108. The bisector of an angle is the locus of all points within the angle equally distant from its sides.

TO PROVE

the angle AOB and its bisector OC.

OC is the locus of all points equally distant from

AO and BO.

It is necessary to prove :

I. That every point in OC satisfies the condition of being equally distant from AO and BO.

II. That any point without OC is unequally distant from AO and BO.

I. (Fig. 1.) Take P, any point in OC. perpendicular to OB and OA.

In the right triangles POM and PON

Hence

Therefore

OP=OP,

angle POM-angle PON.

Draw PM and PN

triangle POM=triangle PON.

[Having the hypotenuse and an acute angle respectively equal.]

PM-PN.

[Being homologous sides of equal triangles.]

Iden.

Hyp.

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II. (Fig. 2.) Take X, any point within the angle, but not in OC. Draw XM and XN perpendicular to OB and OA.

One of these lines, as XM, must cut OC in some point, as D. Draw DK perpendicular to OA and join XK.

OUTLINE PROOF: XN< XK < XD + DK = XD+DM= XM; hence XN< XM.

109. COR. The three bisectors of the angles of a triangle meet in a common point.

A

B

Hint.-Show that the intersection of two of the lines must lie on the third as in Proposition XXXIII.

110. Remark.-This point is the centre of the triangle so far as its sides are concerned-that is, it is equally distant from the sides.

111. Exercise.--What is the locus of all points equally distant from two intersecting straight lines?

112. Exercise.-What is the locus of all points at a given distance from a fixed straight line of indefinite length?

113. Exercise.--What is the locus of all points at a given distance from a given line of a definite length?

PARALLELOGRAMS

114. Defs.—A parallelogram is a quadrilateral whose opposite sides are parallel.

A rhombus is a quadrilateral whose sides are all equal and whose angles are oblique.

A rectangle is a parallelogram whose angles are all right angles.

A square is a rectangle whose sides are all equal.

115. Def.-A diagonal of a quadrilateral is a straight line joining opposite vertices.

116. A diagonal of a parallelogram divides it into two equal triangles.

GIVEN

the parallelogram ABCD and the diagonal AC.

TO PROVE that the triangles ABC and ACD are equal.

Q. E. D.

[Having a side and two adjacent angles in each respectively equal.]

117. COR. I. In any parallelogram the opposite sides and angles are equal.

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FIG. I

I

FIG. 2

118. COR. II. Parallels comprehended between parallels are equal. [Fig. 1.]

119. COR. III. Parallels are everywhere equally distant. [Fig. 2.]

Hint.-Apply §§ 33, 36, 118.

PROPOSITION XXXVI.

THEOREM

120. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.

GIVEN―any quadrilateral having its opposite sides equal, viz.: ABCD, and AD=BC.

[When two straight lines (BC and AD) are cut by a third straight line (AC) making the alternate-interior angles (x and x') equal, the straight lines are parallel.]

In like manner, using y and y', we may prove AB parallel to CD.

Therefore ABCD, having its opposite sides parallel, is a parallelogram.

Q. E. D.

121. A “parallel ruler" is formed by two rulers (MN and M'N'), usually of wood pivoted to two metal strips (AA' and BB'), under the following conditions:

(1.) The distances on the rulers between pivots are equal: i. e., AB A'B'.

(2.) The distances on the strips between pivots are equal; i. e., AA'=BB'.

(3.) In each ruler the edge is parallel to the line of pivots; i. e., AB is parallel to MN, and A'B' is parallel to M'N'.

122. Exercise.--Prove: (1.) the quadrilateral whose vertices are the pivots (i. e., the figure ABB'A') is always a parallelogram, whether the ruler be closed or opened.

(2.) The edges of the rulers are always parallel (i. e., MN and M'N' are parallel).