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Book I.

IX. mThe whole is greater than its part.

X.
Two straight lines cannot inclose a space.

XI.
All right angles are equal to one another.

XII.
“ If a straight line meets two straight lines, so as to make the two
"interior angles on the same side of it taken together less than

two right angles, these straight lines being continually produc" ed shall at length meet upon that side on which are the angles “ which are less than two right angles. See the notes on Prop. 6 29. of Book I,"

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PROPOSITIONI. PROBLEM.

Book I. O describe an equilateral triangle upon a given fi

nite straight line. Let AB be the given straight line, it is required to describe an equilateral triangle upon it.

From the center A, at the distance AB describe the circle

a. 3d Poftu

late. BCD, and from the center B, at the distance BA describe the circle

D ACE; and from the point C in

B E which the circles cut one another draw the straight lines 6 CA, CB

b. 2d Post. to the points A, B. ABC shall be an equilateral triangle.

Because the point A is the center of the circle BCD, AC is equal to AB. and because the point B is the center of the circle ACE, c. 15th DeBC is equal to BA. but it has been proved that CA is equal to finition. AB; therefore CA, CB are each of them equal to AB. but things which are equal to the same are equal to one anotherd; therefore d. 11 AxiCA is equal to CB, wherefore CA, AB, BC are equal to one another, and the triangle ABC is therefore equilateral, and it is described upon the given straight line AB, Which was required to be done.

PROP. II. PROB.
FRO
ROM a given point to draw a straight line equal to

a given straight line.
Let A be the given point, and BC the given straight line; it is re-
quired to draw from the point A a straight line equal to BC.

From the point A to B draw the straight line AB; and upon it de

K scribe the equilateral triangle DAB,

H and produce the straight lines DA, DB to E and F; from the center B, at the distance BC described the

d. 3. Poit. circle CGH, and from the center D,

B

E at the distance DG describe the

G circle GKL. AL shall be equal to BC.

om.

a. I Port.

b. i.i.

c. a. Post.

Book I. Because the point B is the center of the circle CGH, BC is equal o

to BG. and because D is the center of the circle GKL, DL is equal ¢. 15. Def. to DG, and DA, DB parts of them are equal; therefore the remainder f. 3. Ax. AL is equal to the remainder f BG. but it has been shewn that BC

is equal to BG; wherefore AL and BC are each of them equal to BG. and things that are equal to the same are equal to one another ; therefore the Atraight line AL is equal to BC. Whercfore from the given point A a straight line AL has been drawn equal to the given straight line BC. Which was to be done.

FRO

2. 3. I.

PROP. III. PROB.
ROM the greater of two given straight lines to cut

off a part equal to the less.
Let AB and C be the two gi-

D
ven straight lines, whereof AB is
the greater. It is required to cut off
from AB, the greater, a part equal

С
to C the less.

А. E B From the point A draw the straight line AD equal to C; and

from the center A, and at the dib. 3. Post. ftance AD describe b the circle DEF. and because A is the center of

the circle DEF, AE shall be equal to AD, but the straight line C is

likewise equal to 'AD. whence AE and C are each of them equal to c. 1. Ax.

AD. wherefore the straight line AE is equal to‘C, and from AB the
greater of two straight lines, a part AE has been cut off equal to C
the less. Which was to be done.

PROP. IV. THEOREM.
Fewo triangles have two sides of the one equal to two

fides of the other, each to each; and have likewise the angles contained by those sides equal to one another: they shall likewise have their bases, or third fides, equal; and the two triangles shall be equal; and their other angles shall be equal, each to each, viz. those to which the equal fides are opposite.

Let ABC, DEF be two triangles which have the two sides AB, AC equal to the two sides DE, DF, cach to each, viz. AB to DE,

IF

CE

and AC to DF; and the angle

D

Book 1. A BAC equal to the angle EDF. the base BC shall be equal to the base EF; and the triangle ABC to the triangle DEF; and the other angles, to which the equal sides are opposite, shall be equal, each to each, viz. the angle ABC to the B

F angle DEF, and the angle ACB to DFE.

For if the triangle ABC be applied to DEF so that the point A may be on D, and the straight line AB upon DE; the point B shall coincide with the point E, because AB is equal to DE, and AB coinciding with DE, AC shall coincide with DF, because the angle BAC is equal to the angle EDF wherefore also the point C shall coincide with the point F, because the straight line AC is equal to DF but the point B coincides with the point E; wherefore the base BC shall coincide with the base EF, because the point B coinciding with E, and C with F, if the base' BC does not coincide with the base EF, two straight lines would inclose a space, which is impossible'. There-a. 10. Ax. fore the base BC shall coincide with the base EF, and be equal to it. Wherefore the whole triangle ABC Mall coincide with the whole triangle DEF, and be equal to it; and the other angles of the one shall coincide with the remaining angles of the other, and be equal to them, viz. the angle ABC to the angle DEF, and the angle ACB to DFE. Therefore if two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by those sides equal to one another ; their bases shall likewise be equal, and the triangles be equal, and their other angles to which the equal fides are opposite, shall be equal, each to each. Which was to be demonstrated.

TH

PROP. V. THEOR.
HE angles at the base of an Isosceles triangle are e-

qual to one another; and if the equal sides be produced, the angles upon the other side of the base shall be equal.

Let ABC be an Isosceles triangle, of which the side AB is equal

a. 3. 1.

b. 4. 1.

Book I. to AC, and let the straight lines AB, AC be produced to D and E.

the angle ABC shall be equal to the angle ACB, and the angle CBD to the angle BCE.

In BD take any point F, and from AE, the greater, cut off AG equal to AF, the less, and join FC, GB.

Because AF is equal to AG, and AB to AC; the two sides FA,
AC are equal to the two GA, AB, each to each ; and they contain the
angle FAG common to the two tri-
angles AFC, AGB; therefore the base

A
FC is equal to the base GB, and the
triangle AFC to the triangle AGB;
and the remaining angles of the one
are equal to the remaining angles
of the other, each to each, to which B

с
the equal fides are opposite; viz. the
angle ACF to the angle ABG, and the

F

G angle AFC to the angle AGB. and because the whole AF is equal to the D whole AG, of which the parts AB, AC

E €. 3. Ax. are equal; the remainder BF shall be equal to the remainder CG.

and FC was proved to be equal to GB; therefore the two sides BF, FC are equal to the two CG, GB, each to each; and the angle BFC is equal to the angle CGB; and the base BC is common to the two triangles BFC, CGB; wherefore the triangles are equal', and their remaining angles, each to each, to which the equal fides are opposite. therefore the angle FBC is equal to the angle GCB,and the angle BCF to the angle CBG, and since it has been demonstrated that the whole angle ABG is equal to the whole ACF, the parts of which, the angles CBG, BCF are also equal; the remaining angle ABC is therefore equal to the remaining angle ACB, which are the angles at the base of the triangle ABC, and it has also been proved that the angle FBC is equal to the angle GCB, which are the angles upon the other side of the base. Therefore the angles at the base, &c. Q. E. D. COROLLARY.Hence every equilateral triangle is also equiangular.

PROP. VI. THEOR,

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F two angles of a triangle be equal to one another, the

fides also which subtend, or are opposite to, the equal angles shall be equal to one another.

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