangent to both sides of an angle is the bisector of the angle. SUGGESTION. - Endeavor to apply the theorem in § 108. 8. If a circle is inscribed in a right triangle, the sum of the legs equals the sum of the hypotenuse and the diameter of the circle. C |