Proposition IX 205. 1. Draw a circle and one of its radii; also a line perpendicular to the radius at its extremity. Is this line a tangent or a secant? 2. Draw a tangent to a circle and a radius to the point of contact. What kind of an angle is formed by these lines? Theorem. A line perpendicular to a radius at its extremity is tangent to the circle; conversely, a tangent is perpendicular to the radius drawn to the point of contact. Data: A circle whose center is 0; any radius, as OD; and a line AB perpendicular to OD at D. To prove AB tangent to the circle. A D E Proof. From o draw any other line to AB, as OE. OD < OE. B Why? Since every point in the circumference is at a distance equal to OD from the center, and E is at a greater distance, E is without the circumference. Therefore, every point of AB, except D, is without the circum rerence. Hence, § 184, AB is tangent to the circle at D. Q.E.D. Conversely: Data: Any tangent to this circle, as AB, and the cadius drawn to the point of contact, as OD. Proof. Ax. 17, every point of AB, except D, is without the tircumference. .. OD is the shortest line that can be drawn between 0 and AB. Hence, § 61, Therefore, etc. OD LAB. Q.E.D. Ex. 195. If in a circle a chord is perpendicular to a radius at any point, how does it compare in length with any other chord which can be drawn through that point? Ex. 196. If tangents are drawn through the extremities of a diameter, what is their direction with reference to each other? Proposition X 206. 1. Draw a circle, a tangent to it, and a chord parallel to the tanHow do the arcs intercepted between the point of tangency and the extremities of the chord compare? gent. 2. Draw a circle and two parallel secants or chords. How do the intercepted arcs compare? Theorem. Parallel lines intercept equal arcs on a circumference. Data: A circle whose center is 0, and any two parallel lines, as AB and CD, intercepting arcs on the circumference. To prove that the arcs intercepted by AB and CD are equal. E A B D D Proof. Case I. When AB is a tangent and CD is a chord. Draw EF || AB, and tangent to the circum E Proposition XI 207. Select three points not in the same straight line. circumferences can be passed through them? How many Theorem. Through three points not in the same straight line one circumference can be drawn, and only one. Data: Any three points not in the same straight line, as A, B, and C. To prove that one circumference can be drawn through A, B, and C, and only one. Proof. Draw AB, BC, and AC; and at their middle points, E, F, and G, respectively, erect perpendiculars. E § 170, these perpendiculars meet in a point, as 0, which is equidistant from A, B, and C; .. § 173, a circumference described from O as a center, and with a radius equal to the distance 04, passes through the points A, B, and C; and, since the perpendiculars intersect in but one point, there can be but one center, and consequently but one circumference passing through the points A, B, and C. Q.E.D. 208. Cor. I. Circles circumscribing equal triangles are equal. Cor. II. Two circumferences can intersect in only two points. If two circumferences have three points in common, they coincide and form one circumference. Proposition XII 209. Draw a circle and from a point outside draw two tangents. How do the tangents compare in length? Theorem. The tangents drawn to a circle from a point without are equal. Data: A circle whose center is O; any point without it, as 4; and AB and AC the tangents to the circle at the points B and C, respectively. B A 210. The line which joins the centers of two circles is called their line of centers. 211. A common tangent to two circles which cuts their line of centers is called a common interior tangent; one which does not cut their line of centers is called a common exterior tangent. Proposition XIII 212. Draw two intersecting circles and a chord that is common to both. What kind of an angle does a line joining their centers make with this common chord? Theorem. If the circumferences of two circles intersect, the line of centers is perpendicular to the common chord at its middle point. Data: Two circles whose centers are O and P, whose circumferences intersect at 4 and B; AB the common chord; and OP the line of centers. To prove OP perpendicular to AB at its middle point. Proof. § 173, P is equidistant from A and B, and also, o is equidistant from A and B ; P .. § 104, both P and o lie in the perpendicular bisector of AB. Hence, Ax. 11, OP coincides with this perpendicular bisector; that is, Therefore, etc. OP LAB at its middle point. Q.E.D. MILNE'S GEOM.- -7 Proposition XIV 213. Draw two circles tangent to each other, and a line joining their centers. Through what point will this line pass? Theorem. If two circles are tangent to each other, their line of centers passes through the point of contact. Proof. At C draw the common tangent AB. C Ax. 17, .. § 205, also, hence, § 58, OP passes through C. LOCA + PCA Q.E.D. MEASUREMENT 214. The theorems thus far presented and proved have usually established only the equality or inequality of two magnitudes, but it is sometimes desirable to measure accurately the magnitudes that are given. A magnitude is measured when we find how many times it contains another magnitude of the same kind, called the unit of measure. The number which expresses how many times a magnitude contains a unit of measure is called its numerical measure. 215. The relation of two magnitudes which is determined by finding how many times one contains the other, or what part one is of the other, is called their Ratio. |