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Let ABC, DEF be two isosceles As, having the sides

AB, AC of the one

other, and also

=

=

to each

A

to the sides

Also let

DE, DF of the other.

the included / BAC = included

LEDF; then the 4S B and C

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respectively
(Prop. IV.) But since AB = DF,
and AC DE, if the A DEF be B
turned over, and applied to the ▲
DF may coincide with AB, and DE
will coincide with B, and the
Prop. IV.; .. the

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ABC, so that the with AC, the F with c, as in

B and c are each = each other.-Q. E. D.

E

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In forming roofs, brackets, tripods, and in various operations of Carpentry, this principle is constantly applicable.

PROP. VI. THEOR.

GEN. ENUN. If two angles of a triangle be equal to one another, the sides also which subtend, or are opposite to, the equal angles, shall be equal to one another.

PART. ENUN.-Let ABC be a▲, of which the ABC ACB, then the side

AB AC.

CONST.-For if AB be not = AC, one must be > the other.

D

A

Suppose AB to be the >, and that from it is cut off DB = AC, B the <. (Prop. III.) Join DC (Post. 1). DEMONST.-1. Show the absurdity resulting from the above supposition; viz., that ▲ DBC would = ▲ ABC.

Since in AS DBC, ACB, the side DB = AC, and BC is common to both, ... the two DB, BC = the two AC, CB, each to each; and the DBC = ACB (Hyp.); .. base DC = base AB, and the area of DBC area of ▲ ACB: i.e. the <= >, which is absurd.

is

2. Hence AB is not unequal to AC: i. e. it = to it.

Wherefore, if two s of a triangle, &c.— Q. E. D.

COR.-Hence every equiangular

equilateral.

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To prove this corollary, let the 4s of the ▲ ABC be all equal to one another :

then because the

ABC

AB AC; and because the

=

LACB,..
ABC =

BAC, .. BC = AC; .. AB = AC = BC:
i. e. the ABC is equilateral.-Q. E. D. B

is also

C

The Sixth Proposition is proved by what is called a reductio ad absurdum; that is, by assuming an opposite conclusion to that which is enunciated, and showing it to be inconsistent with the hypothesis. It is the converse of the former part of the fifth; and though in this case, as in others, Euclid has only demonstrated what was necessary to the proof of subsequent propositions, the second part is also convertible. It is not, however, to be supposed that, in taking the conclusion for an hypothesis, the original hypothesis becomes a necessary conclusion. Instances will be noticed in which this is not the case. In the meantime, the converse of the latter part of the Fifth Proposition may be thus demonstrated.

PROP. C. THEOR.

GEN. ENUN.-If, when two sides of a triangle are produced, the angles below the base are equal to each other, the two sides are also equal to one another.

PART. ENUN.-Let ABC be a A; and the sides AB, AC being produced to D and E, let the DBC = LECB; then the side AB shall the side AC.

=

CONST.-In BD take any pt. F, and from CE cut off CG = BF. (Prop. III.) Join BG, CF, FG.

DEMONST.-1. Prove that the BCF
LCBG.

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Because BF CG, and BC is common to the ▲ FBC, GCB, .. the two sides FB, BC = GC, CB, each to each; and the contained / FBC = Z GCB (Hyp.); ... the base FC = base GB, and the BCF (Prop. IV.)

2. Prove that

But BCG =

GCF = LFBG.

= LCBG.

CBF (Hyp.); .. by subtraction,

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Again, in ▲ FBG, GCF, the sides FB, BG = GC, CF, each to each; and the FBG =

Z CGF. (Prop. IV.)

4. Prove AF = AG.

GCF;.. BFG =

Now these are the 4s at the base of the ▲ AFG; .. AF AG. (Prop. VI.)

5. Prove AB = AC.

But BF CG (Const.); .. by subtraction, Ab = ac. Wherefore, if, when two sides of a A &c.-Q. E. D.

PROP. VII. THEOR.

GEN. ENUN.-Upon the same base, and on the same side of it, there cannot be two triangles that have their sides, which are terminated in one extremity of the base, equal to one another, and likewise those which are terminated in the other extremity.

ADB

PART. ENUN.-Let the two As ACB, S be upon the same base AB, and the same side of it, and let the sides CA, DA, terminated in A, be; then the sides CB, DB, terminated in B, cannot be =.

CONST.-Join CD.

DEMONST. (By reduct. ad absurdum.—If possible, suppose CB = DB.

Case 1. When the vertex of each is without the other A.

1. Prove that . BDC is >

BCD.

Because AC = Ad (Hyp.),

▲ ACD = ▲ Adc. (Prop. V.) But ACD > < BCD (Ax. 9);

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D

B

.. < ADC > < BCD. Much more is BDC

> BCD.

2. Prove that, also, on the above supposition,

BDC BCD.

Now if BC = BD, .'. ≤ BDC =

(Prop. V.)

LBCD.

3. But it has been shown that BDC is > < BCD, which is impossible.

Case 2. When one of the vertices, as D, is within the other ▲ ACB.

CONST.-Produce AC, AD to E and F, and

join CD.

DEMONST. (Ad abs.)

BDC is > < BCD.

1. Prove that

Because AC = AD (Hyp.) in acd, .'.

A

ACD,

D

<S ECD, FDC, upon the other side of the base

BCD

CD, are equal. (Prop. V.)
But the ECD >
(Ax. 9); .. ▲ FDC > <
BCD. Much more is

BDCBCD.

2. Also that they are =.

BCD, BC = BD, .'.

Now, if in
BCD. (Prop. V.)

E

F

D

B

BDC =

3. But it has been shown that BDC is > < BCD, which is impossible.

The third case, in which the vertex of one A is upon a side of the other, needs no demonstration. (See figure to Prop. VI.)

Wherefore, upon the same base, and on the same side of it, &c.-Q. E. D.

In the same way it may be proved that, if the sides BC, BD are equal, then AC, AD cannot be equal; and the student will do well to work out the demonstration. The proof includes what is called a demonstration á fortiori.

C

Euclid has established this theorem for the sole purpose of proving the next, to which it may therefore be considered as a Lemma; i. e. a proposition which is merely preparatory to the demonstration of another. The limitation that the two triangles should be upon the same side of the base is manifestly essential; for upon A the other side of the base AB, a ▲ ABD may be constructed in every respect similar and = to the ▲ ABC. (Prop. IV.)

D

B

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