Lawrence S. (Lawrence Sluter) Benson.

# Geometry : the elements of Euclid and Legendre simplified and arranged to exclude from geomtrical reasoning the reductio ad absurdum : with the elements of plane and spherical trigonometry, and exercises in elementary geometry and trigonometry / /c By Lawrence S. Benson online

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BOOK IV,] EUCLID AND LEGENDRE. 101

Prop. IY. â€” Theor. â€” If four numbers be proportionals, they
are also proportionals when taken alternately.

If A : B : : C : D ; tlien, alternately, A : C : : B : D.

For (IV, 2, cor. 1) AD=;:BC ; whence, by dividing by C and

A B

D we get 7s =yt; or (IV. 2, scho. 2) A : C : : B : D, There-

fore, if four numbers, etc,

Scho. When the first and second terms are not of the same
kind as the third and fourth, the terms can not be taken altern-
ately, as ratios would thus be instituted between heterogene-
ous macrnitudes.

o

Prop, V. â€” Theor. â€” Ejual numbers have the same ratio to
the same number ^ and the same has the same ratio to equal
numbers.

Let A and B be equal numbers, and C a third ; then A : C : :
B : C, and C : A : : C : B.

A B

For, A and B being equal, the fractions -^ and y, are also

equal, or, which is the same, A : C : : B : C ; and, by inversion,
(IV. 3) C : A : : C : B, Therefore, equal numbers, etc.

Prop. VI. â€” Theor. â€” JVmnbers which have the same ratio
to the same nwnber are equal ; and those to which the same
has the same ratio are equal.

If A : C : : B : C, or if C : A : : C : B, A is equal to B.

A B

For, since â€”7=., ,by multiplying by C we get A = B.

The proof of the second part is the same as this, since, by in-
version (IV. 3), the second analogy becomes the same as the
first. Therefore, numbers, etc.

Prop. VII. â€” Theor. â€” Ratios that are equal to the sameratio

are equal to one another.

If A : B : : C : D, and E : F : : C : D ; then A : B : : E : F.

-. . AC , E C . â€ž ,^ . A E . ^

ror, smce p=-r., and = - ,therciore (1. ax. 1) p=p, ; that ,

is, (IV. 2, scho. 2) A : B : : E : F. Therefore, ratios, etc.

102 THE ELEMENTS OF [bOOK 17.

Pkop. Vin. â€” Theok. â€” Ofnumhers which are proportionals^
as any one of the antecedents is to its consequent^ so are all the
antecedents taken together to all the conseqzients.

If A : B : : C : D : : E : F ; then A : B : : A+C+E : B+D
+F.

ACE

Since :^==: ==-7, put each fraction equal to q, and multiply

hj the denominators ; then A=zBq, C=:Dq, and E^Fg-.
Hence, by addition, A+C-}-E = (B + D-f F)^'/ and by dividing

by B+D+F, we get ^=WTWZw' ^^^ ?â€” g 5 ^^^ there-

foi'e 4"= t"!"^tS > or A : B : : A+C + E : B+D + F. There-
a Â±>+jj+i^

fore, etc.

Prop. IX. â€” Theor. â€” 3fagmtudes have the same ratio to one
another that their like multiples have.

Let A and B be two magnitudes ; then, n being a whole
number, A : B : : nA : nB.

For :jj=â€” ^, or A : B : : nA : wB. Therefore, magnitudes,
etc.

Prop. X. â€” ^Theor. â€” If four mimbers be prop>ortionals ; then
(1) hy composition, the sum of the first and seco7\d is to the
second, as the sum of the third and fourth to the fourth ; and
(2), 5y division, the excess of the first above the second is to the
second, as the excess of the third above the fourth is to the
fourth.

If A : B : : C : D ; then, by composition, A+B : B : : C+D:
D; and by division, Aâ€” B : B : : C â€” D : D.

1. Since (hyp.) vi = i-:i and since -rr = ^: add the latter frac-

^ "^ ^ ' B D B D

tions to the former, each to each, and there results â€” rr~ =

5^, or A+B : B : : C +D : D.

2. By subtracting the latter pair of the same fractions from

the former, each from each, we obtain â€” =â€”- = â€”=-â€” ; or A â€”

B i)

B : B : : Câ€” D : D. If, therefore, etc.

BOOK rV.] EUCLID AND LEGENDRE. 103

Cor. By dividing the fractions which were found above by-
addition, by those which were found by subtraction, we get

:^^=^^; or (IV. 2, scho. 2) A+B : A-B : : C+D : C-

D; that is, if four numbers be proportional, the sura of the fii-st
and second terms is to their difference, as the sum of the third
and fourth terms is to their difference. It is evident, that if B
be greater than A, the analogy would become B+A : Bâ€” A : :
D + C : Dâ€” C.

Prop. XI. â€” Theoe. â€” If foumumhers he proportional ; then,,
by conversion, the first is to its excess abuve the second^ as the
third to its excess above the fourth.

If A : B : : C : D ; then, by conversion, A : A â€” B : : C :
Câ€” D.

x^ â€˘ n ;,- .BD -. AC.

Jbor, smce (hyp. and mver.) â€”=^^ and smce â€” = -; take

the former fractions from the latter, each from each, and there

remains â€” -r â€” = â€” ^ â€” , or (by inver.) A:A â€” B:: C:C â€” D.

Therefore, if four numbers, etc.

Prop. XII. â€” Theor. â€” If there be members forming two or
more analogies which have common consequents^ the sum of
all the first antecedents is to their common consequent, as the
sum of all the other antecedents is to their common consequent.

If A : B : : C : D, and E : B : : F : D ; then A + E : .B : :
C+F : D.

For (hyp.) -j5=â€” ,and p = Tx; and hence, by addition, â€” -^â€”
=5^, or A+E : B : : C + F : D. If, therefore, etc.

Prop. XIII. â€” Theor. â€” If there be three or more niiyn^ers,
and OS many others, which, taken two and two in order, have
the sa7ne ratio ; then, ex aequo, the first has to the last of the
first rank the sayne ratio that the first has to the last of the
second rank.

If the two ranks of numbers. A, B, C, D, and E, F, G, II, be

104 THE ELKMENT8 OF [bOOK IV.

Buch that A : B : : E : F, B : C : : F : G, and C : D : : G : 11 ;
then A : D : : E : H.

A E B F C C

For, since (hyp.) - =-, - = - and ^ = - ; by multiplying

together the first, third, and fifth fractions, and the second,
fourth, and sixth, we obtain jTnn^KnxT 5 ^r, by dividing the

terras of the first of these fractions by BC, and those of the

A E

second by FG, j)=g, or A : D : : E : II. Therefore, if there

be three, etc.

This proposition might also be enunciated tlius : If there be
numbers forming two or more analogies, such that the conse-
quents in each are the antecedents in the one immediately fol-
lowing it, an analogy will be obtained by taking the antece-
dents of the first analogy and the consequents to the last for
its antecedents and consequents.

Prop. XIV. â€” Theok. â€” If there be three or more numbers,
and as many others, which, taken two and two in a cross order,
have the same ratio ; then, ex tequo inversely, the first has to
the last of the first rank the same ratio uhlch the first has to
the last of the second ranTc.

If the two ranks of numbers. A, B, C, D, and E, F, G, H,be
such that A : B : : G : H, B : C : : F : G, and C : D : : E : F ;
then, ex mqrco inversely, A : D : : E : H.

A C B F P F

For, since (hyp.) -^=^, ^=^,and ^ =^, by multiplying to-
gether the fractions as in the preceding proposition, we get

ABC GFE , ,,..,., . , . ^

^pY)=TT7TT^j whence, by dividing the terms of the first of

these fractions by BC, and those of the second by GF, we ob-

A E
tain jx = fT, or A : D : : E : H. If, therefore, etc.

This proposition may also be enunciated thus: If there be
numbers forming two or more analogies, such that the means
of each are the extremes of the one immediatelj' following it,
another analogy may be obtained by taking the extremes of
the first analogy and the means of the last for its extremes and
means.

BOOK IV.] EUCLID AND LEGENDKE. 105

Prop. XV. â€” Theor. â€” If there he numbers forming tico or
more analogies^ the products of their corresponding terms are
proportionals.

If A : B : : C : D, E : F : : G : II, and K : L : : M : N ; then
AEK : BFL : : CGM : DUN.

^ ,, - A C E G TC M ^ . .

For (hyp.) j^ =0, ^=|j>' aucl jy = ^; and taking the pro-
ducts of the corresponduig terms of these fractions, we obtain

A FTC OC \f

^^=^^., or AEK : BFL :: CGM : DHN. Therefore, if

Bi'L DHIS'

there be numbers, etc.

Cor. 1. Hence, if there be two analoj^ies consistinsr of the
sane terms, A, B, C, D, we have A" : B' : : C^ ; D^ ; if there be
three, we have A' : B' : : C* : D', etc. ; and it thus appears,
that like powers of proportional numbers are themselves pro-
portional.

Cor. 2. Like roots of proportional numbers are proportional.
Thus, if A : B : : C : D, let 4/ A : VB : : VC : VE. Then, by
the preceding corollary, A : B : : C : E. But (hyp.) A : B : :
C : D ; and therefore (IV. 7) C : E : : C : D, and (IV. 6) E =
D, and consequently V : A -/ B : : VC : VE, or VD.

Prob. XVI. â€” Theor. â€” The sum of the greatest and least of
four p. oportional members is greater than the sum of the other
two.

If A : B : : C : D, and if A be the greatest, and therefore
(IV. 2, cor. 2) D the least ; A and D are together greater than
B and C.

For (by conversion) A : A â€” B :: C : C â€” D, and, altern-
ately, A : C : : Aâ€” B : Câ€” D. But (hyp.) A>C, and there-
fore' (IV. 2, cor. 2) Aâ€” B > C â€” D. To each of these add B ;
then A>B + Câ€” D. Add again, D; then, A-hD>B4-C.
Therefore, etc.

Cor. Hence the mean of three proportional numbers is less
than half the sum of the extremes.

Prob. XVII. â€” Theor. â€” In numbers which are continual
proportiojials, the first is to the third as the second power of
the first to the second power of the second y the first to the

106 THE ELEMENTS OF [bOOK IV.

fourth as the third power of the first to the third power of the
second ; the first to the fifth as the fourth power of the first to
the fourth power of the second ; and so on.
1 If A, Vy, C, D, E, etc., be continual proportionals; A : C : :
A^ B^ ; A : D : : A^ : B'; A : E : : A^ : B', etc.

For, since (IV. def. 8) A : B : : B : C, and since A : B : :
A : B, we have (IV. 15) A^ : B^ : : AB : BC, or, dividing the
third and fourth terms by B, A* : B' : : A : C.
Again : since A* : B' : : A : C,

and A : B : : C : D we have (IV. 15)
A' : B' : : AC : CD, or dividing the third and fourth terms
by C, A^ : B^ : : A : D ; and so on, as far as we please. There-
foi'e, etc.

Cor. Hence (IV. defs. 11 and 12) the ratio which is duplicate
of that of any two numbers, is the same as the ratio of their
squares ; that which is triplicate of their ratio, the same as the
ratio of their cubes, etc.

Prop, XVIII. â€” ^Theor. â€” A ratio xcMch is compounded of
other ratios, is the same as the ratio of the products of their
homologous terms.

Let the ratio of A to D be compounded of the ratios of A to
B, B to C, and C to D ; the ratio of A to D is tlie same as that
of ABC, the product of the antecedents, to BCD, the product
of the consequents.

For, since A : D : : A : D, multiply the terms of the sec-
ond ratio by BC ; then (IV. 9) A : D : : ABC : BCD. There-
fore, etc.

Prop. XIX. â€” Theor. â€” In numbers which are continual
pro2)ortionalsj the difference of the first and second is to the
first, as the difference of the first and last is to the sum of all
t/ie terms excejjt the last.

If A, B, C, D, E be continual proportionals, A â€” B : A : :
Aâ€” E: A + B + C + D.

For, since (hyp.) A : B : : B : C : : C : D : : D : E, we
have (IV. 8) A : B

(conv.) A : A â€” B :
Aâ€” B : A :: Aâ€” E

A+B+C+D : B + C + D + E. Hence
A + B + C + D : Aâ€” E; and (inver.)
A+B + C + I).

BOOK IV.] EUCLID AND LEGENDRE. 107

It is evident that if A were the least term, and E the great-
est, we should get in a similar manner, B^ â€” A : A : : E â€” A :
A-f-B-f-C+D. Therefore, in numbers, etc.

Cot. If the series be an infinite decreasing one, the last term
will vanish, and if S be put to denote the sum of the series, the
analogy will become A â€” B : A : : A : S ; and this, if rA be
put instead of B, and the first and second terms be divided by
A, will be changed into 1 â€” r : 1 : : A : S. The number r is
called the common ratio, or common m,ultiplier, of the series,
as by multiplying any term by it, the succeeding one is ob-
tained.

END OP BOOK FOURTH.

BOOK FIFTH.

DEFINITIONS.

1. Similar rectilineal figures are those which have their
several angles equal, each to each, and the sides about the
equal angles proportionals.

2. Two magnitudes are said to be reciprocally proportional
to two others, when one of the first pair is to one of the second,
as the remainincc one of the second is to the remaining one of
the first.

3. A straight line is said to be cut in extreme andmean ratio^
â– when the whole is to one of the segments as that segment is to
the other.

4. The altitude of any figure is the straight line drawn from
its vertex perpendicular to its base.

5. A sti-aight line is said to be cut harmonically, Avhen it is
divided into three segments, such that the whole line is to one
of the extreme segments as the other extreme segment is to the
middle one.

PROPOSITIONS.

Prop. I. â€” Theor. â€” Triangles and parallelograms of the
same altitude are one to another as their bases. ^

Let the triangles ABC, ACD, and the parallelograms EC,
CF have the same altitude, viz., the perpendicular drawn from
the point A to BD ; then, as the base BC is to the base CD, so
is the triangle ARC to the triangle ACD, and the parallelo-
gram EC to the parallelogram CF.

Produce BD both ways, and take any number of straight
lines BG, Gil, each equal to BC ; and any number DK, KL,
each equal to CD; and join AG, AH, AK, AL. Then, because
CB, BG, Gil are all equal, the triangles ABC, AGB, AUG are
(L 15, cor.) all equal. Therefore, whatever multiple the base

BOOK v.]

EUCLID AND LEGENDRE.

loa

HC is of BC, the same multiple is the triangle AUG of ABC.
For the same reason, wliatver multiple LC is of CD, the same
multiple is the triangle ALC
of ADC. Also, if the'base HC -
be equal to CL, the triangle
AHC is equal (I, 15, cor.) to
ALC ; and if the base HC be
greater than CL, likewise (L
15, cor. 6) the ti-iangle AHC is

greater than ALC; and if less, less. Therefore, since there are
four magnitudes, viz., the two bases, BC, CD, and the two tri-
angles ABC, ACD; and of the base BC, and the triangle ABC,
the first and third, any like multiples whatever have been taken,
viz., the base HC, and the triangle AHC ; and of the base CD,
and the triangle ACD, the second and fourth, have been taken
any like multiples whatever, viz., the base CL, and the triangle
ALC ; and that it has been shown that, if the base HC be
greater than CL, the triangle AHC is greater than ALC ; if
equal, equal; and if less, less; therefore (IV. def. 5) as the
base BC is to the base CD, so is the triangle ABC to the trian-
gle ACD.

Again : because (L 15, cor.) the parallelogram CE is double
of the triangle ABC, and the parallelooi-ani CP"" of the triangle
ACD, and that (IV. 9) magnitudes have the same ratio which
their like multiples have ; as the triangle ABC is to the trian-
gle ACD, so is the parallelogram EC to the parallelogram CF.
But it has been shovvn, that BC is to CD, as the triangle ABC
to the triangle ACD ; and as the tiiansjfle ABC is to the trian-
gle ACD, so is the parallelogram EC to the parallelogram CF;
therefore (TV. 7) as the base BC is to the base CD, so is the
parallelogram EC to the parallelogram CF. Wheiefore, trian-
gles, etc.

Scho. This proposition may be briefly demonstiated thus:

Let a perpendicular drawn from A to BD be called P. Then,

â– J^P.BC will be equivalent to the area of the triangle ABC, and

Ap.CD that of ACD. Dividing, therefore, the former of these

, t, , 1 il'-KC BC ABC ,â€ž, â€ž

equals by the latter, we get yp-^ or, qy)~A<Jd' Â°^" ^ '

scbo. 2) BC : CD : : ABC : ACD. In extending this method

110 THE ELEMENTS OF [BOOK V.

of proof to the parallelograms, we have merely to hbc P instead
ofiP.

Cor. 1. From this it is plain, that triangles and parallelo-
grams which have equal altitudes, are one to another as their
bases.

Let the figures be placed so as to have their bases in the
same straight line ; and perpendiculars being drawn from the
vertices of the triangles to the bases, the straight line Avhich
joins the vertices is parallel (I, 15, cor.) to that in Avhich their
bases are, because the perpendiculars ai-e both equal and paral-
lel to one another. Then, if the same construction be made as
in the proposition, the demonstration will be the same.

Cor. 2. Hence, if A, B, C be any three straight lines, we have
A : B : : A.C : B.C.

Cor. 3. So, likewise, if the straight lines A, B, C, D be pro-
portional, and E and F be any other straight lines, we shall
have, according to the preceding corollary, and the seventh
proposition of the fourth book, A.E : BE : : C.F ; D.F.

Prop. II. â€” Theor. â€” If a straight line he parallel to the base
of a triangle, it cuts the other sides, or those produced, propor-
tionally, and the segments between the parallel and the base are
homol'jgous to one another ; and (2) if the sides of a triangle,
or the sides produced, be cut proportionally, so that the seg-
ments between the points of section and the base are homologous
to one another, the straight line which jo. ns the points of sec-
tion is parallel to the base.

The enunciation of this proposition which is given by Dr.
Simson and others, is defective, and might lead to error in
its application, as it does not point out what lines are homolo-
gous to one another in the analogies.

It is plain that, instead of one proposition, this is in reality
two, which are converses of one another.

1. Let DE be parallel to BC, one of the sides of the triangle
ABC ; BD : DA : : CE : EA.

Join BE, CD. Then (L 15, cor.) the triangles BDE, CDE
are equivalent, because they are on the same base DE, and be-
tween the same parallels. DE, BC. Now ADE is another tri-

BOOK v.] EUCLID AND LEGENDRK 111

angle, and (IV. 5) equal magnitudes have to the same the same
ratio; therefore, as the triangle

BDE to ADE, so is the triangle ^ ^ Â»

(V. ]) as the triangle BDE
to ADE, so is BD to DA ;

Because, having the same alti-
tude, viz., the perpendicular
drawn from E to AB, they are
to one another as their bases; ^ c B O

and for the same reason,

as the triangle CDE to ADE, so is CE to EA.

Therefore (IV. 1) as BD : DA : : CE : E A.

2. Next, let the sides AB, AC of the triangle ABC, or those
produced, be cut proportionally in the points D, E ; that is, so
that BD : DA : : CE : EA, and join DE ; DE is parallel to BC.

The same construction being made, because (hyp.)

as BD : DA : : CE : EA ; and (V. 1)

as BD to DA, so is the triangle BDE to the triangle ADE; and
as CE to EA, so is the triangle CDE to ADE ; therefore (IV.
7) the triangle BDE is to ADE, as the triangle CDE to ADE ;
that is, the triangles BDE, CDE have the same ratio to ADE ;
and therefore (IV. 6) the triangles BDE, CDE are equal ; and
they are on the same base DE, and on the same side of it ;
therefore (I. 15, cor.) DE is parallel to BC. Wherefore, if a
straight line, etc.

Cor. The triangles which two intersecting straight lines
form with two parallel ones, have their sides which are on the
intersecting lines proportional ; and those sides are homologous
which are in the same straight line; and (2), conversely, if two
straight lines form with two intersecting ones triangles which
have their sides that are on the intersecting lines proportional,
the sides which are in the same Rtrai<irht line with one another
being homologous, those straight lines ai-e parallel.

1. Let DE and BC (first and second figures) be the parallels,
and let them be cut by the straight lines BD, CE, which inter-
sect each other in A; then BA : AC : : DA : AE. For, since
BD : DA : : CE : EA, we have, by composition in the first fig-

112 THE ELEMENTS OF [BOOK V.

tire, and by division in the second, BA : DA : : CA : EA, and,
alternately, BA : AC : : DA : AE.

2. But if BA : AC : : DA : AE, DE and BC are parallel.
For, alternately, BA : DA :: CA : EA ; then, in the first figure
by division, and in the second by composition, we have BD :
DA : : CE : EA ; and therefore, by the second part of this
proposition, DE is pai-allel to BC.

Prop. III. â€” TheoPw â€” Tlie sides about the equal aiigles of
equiangular triangles are proportionals ; and those ichich are
ojyposite to the equal angles are homologous sides^ that is, are
the antecedents or consequents of the nitios.

Let ABC, DCE be equiangidar triangles, having the angle
ABC equal to DCE, and ACB to DEC, and consequently (I.
20, cor. 5) BAC equal to CDE ; the sides about the equal
angles are proportionals ; and those are the homologous sides
which are opposite to the equal angles.

Let the triangles be placed on the same side of a straight
line BE, so that sides BC, CE, which are opposite to equal
ano-les, may be in that straiu^ht line and contii^nous to one an-
other; and so that neither the equal angles ABC, DCE, nor
ACB, DEC at the extremities of those sides may be adjacent.

Then, because (I. 20) the angles
ABC, ACB are together less than
two right angles, ABC and DEC,
which (hyp.) is equal to ACB, are
also less than two right angles ;
wherefore (I. 19) BA, ED will meet,
if produced ; let them be produced
and meet in F. Again : because
the angle ABC is equal to DCE, BF is parallel (L 16, cor.) to
CD ; and, because the angle ACB is equal to DEC, AC is par-
allel to FE. Therefore, FACD is (I. def 15) a parallelogram;
and consequently (L 15, cor.) AF is equal to CD, and AC to
FD. Now (V. 2) because AC is parallel to FE, one of the
sides of the triangle FBE,

BA : AF : : BC : CE.
But AF is equal to CD ; therefore (IV. 5)

as BA : CD : : BC : CE,

BOOK v.] EUCLID AND LI GKNDRE. 113

and alternately (IV. 4) as AB : BC : : DC : CE.
Ai^ain : (V. 2) because CD is parallel to 15F, as

BC : CE : : FD : DE; but FD is equal to AC; therefore,
as BC : CE : : AC : DE ; and, alternately,
as BC : CA : : CE : ED.
Therefore, because it has been proved that

AB : BC : : DC : CE, and as BC : CA : : CE : ED;
ex mquo (IV. 13), BA : AC : : CD : DE. Therefore, the
Bides, etc.

Scho. 1. Hence (V, def l) equianirnlar triangles are similar.

Cor. If two angles of one triangle be respectively equal to
two angles of another, their sides are proportional, and the
sides opposite to equal angles are homologous. For (I. 20, cor.
5) the remaining angles are equal, and therefore the triangles
are equiangular.

Scho. 2. In a similar manner we may produce a given
straight line, so that the whole line so produced may have to
the part produced the ratio of two given straight lines. Thus,
if BA be the line to be produced, make at B an angle of any
magnitude, and take BE and CE equal to the other given lines;
join AC, and draw FE parallel to it. Then, since FE is paral-
lel to AC, a side of the triangle ABC, we have (V. 2) BF :
AF : : BE : CE, so that BF has to AF the given ratio.

Prop. IV. â€” Theor. â€” The straight line which bisects an aru-
gle of a triangle, divides the opposite side into segments ichich
have the same ratio to one another as the adjacent sides of the
triangle have ; and (2) if the segm.ents of the base have the
same ratio as the adjacent sides, the straight line draicn from
the vertex to the point of section, bisects the vertical angle.

1. Let the angle BAC of the triangle ABC be bisected by
the straight line AD ; then BD : DC : : BA : AC.

Through C draw (I. 18) CE par-
allel to DA; then (I. 16, cor. l)
BA produced will meet CE; let
them meet in E. Because AC
meets the parallels AD, EC, the g D c

angle ACE is equal (I. 16) to the

alternate angle CAD ; and because BAE meets the same paral-
8

114

THE ELEMENTS OF

[book

lels, the angle E is equal (I. ] 6, part 2) to BAD ; therefore (I.
ax. 1) the angles ACE, AEC are equal, because they are re-
spectively equal to the equal angles, DAC, DAB; and conse-
quently AE is equal (I. 1, cor.) to AC. Now (V. 2) because
AD is parallel to EC, one of the sides of the triangle BCE,
BD : DC : : BA : AE ; but AE is equal to AC; therefore (IV.
5) BD : DC : : BA : AC.

2. Let now BD : DC : : BA : AC, and join AD ; the angle

The same construction being made, because
(hyp.) BD : DC : : BA : AC; and
(V. 2) BD : DC : : BA : AE,
since AD is parallel to EC ; therefore (IV. 1) BA : AC : : BA :
AE; consequently (IV. 6) AC is equal to AE ; and (I. 1) the
angles AEC, ACE are therefore equal. But (I. 16) the angle
BAD is equal to E, and DAC to ACE ; wherefore, also, BAD
is equal (I. ax. 1) to DAC; and therefore the angle BAC is
bisected by AD. The straight line, therefore, etc.

And if an exterior angle of a triangle be bisected by a straight
line which also cuts the base produced, the segments between
the bisecting line and the extremities of the base have the
same ratio to one another as the other sides of the triangle
have ; and (2) if the segments of the base produced have the
same ratio which the other sides of the triangle have, the
straight line drawn fiom the vertex to the point of section