165. Theorem. The two tangents to a circle from a point outside of the circle are equal. A B Hypothesis. AB and AC are tangents to circle with center O at B and C, respectively. Conclusion. AB= AC. 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 A AOB if it is first proved that ≤ C and ▲ B are right angles and that OC = OB. Hence, begin by drawing OA, OB, and OC, 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, AO bisects BC. 3. In the figure of § 165, AO is the perpendicular bisector of the chord joining the points of contact of the tangents. M B 4. The instrument called a center square is used for locating the centers of circular objects. It consists of a steel blade M, upon which slides an attachment N with two prongs, AC and AD. The edge AB of the blade M bisects the angle between the prongs AC and AD. When the center of a circular object is to be found, the instrument is placed so that the prongs AC and AD are tangent to it. Prove that when this is done AB passes over the center of the object. D 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, AC 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. 9. AB is tangent to the circle with center O at M. AC and BD are tangent at C and D, respectively. Prove that AB = AC + BD. 10. If an isosceles triangle is circumscribed about a circle, the base is bisected at the point of contact. 11. If a circle is inscribed in an equilateral triangle, the three sides are bisected at the points of contact. 12. In any circumscribed quadrilateral, the sum of two opposite sides equals the sum of the other two opposite sides. A 13. The perimeter of any circumscribed trapezoid is equal to four times the line-segment which joins the middle points of the two nonparallel sides. 14. A parallelogram circumscribed about a circle is either a rhombus or a square. 15. In any circumscribed hexagon, the sum of one set of alternate sides equals the sum of the other set, that is AB+ CD + EF = BC + DE + FA. 16. In any circumscribed octagon, the sum of one set of alternate sides equals the sum of the other set. E B 17. The sides of a triangle are 5 in., 6 in., and 8 in. Find the lengths of the segments into which the sides are divided at the points of contact with the inscribed circle. SUGGESTION. quantities. Form three equations and solve for the three unknown 167. Line of centers. The straight line joining the centers of two circles is called the line of centers of the circles. 168. Theorem. The line of centers of two intersecting circles is the perpendicular bisector of their common chord. M B Hypothesis. A and B are the centers of two intersecting circles, AB is the line of centers, and MN is the common chord. Conclusion. AB is the perpendicular bisector of MN. Suggestions. The conclusion will follow from § 107, if A and B are both proved equidistant from M and N. begin by proving the latter. Write the proof in full. Hence 169. Common tangents. A straight line which is tangent to each of two circles is called a common tangent of the circles. If the circles lie on the same side of the common tangent, it is called a common external tangent, as AB. If the circles lie on the opposite sides of the common tangent, it is called a common internal tangent, as CD. EXERCISES 1. What is the largest number of common tangents which two circles may have? 2. Draw a figure showing two circles which have only three common tangents. 3. Draw a figure showing two circles which have only two common tangents. 4. Draw a figure showing two circles which have only one common tangent. 5. Draw two circles which have no common tangent. 6. Prove that the common internal tangents of two circles are equal. 7. Prove that the common external tangents of two equal circles are equal. 8. Prove that the common external tangents of two unequal circles are equal. 9. What machinery have you seen which illustrates common tangents to two circles? - 170. Tangent circles. Two circles both of which are tangent to the same line at the same point are called tangent circles. The point of contact of the circles and line is called the point of contact of the circles. @ TANGENT INTERNALLY ∞ TANGENT EXTERNALLY Two circles are said to be tangent internally when they lie on the same side of the common tangent line, and tangent externally when they lie on opposite sides of it. |