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PROBLEM H.

To divide a straight line into any given number of equal parts.

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Let it be required to divide the straight line AB into five equal parts.

Through A draw any other straight line AC:
from it cut off any part AD,

and also DE, EF, FG, GH each equal AD;

join HB, and through D, E, F, G draw lines | to HB

and cutting AB in X, Y, Z, W;

then shall AB be divided into five equal parts.

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Similarly YZ, &c. are each = AX,

and ... AB is divided into five equal parts.

(I. 26)

In like manner may a straight line be divided into any

given number of equal parts.

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Let AB be the given straight line.

It is required to describe a square on AB.

Through A draw AC 1 to AB,

from AC cut off AH = AB. Through H draw HK || to AB,

and through B draw BK || to AH.

Then ABKH shall be the square required.

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(1. 27).

.. all the 4s of AK are rights;

.. the AK is both equilateral and rectangular,

and is.. a square.

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Let the s ABCD, EBCF be on the same base BC,

and between the same parallels AF, BC.

Then shall the ABCD be the EBCF.

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Also AB = DC and BE = CF;

.. AABE = ▲ DCF

Now from the quadrilateral ABCF take the ▲ ABE, and from the same figure take the ▲ DCF; .. the remainder, the EBCF, is = the ABCD.

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If the sides opposite the base terminate in the same. point D,

then each of the □s is double of the ▲ DBC; (1. 26)

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DEFINITIONS.

The greatest perpendicular which can be let fall from a point in the boundary of a figure on the base or base produced is called the altitude of the figure.

The perpendicular let fall on the base or base produced of a ▲ from the vertex is called the altitude of the a.

Perpendiculars let fall on the base or base produced of a from points in the opposite side are equal to one another, and any one of them is spoken of as the altitude of the .

If any one of the sides of a rectangle be taken as the base, then either of the adjacent sides is equal to the altitude.

... rectangles, whose bases and altitudes are equal, are themselves equal in all respects.

(I. 29)

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i.e. a is the rectangle having the same base and altitude; but all rectangles having equal bases and altitudes are equal.

..a is any rectangle of equal base and altitude.

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... all parallelograms, having equal bases and altitudes, are equal.

Thus parallelograms upon equal bases and between the same parallels are equal.

PROPOSITION XXXII.

If a parallelogram and a triangle be on the same base and between the same parallels, the parallelogram is double of the triangle.

Let

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the EBCD and the ▲ ABC be on the same

base BC, and between the same parallels.

Then shall

EBCD be double of a ABC.

Δ

(1. 26)

If the vertex A of the ▲ ABC is at D or E the proposition is evident. But if not through A draw AX || to EB or DC, meeting the base or base produced in X.

Then EX is a ☐

and is ... double of ▲ ABX, (1. 26) and is .. double of ▲ ACX, (1. 26) ..□ EC is double of ▲ ABC.

also DX is a

COR. A triangle is half of the rectangle on the same base and between the same parallels.

i.e. a triangle is half the rectangle having the same base and altitude.

.. a triangle is half of any rectangle of equal base and altitude.

.. all triangles having equal bases and altitudes are equal.

Thus triangles on the same or equal bases and between the same parallels are equal.

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