(2) If two circumferences meet on their center line, the center sect equals the sum, or the difference, of the radii, according as the center of one circle is outside, or inside, the other.

(3) *If two circumferences meet at a point not on their center line, the center sect is less than the sum, and greater than the difference, of the radii.

If the figures are drawn, the reasons for the statements will be found to depend upon the inequality axiom, and the length relations between the sum or difference of two sides of a triangle and the third side.

170. Intersection of Two Circumferences. (Converses of § 169.)

*If the center sect of two circles

(1) is greater than the sum of the radii, the circumfer ences do not meet, and lie outside each other.

(2) is less than the difference of the radii, the circumferences do not meet, and one circle is within the other.

(3) equals the sum of the radii, the circumferences meet on the center line, and are outside of each other.

(4) equals the difference of the radii, the circumferences meet on the center line, and one circle is inside the other.

(5) is less than the sum and greater than the difference of the radii, the circumferences meet at a point not on the center line.

These statements are used in all cases where it is necessary to show whether or not two circumferences meet.

NOTE. There are five values for the center sect: less than the difference of the radii, equal to that difference, between the difference and the sum, equal to the sum, and greater than the sum. The pupil should understand the results from each of these lengths.

171. *Two circumferences that meet at a point not on their center line, intersect twice.

If the circum

ferences of Os o

and o meet at P,
a second triangle
oo'q could be con- o
structed congruent
to oo'p, by drawing
angles on the other

side of oo' equal respectively to O'op and Zoo'P. Why does this show that Q is a second point of intersection?

NOTE. To show that two circumferences intersect twice it is necessary to show.

(1) the center sect less than the sum of the radii.

(2) the center sect greater than the difference of the radii.

172. Common Chord. If two circumferences meet twice, the line joining the points of intersection of the circles is called their common chord.

*If two circumferences intersect twice, their center line is the perpendicular bisector of their common chord. Use equidistance.

173. Tangency of Circles. Two circumferences that meet at one point only are said to be tangent to each other, or to touch each other. The point where they meet is called the point of tangency, or the point of contact. The circles also are sometimes spoken of as tangent to each other.

*If two circumferences meet at a point on their center line, they are tangent to each other. If they meet at a

second point on the center line, they are identical; if at a second point which is not on the center line, the center sect would have two different lengths, which is impossible [§ 169 (2), (3)].

*Two circles tangent to the same line at the same point are tangent to each other.

174. The shortest sect from a given point to a circumference is a sect of the line from that point through the center. Why? Prove it when the point is inside and when it is outside.

175.

ORAL AND REVIEW QUESTIONS

If the radii of two circles are 10 ft. and 6 ft. long, tell what is known about the position of the circles if the center sect is (1) 12 ft. long, (2) 16 ft. long, (3) 4 ft. long, (4) 2 ft. long, (5) 20 ft. long. Why is a circumference a curve? Why has it but one center? How many points not in a straight line determine a circumference? Can a circumference always be drawn through four points? Can it always be drawn tangent to three lines? When are circles tangent? When is a line tangent to a circle? What is known about the center line of two intersecting circumferences and their common chord? What relation is there between the distance of a line from the center of a circle and the intersection of the line and circumference. If it is necessary to draw two circles which are tangent externally (i.e. outside of each other), how long must the center sect be made? If it is necessary to have their circumferences meet twice?

SECTION II. CONSTRUCTIONS

176. Postulates. In plane geometry, the use of two instruments for construction is allowed, or postulated. These instruments are the straight edge (or ruler, except that no measurements may be taken with it) and the compass.

The postulates that allow the use of these instruments

are:

(1) A straight line may be drawn from any one point to any other point.

(2) A sect may be produced to any length in that line. (3) A circumference may be described with any point as a center and any sect as a radius.

177. Constructions. In the theorems of Book I, auxiliary lines have been added to the given figure, but only as representations of existing lines about which it was necessary to reason in order to establish the proof. The question of the accuracy of the drawing of these lines had no bearing on the truth of the theorem, for the reasoning was entirely about the figures that the lines. were taken to represent. In the problems that are to be done, the proof consists in showing that the figure constructed is-within the limits of accuracy of the instruments used-a correctly drawn figure according to the requirements of the proposition. The theorem proves a fact; the construction makes a required figure, then proves the correctness of the method of construction.

To discover the

178. Analysis of a Construction. method of drawing a required construction, first draw the completed figure as accurately as possible, without actually constructing it. By the study of the completed figure, attempt to find what lines must be constructed in order to make the figure in such a way that it can be shown to be the required one. This is really working backwards from the completed figure in an attempt to find upon what it is based. Having analyzed the figure, draw the lines found necessary, and so build up a figure that can be proved to be

correct.

Do not neglect the classification method, for it will make the analysis method unnecessary in many cases. For example, a perpendicular can be obtained by finding two points equidistant from the ends of the sect to which the line is to be perpendicular.

The cir

179. Construction Uses of the Circumference. cumference is the locus of points at a certain fixed distance from a fixed point. The definition shows this fact, and it is of great importance, both in proofs, and in construction work. The two following hints will show common uses of the circumference.

To draw a line of a given length from a given point, use the compass to draw a circumference around the given point, with the given sect as a radius. This is called describing a circumference with a certain center and radius.

To construct a point equidistant from two given points, use each of the given points as a center, and use the same radius. What relation must hold between the center sect and the radii?