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Note.-Propositions I, III, and V hold if the triangles are situated thus:

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having the sides AB, BC, CA corresponding to DE, EF, FD.

In propositions I and III it will be necessary, after taking up ▲ ABC, to reverse it before applying it to ▲ DEF.

In proposition V, the ▲ ABC must be applied without being reversed.

The demonstrations will be the same as before.

DEFINITIONS.

An equilateral triangle is one which has all its sides equal.

A circle is a plane figure contained by one line called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal. That point is called the centre of the circle, and those straight lines radii.

It is taken for granted that a circle can be described having any centre and radius equal to any given straight line.

PROBLEM A.

Describe an equilateral triangle upon a given straight line.

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Let AB be the given straight line.

It is required to describe an equilateral triangle upon AB.

With A as centre, and radius AB, describe the circle BCD, and with B as centre and radius BA describe the circle ACE.

and

Let these circles intersect in C; join CA, CB. Then ABC shall be the equilateral triangle required.

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.. ABC is an equilateral triangle, and it has been described upon AB.

Q. E. F.

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With centre A describe a circle cutting AD, AE in the points B and C.

With centres B and C describe equal circles cutting one another in F

Join AF; then ▲ DAE shall be bisected by AF.

Then

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Join BF, CF.

in the As ABF, ACF,

BA is CA, BF= CF, and AF is common ;

.. the As ABF, ACF are equal in all respects,

and

..the BAF is the CAF,

=

... DAE has been bisected by AF.

(I. 5)

Q. E. F.

PROBLEM C.

To bisect a given straight line.

Let AB be the given straight line.

It is required to bisect it.

With centres A and B describe two circles having equal radii intersecting in C, D.

Join CD cutting AB in E;

then AB shall be bisected in E.

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and

AC, CD, DA are respectively = BC, CD, DB,
..AS ACD, BCD are equal in all respects,

..LACD is = L BCD.

Hence in the As ACE, BCE,

(1.5)

AC, CE and the included ACE are respectively = BC, CE and the included ▲ BCE;

...

AS ACE, BCE are equal in all respects,

and .. AE is = BE;

(1. 1)

.... AB has been bisected in E.

Q. E. F.

RIGHT ANGLES.

DEFINITION.

When one straight line, standing on another straight line, makes the adjacent angles equal to one another, each of them is called a right angle; and the straight line which stands upon the other is said to be perpendicular to it.

PROPOSITION VI.

If two straight lines cut one another the opposite angles shall be equal.

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Let two intersecting straight lines form the four angles A, B, H, K.

Then shall the opposite angles A, B be one another.

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For if the figure were taken up, reversed, and placed so that each of the arms of H might fall along the former position of the other arm; then each of these lines produced would fall along the former position of the other.

Thus the arms of A would fall along the former positions of the arms of B ;

.. the angles A, B are equal to one another."

Similarly the angles H, K may be proved equal.

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