ARGUMENT

REASONS

6. .. BF is the bisector of 6. The bisector of an is

ZABC.

Q.E.D.

the line which divides the into two equal 4. § 53.

III. Discussion

This construction is always possible, and there is only one solution.

Ex. 100. Draw an obtuse angle and divide it into: (a) four equal angles; (b) eight equal angles.

Ex. 101. Construct the bisector of the vertex angle of an isosceles triangle.

Ex. 102.

Draw two intersecting lines and construct the bisectors of the four angles formed.

Ex. 103. Bisect an angle between two bisectors in Ex. 102, and find the number of degrees in each angle.

Ex. 104. Construct the bisector of an exterior angle at the base of an isosceles triangle.

Ex. 105. Construct the bisectors of the three angles of any triangle. What can you infer about them? Can the correctness of this inference be proved by making a careful construction?

128. Def. The distance between two points B is the length of the straight line joining them. Thus if three points, A, B, and C, are so located that AB = AC, A is said to be equidistant from B and C.

C

Ex. 106. Find all the points on the blackboard which are one foot from a fixed point, P, on the blackboard.

A

Ex. 107. Draw a line, AB, on the blackboard and mark some point near the line, as P. Find all the points in AB that are a foot from P.

Ex. 108. Mark a point, Q, on the blackboard. Find all the points on the blackboard which are: (a) ten inches from Q; (b) four inches from Q. How far are the points of (b) from the points of (a), if the distance is measured on a line through Q?

LOCI

129. In many geometric problems it is necessary to locate all points which satisfy certain prescribed conditions, or to determine the path traced by a point which moves according to certain fixed laws. Thus, the points in a plane two inches from a given point are in the circumference of a circle whose center is the given point and whose radius is two inches.

Again, let it be required to find all points in a plane two inches from one fixed point and three inches from another. All points two inches from the fixed point P are in the circumference of the circle LMS, having P for center and having a radius equal to two inches. All points three inches from the fixed point are in the circumference of the circle LRT, having Q for center and having a radius equal to three inches. If the two circles are

S

2 in.

M

3 in.

T

R

wholly outside of each other, there will be no points satisfying the two prescribed conditions; if the two circumferences touch, but do not intersect, there will be one point; if the two circumferences intersect, there will be two points. It will be proved later (§ 324) that there cannot be more than two points which satisfy both of the given conditions.

130. Def. A figure is the locus of all points which satisfy one or more given conditions, if all points in the figure satisfy the given conditions and if these conditions are satisfied by no other points.

A locus, then, is an assemblage of points which obey one or more definite laws.

It is often convenient to locate these points by thinking of them as the path traced by a moving point the motion of which is controlled by certain fixed laws.

131. In plane geometry a locus may be composed of one or more points or of one or more lines, or of any combination of points and lines.

132. Questions.* — What is the locus of all points in space two inches from a given point?

What is the locus of all points in space two inches from a given plane? What is the locus of all points in space such that perpendiculars from them to a given plane shall be equal to a given line? What is the locus of all points on the surface of the earth midway between the north and south poles? 231° from the equator? 231° from the north pole? 90° from the equator? What is the locus of a gas jet four feet from the ceiling of this room? four feet from the ceiling and five feet from a side wall? four feet from the ceiling, five feet from a side wall, and six feet from an end wall?

Ex. 109. Given an unlimited line AB and a point P. Find all points in AB which are also: (a) three inches from P; (b) at a given distance, a, from P.

Ex. 110. Given a circle with center O and radius six inches. State, without proof, the locus: (a) of all points four inches from 0; (b) of all points five inches from the circumference of the circle, measured on the radius or radius prolonged.

Ex. 111. Given the base and one adjacent angle of a triangle, what is the locus of the vertex of the angle opposite the base? (State without proof.)

Ex. 112. Given the base and one other side of a triangle, what is the locus of the vertex of the angle opposite the base? (State without proof.)

Ex. 113. Given the base and the other two sides of a triangle, what is the locus of the vertex of the angle opposite the base?

Ex. 114.

Given the base of a triangle and the median to the base, what is the locus of the end of the median which is remote from the base?

Ex. 115. Given the base of a triangle, one other side, and the median to the base, what is the locus of the vertex of the angle opposite the base ?

133. Question. In which of the exercises above was a triangle determined?

* In order to develop the imagination of the student the authors deem it advisable in this article to introduce questions involving loci in space. It should be noted that no proofs of answers to these questions are demanded.

PROPOSITION IX. THEOREM

134. Every point in the perpendicular bisector of a line is equidistant from the ends of that line.

Given line AB, its bisector CD, and P any point in CD.

Ex. 116. Four villages are so located that B is 25 miles east of A, C 20 miles north of A, and D 20 miles south of A. Prove that B is as far from C as it is from D.

Ex. 117. In a given circumference, find the points equidistant from two given points, 4 and B,

135. Def. One theorem is the converse of another when the conclusion of the first is the hypothesis of the second, and the hypothesis of the first is the conclusion of the second.

The converse of a truth is not always true; thus," All men are bipeds" is true, but the converse, “ All bipeds are men,” is false. "All right angles are equal" is true, but "All equal angles are right angles" is false.

136. Def. One theorem is the opposite of another when the hypothesis of the first is the contradiction of the hypothesis of the second, and the conclusion of the first is the contradiction of the conclusion of the second.

The opposite of a truth is not always true; thus, "If a man lives in the city of New York, he lives in New York State," is true, but the opposite, "If a man does not live in the city of New York, he does not live in New York State," is false.

137. Note. If the converse of a proposition is true, the opposite also is true; so, too, if the opposite of a proposition is true, the converse also is true.

This may be evident to the student after a consideration of the following type forms:

If (2) is true, then (3) must be true. Again, if (3) is true, then (2) must be true.

138. A necessary and sufficient test of the completeness of a definition is that its converse shall also be true. Hence a definition may be quoted as the reason for a converse or for an opposite as well as for a direct statement in an argument.

Ex. 118. State the converse of the definition for equal figures; straight line; plane surface.

Ex. 119. State the converse of: If one straight line meets another straight line, the sum of the two adjacent angles is two right angles.

Ex. 120. State the converse and opposite of Prop. IX.

Ex. 121. State the converse of Prop. I. Is it true?