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PROPOSITION XI. 270. Equimultiples of two quantities are in the same ratio as the quantities themselves.

Let a and 6 be any two quantities.
We are to prove ma : mb :: a : b.
Now

a

a

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PROPOSITION XII. 271. If two quantities be increased or diminished by like parts of each, the results will be in the same ratio as the quantities themselves.

Let a and 6 be any two quantities.
We are to prove a + 2a : 6 + 2.0 :: a : b.
In the proportion,

ma : mb :: a : 6,
substitute for m, 1 + ?

Then

(1 + $) a : (1 + %) 0 :: 2 : 0,

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or
+ Pa : 6 + PO :: a : 6.

Q. E. D.
272. DEF. Euclid's test of a proportion is as follows:-

“The first of four magnitudes is said to have the same ratio. to the second which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth;

“If the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth ; or,

“If the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth ; or,

“ If the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth.”

PROPOSITION XIII.

273. If four quantities be proportional according to the algebraical definition, they will also be proportional according to the geometrical definition. Let a, b, c, d be proportional according to the alge

braical definition; that

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Now from the nature of fractions,
if ma be less than nb, mc will also be less than nd;
if ma be equal to nb, mc will also be equal to nd;.
if ma be greater than rb, mc will also be greater than nd.

i. a, b, c, d are proportionals according to the geometrical definition.

Q. E. D.

EXERCISES.

1. Show that the straight line which bisects the external vertical angle of an isosceles triangle is parallel to the base.

2. A straight line is drawn terminated by two parallel straight lines ; through its middle point any straight line is drawn and terminated by the parallel straight lines. Show that the second straight line is bisected at the middle point of the first.

3. Show that the angle between the bisector of the angle A of the triangle A B C and the perpendicular let fall from A on BC is equal to one-half the difference between the angles B and C.

4. In any right triangle show that the straight line drawn from the vertex of the right angle to the middle of the hypotenuse is equal to one-half the hypotenuse.

5. Two tangents are drawn to a circle at opposite extremities of a diameter, and cut off from a third tangent a portion A B. If C be the centre of the circle, show that AC B is a right angle.

6. Show that the sum of the three perpendiculars from any point within an equilateral triangle to the sides is equal to the altitude of the triangle.

7. Show that the least chord which can be drawn through a given point within a circle is perpendicular to the diameter drawn through the point.

8. Show that the angle contained by two tangents at the extremities of a chord is twice the angle contained by the chord and the diameter drawn from either extremity of the chord.

9. If a circle can be inscribed in a quadrilateral; show that the sum of two opposite sides of the quadrilateral is equal to the sum of the other two sides.

10. If the sum of two opposite sides of a quadrilateral be equal to the sum of the other two sides ; show that a circle can be inscribed in the quadrilateral.

ON PROPORTIONAL LINES.

PROPOSITION I. THEOREM. 274. If a series of parallels intersecting any two séraight lines intercept equal parts on one of these lines, they will intercept equal parts on the other also.

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Let the series of parallels A A', B B', CC", D D', E E',

intercept on H' K' equal parts A' B', B'C', C'D', etc.
We are to prove
they intercept on H K equal parts A B, BC, C D, etc.
At points A and B draw A m and B n || to H' K'.
A m = A' B',

§ 135
(parallels comprehended between parallels are equal).
Bn= B'C',

§ 135 1. Am = B n. In the A B Am and C Bn,

ZA = B, (having their sides respectively II and lying in the same direction from

the vertices). Zm= Zn,

$ 77 A m = B 1,

.. A B Am = AC Bn, (having a side and two adj. As of the one equal respectively to a side and

two adj. És of the other).

.. A B = BC,

(being homologous sides of equal A).
In like manner we may prove BC=CD, etc.

Q. E. D.

§ 77

and

§ 107

PROPOSITION II. THEOREM. 275. If a line be drawn through two sides of a triangle parallel to the third side, it divides those sides proportionally.

Fig. 1.

Fig. 2.
In the triangle A B C let EF be drawn parallel to BC.

EB FC
We are to prove i

A E AF
CASE I. – When A E and EB (Fig. 1) are commensurable.
Find a common measure of A E and E B, namely B m.
Suppose Bm to be contained in B E three times,

and in A E five times.
Then

EB _ 3

A E 5° At the several points of division on B E and A E draw straight lines II to B C.

These lines will divide A C into eight equal parts, of which FC will contain three, and A F will contain five, $ 274 (if parallels intersecting any two straight lines intercept equal parts on one of these lines, they will intercept equal parts on the other also).

· FC 3

er eo er! Co

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A E = 5

_FC

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Ax. 1

AF

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