To describe an equilateral triangle upon a given finite straight line.

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

From the centre A, at the distance AB,

describe the circle BCD, and * 3 Pos- from the centre B, at the dis

tulate.

tance BA, describe the circle D A BE ACE; and from the point C, in which the circles cut one another, draw the straight lines CA, CB, to the

#1 Post.

finition.

points A, B; ABC shall be an equilateral triangle. Because the point A is the centre of the circle BCD, AC is equal to AB; and because the point B is the cen12 De tre of the circle ACE, BC is equal to BA: But it has been proved that CA is equal to AB; therefore CA, CB, are each of them equal to AB; but things which are equal to the same thing are equal to one another; therefore CA 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.

*1st Axiom.

PROP. II. PROB.

From 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 required to draw from the point A a straight line equal to BC.

#1 Post. From the point A to B draw* the straight line *Prop. 1. AB; and upon it describe the equilateral triangle DAB. From the centre B, at the distance BC, #3 Post. describe the circle CGH, and produce DB to meet *2 Post. it in G. From the centre D, at the distance DG, describe the circle GKL, and produce DA to meet it in L; AL shall be equal to BC.

*12 Def.

Because the point B is the centre of the circle CGH, BC is equal to BG; and because D is the centre of the *By con- circle GKL, DL is equal to DG; struction and DA, DB, parts of them, are *3 Ax. equal; therefore the remainder* AL

is equal to the remainder BG: but

H

+ 1 Ax.

it has been shewn, that BC is equal to BG; wherefore AL and BC are each of them equal to BG: and things which are equal to the same thing are equalf to one another; therefore the straight line AL is equal to BC. Wherefore, from the given point A, a straight line AL has been drawn equal to the given straight line BC. Which was to be done.

The triangle DBA, employed in this problem, might evidently have been isosceles only, having merely DB equal to DA; for no consequence is deduced from the equality of BA with each of these. Euclid has not, however, shewn how to construct a triangle on BA except under the restrictions that all three of its sides are equal; and as his plan is never to assume any construction as practicable, till he has, by aid of the postulates, actually shewn how to effect it, we at once see the reason, and the propriety, of his directing an equilateral triangle to be described upon AB; and we subjoin these remarks solely for the purpose of pointing out to the student the restrictions which are non-essential to the argument.

PROP. III. PROB.

From the greater of two given straight lines to cut off a part equal to the less.

Let AB and C be the two given straight lines, whereof AB is the greater. It is required to cut off from AB, the greater, a part equal to C, the less.

* 2. 1.

From the point A draw the straight line AD equal to C; and

D

A

VE B

F

*3 Post. from the centre A, and at the distance AD, describe* the circle DEF: AE shall be equal to C.

Because A is the centre of the circle DEF, AE is +12 Def, equal to AD; but the straight line C is likewise + Const. equal to AD; whence AE and C are each of them #1 Ax. equal to AD:wherefore the straight lineAE 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.

C

PROP. IV. THEOREM.

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; they shall likewise have their bases, or third sides, equal; and the two triangles shall be equal; and their other angles shall be equal, each to euch, viz. those to which the equal sides are opposite.

A

Let ABC, DEF be two triangles, which have the two sides AB, AC, equal to the two sides DE, DF, each to each, viz. AB to DE, and AC to 'DF; and the angle 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 angle DEF, and the angle ACB to DFE.

B

CE

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

',

+Hyp.

equal to DE: and AB coinciding with DE, AC Hyp shall fall upon 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 +Hyp. was proved to coincide 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 did not coincide with the base EF, two straight lines would enclose a space, which is impossible. Therefore the base BC coincides with the base EF, and therefore is equalt to it. Wherefore the whole triangle ABC coincides with the whole triangle DEF, and is equal to it; and the other angles of the one coincide with the remaining angles of the other, and are equal to them, viz. the angle ABC to the angle DEF, and the angle ACB to

#10 Ax.

18 Ax.

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 shall be equal, and their other angles to which the equal sides are opposite shall be equal, each to each. Which was to be demonstrated.

PROP. V. THEOR.

The angles at the base of an isosceles triangle are equal 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 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.

+Const.

A

In BD take any point F, and from AE the greater, #3. 1. cut off AG equal to AF the less, and join FC, GB. Because AF is equal to† AG, and AB to AC, ¡Hyp. the two sides FA, AC are equal to the two GA, AB, each to each; and they contain the angle at A common to the two triangles AFC, AGB; therefore the base FC is equal #4. 1. 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 the equal sides are opposite; viz. the angle ACF to the angle D

#4. 1.

B

F

ABG, and the angle AFC to the angle AGB: and because the whole AF is equal to the whole AG, of which the parts AB, AC, are equal; the remainder BF is 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 was proved to be equal to the angle CGB, wherefore the two triangles BFC, CGB; are equal, and their remaining angles each to each, to which the equal sides 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; therefore

the remaining angle ABC is equal† to the remaining +3 Ax. 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.

lar.

COR.-Hence every equilateral triangle is also equiangu

PROP. VI. THEOR.

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.

#3. 1.

Let ABC be a triangle having the angle ABC equal to the angle ACB: the side AB shall be equal to the side AC. For, if AB be not equal to AC, one of them is greater than the other: let AB be the greater; and from it cut off BD equal to AC, the less, and join DC: therefore, because in the triangles DBC, ACB, DB is equal to AC, and BC common to both, the two sides, DB, BC are equal to the two AC, CB, each to each; and the angle DBC is equal to the anglet ACB; +Hyp. therefore the base DC is equal to the base AB, and the triangle DBC is equal to the triangle ACB, the less to the greater, which is absurd. Therefore AB is not unequal to AC, that is, it is equal. to it. Wherefore, if two angles, &c. Q.E.D.

*4. 1.

B

A

D

COR.-Hence every equiangular triangle is also equila

teral.