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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.


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.


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.


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.


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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,


..the BAF is the CAF,


... DAE has been bisected by AF.

(I. 5)

Q. E. F.


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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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,


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.



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.


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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