So Propositions 11 and 12 may be shewn to hold for rectilineal figures.

Also since the areas of any figures whatever are equal to the areas of certain rectilineal figures; .. by (v. 2) the above propositions hold for the areas of any figures whatever.

Also all the preceding theorems may be similarly extended to other magnitudes as solids and angles. For solids may be shewn to be proportional to certain straight lines and angles to certain arcs. Moreover

PROPOSITION XIII.

If A, B, H be areas and c, d, k straight lines, such that

BOOK VI.

COMPARISON OF STRAIGHT LINES.

PROPOSITION I.

If a straight line be drawn parallel to one of the sides of a triangle it shall cut the other sides, or these produced, proportionally.

Let DE be drawn | to BC, one of the sides of the ▲ ABC. Then AD: DB as AE: EC.

Join BE, DC.

Then. DE is | to BC; .. ▲ BDE = ▲ CDE; (1.32)

.. ▲ADE: ABDE as ▲ ADE:

but A ADE: ABDE as AD: DB
and ▲ ADE: ▲ CDE as AE: EC

.. AD : DB as AE: EC.

▲ CDE; (v. 2)

}

(v. 1)

(v. 1)

(v. 3)

Conversely,

If the sides AB, AC of the ABC, or these sides produced, be cut proportionally in D, E,

then shall the straight line joining D, E be | to BC.

The same construction being made,

·.· AD : DB as AE: EC,

.. AADE: ABDE as ▲ADE: ▲ CDE;

(v. 3)

.. ABDE is = ▲ CDE;

(V. 4)

(I. 33)

.. DE is to BC.

COR. Similarly it may be proved that,
if DE be drawn || to BC,

then AD AB as AE: AC

and conversely.

PROPOSITION II.

The line bisecting the vertical angle of a triangle divides the base into two segments which have the same ratio to one another which the sides of the triangle have.

R

Let the vertical ▲ BẠC of the ▲ ABC be bisected by

AQ;

then BQ QC as BA AC.

:

Through C draw CR || to QA meeting BA produced

in R.

Now Q4 is to CR; .. BAQ is = LARC,

that

Converse Theorem.

If the side BC of ABC be divided in the point Q so

Through C draw CR || to QA meeting BA produced

.. LACR = L CAQ and ▲ ARC= L BAQ; (1. 21)

..L CAQL BAQ.