If at M, the middle point of the segment AB, an auxiliary line MN be drawn perpendicular to AB (§ 13) it will be perpendicular to DC (§ 19, Ex. 1). But we do not as yet know whether it will bisect DC or not.

If we revolve the portion of the figure that lies on the right of the line MN (§§ 8 and 10) about MN as an axis until the revolved portion coincides with the unrevolved portion, we shall have the segment MB coinciding with the segment MA (because M is by selection the middle point of the segment AB). The point B will fall at A, and the line BC (being perpendicular to MB) will in its revolved position take the direction AD.

Because NC is perpendicular to MN, it will when rotated lie in the direction ND.

The point C will then lie somewhere in the line AD, and somewhere in the line ND. It must lie at their intersection (D). Therefore the segment BC will coincide with the segment AD and be equal to it.

Q. E. D.

3. Show that if two lines are parallel to a third line, they are parallel to each other.

4. Show how perpendiculars to a line may be drawn from points on the line. Let fall perpendiculars from given points to a given line. Through given points draw parallels to a given line.

Suggestion. Use a ruler and a right-angled triangle; and assume the line in a variety of positions.

NOTE. The formal statement of a theorem may be followed by the proof; or the relations leading to conclusions may be presented first and the formal statement of the theorem be presented at the end. When placed before the proof, it is a statement of relations said to exist. The proof is the establishing of these said relations (see § 17). When placed after the determination of relations it is in the form of a conclusion (see § 16).

In general it is better to state the theorem first, so that the student shall have in mind the relation that he is undertaking to establish.

20. THEOREM. If at the middle point of a segment of a line a perpendicular be erected, and if from any point in the perpendicular lines be drawn terminating at the extremities of the segment, they will be equal to each other.

=

FIG. 22.

If PB PA, it can be brought to coincide with PA, and since AB and MP are perpendicular to each other, a rotation of one part of the figure on a line as an axis is suggested.

Proof. Revolve the portion of the figure that lies to the right of the line MP, on MP as an axis, until it coincides with the plane on the left of MP.

§ 10.

The PMB will coincide with the PMA. B will fall at A; P will remain stationary; and the segment PB will coincide with the segment PA, and must therefore be equal to it.

Q. E. D.

Exercises. 1. Show that any point not on the perpendicular bisector will not be equally distant from the extremities of the bisected segment.

2. Show that if a line have two of its points equally distant from the ends of the segment, it will be the bisector of the segment.

Solution. If a perpendicular bisector of the segment were erected it would contain all points that are equally distant from the extremities of the segment. For this reason it would pass through the two points mentioned in the hypothesis; but "two points determine the position of a straight line."

Therefore the line which by hypothesis passed through two points that were equally distant from the extremities of a given segment will coincide with and will be the perpendicular bisector of the given segment.

Q. E. D.

NOTE. The perpendicular bisector of the segment of a line is said to be the locus (place) of the point, when moving so that its distances from the segment ends shall always be equal to each other.

Or it may be described as the locus of a point moving so that the ratio of its distances from two fixed points always equals unity.

A CIRCLE.

21. Definitions. The locus of a moving point, the distance of which from a given point is fixed, is called a circumference.

Ө

FIG. 23.

B

If the line AB rotate about A as a pivot, any point in the line AB, as the point B, in a complete rotation, will remain at a fixed distance from A, and will generate a circumference.

Anything less than a complete rotation will generate an arc. A half rotation will generate a semi-circumference; and a quarter rotation will generate a quadrant.

The point A is called the centre; and the distance of B from A is called the radius.

A point nearer to A than B is, will generate a circumference that will lie entirely within the circumference generated by the point B; and a point at a greater distance from A than B is, will generate a circumference, lying entirely outside of the circumference generated by B.

When AB rotates about A as a pivot, the change of direction makes an angle at A; each point of AB generates a circumference, and the whole line generates the surface of the plane.

The figure bounded by a circumference is called a circle.

CONGRUENT TRIANGLES.

22. Definition. When a geometric figure may be substituted for another and may be made to occupy exactly

the same space which the other did, the figures are said

to be congruent.

THEOREM. If two triangles have the three sides of the one equal to the three sides of the other, each to each, they are congruent.

If the two triangles (p) and (q) having:

be so placed that a pair of equal sides shall lie together, and the triangles be not superimposed, we shall have them placed as in figure (r).

By the terms of the theorem, B and C are two points equally distant from A and E. Draw the auxiliary line AE.

By $ 20, Ex. 2, the line CB will be the perpendicular bisector of the segment AE.

BD

(r)

E

K

FIG. 25.

If the figures to the right of the line CK be revolved on CK as an axis, the point E will fall at A; the segments ED and AB will coincide; the same will be true of EF and AC; DF and BC will remain in coincidence; and all the parts of one triangle (perimeter, angles, and surface) will coincide with the parts of the other.

Q. E. D.

23. THEOREM. If two triangles have two angles and an included side of one equal to two angles and an included side of the other, they are congruent.

The student will make the necessary constructions, and show that the triangles may be placed so as to coincide.