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BOOK IV.

DEFINITIONS.

I. A RECTILINEAL figure is said to be inscribed in another rectilineal figure, when all the angles of the inscribed figure are upon the sides of the figure in which it is inscribed, each upon each.

II. In like manner a figure is said to be described about another figure, when all the sides of the circumscribed figure pass through the angular points of the figure about which it is described, each through each.

Definitions I. and II.

III. A rectilineal figure is said to be inscribed in a circle, when all the angles of the inscribed figure are upon the circumference of the circle.

Definition III.

IV. A rectilineal figure is said to be described about a circle, when each side of the circumscribed figure touches the circumference of the circle.

V. In like manner, a circle is said to be inscribed in a rectilineal figure, when the circumference of the circle touches each side of the figure.

Definitions IV. and V.

VI. A circle is said to be described about a rectilineal figure, when the circumference of the circle passes through all the angular points of the figure about which it is described.

VII. A straight line is said to be placed in a circle, when the extremities of it are in the circumference of the circle,

Definitions VI. and VII.

PROPOSITION I. PROBLEM.

In a given circle to place a straight line equal to a given straight line not greater than the diameter of the circle.

LET abc be the given circle, and d the given straight line, not greater than the diameter of the circle.

d

Draw be the diameter of the circle abc; then, if bc is equal to d,

D

the thing required is done, for in the circle abc a straight line bc is placed equal to d; but, if it is not, bc is greater than d; make ce equal (i. 3) to d, and from the centre c, at the distance ce, describe the circle a ef, and join ca: therefore, because c is the centre of the circle a ef, ca is equal to ce but dis equal to ce; therefore d is equal to ca. Wherefore in the circle a bc,

a straight line is placed equal to the given straight line d, which is not greater than the diameter of the circle. Which was to be done.

PROPOSITION II.-PROBLEM.

In a given circle to inscribe a triangle equiangular to a given triangle. LET abc be the given circle, and def the given triangle; it is required to inscribe in the circle abc a triangle equiangular to the triangle

d

b

g

h def

Draw (iii. 17) the straight line gah touching the circle in the point a, and at the point a, in the straight line ah, make (i. 23) the angle hac equal to the angle def; and at the point a, in the straight line a g, make the angle gab equal ag,

to the angle dfe, and join bc: therefore because hag touches the circle abc, and a c is drawn from the point of contact, the angle h a c is equal (iii. 32) to the angle abc in the alternate segment of the circle; but hac is equal to the angle def; therefore also the angle abc is equal to def. For the same reason, the angle a cb is equal to the angle dfe; therefore the remaining angle bac is equal (i. 32) to the remaining angle edf: wherefore the triangle a bc is equiangular to the triangle def, and it is inscribed in the circle a bc. Which was to be done.

PROPOSITION III.-PROBLEM.

About a given circle to describe a triangle equiangular to a given triangle. LET abc be the given circle, and def the given triangle; it is required to describe a triangle about the circle a b c equiangular to the triangle def.

:

a

1

d

Produce ef both ways to the points g, h, and find the centre k of the circle a bc, and from it draw any straight line k b; at the point k in the straight line k b, make (i. 23) the angle bka equal to the angle deg, and the angle bkc equal to the angle dfh; and through the points a, b, C, draw the straight lines lam, mbn, ncl, touching (iii. 17) the circle abc therefore, because 1m, mn, nl touch the circle a bc in the points a, b, c, to which from the centre are drawn ka, kb, kc, the angles at the points a, b, c, are right (iii. 18) angles: and because the four angles of the quadrilateral figure ambk are equal to four right angles, for it can be divided into two triangles; and that two of them kam, kbm are right angles, the other two akb, amb are equal to two right angles: but the angles deg, def are likewise equal (i. 13) to two right angles; therefore the angles a kb, amb are equal to the angles deg, def, of which a kb is equal to deg; wherefore the remaining angle amb is equal to the remaining angle def. In like manner, the angle lnm may be demonstrated to be equal to dfe; and therefore the remaining angle mln is equal (i. 32) to the remaining angle e df: wherefore the triangle 1mn is equiangular to the triangle def: and it is described about the circle abc. Which was to be done.

m

b

n g

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PROPOSITION IV.—PROBLEM.

To inscribe a circle in a given triangle.

LET the given triangle be a bc; it is required to inscribe a circle in a bc. Bisect (i. 9) the angles abc, bca by the straight lines bd, cd meeting one another in the point d, from which draw (i. 12) de, df, dg per

g

80

pendiculars to a b, bc, ca: and because the angle ebd is equal to the angle fb d, for the angle abc is bisected by bd, and that the right angle bed is equal to the right angle bfd, the two triangles ebd, fbd have two angles of the one equal to two angles of the other, and the side bd, which is opposite to one of the equal angles in each, is common to both; therefore their other sides shall be equal (i. 26); wherefore de is equal to df. For the same reason, dg is equal to df; therefore the three straight lines de, df, dg, are equal to one another, and the circle described from the centre d, at the distance of any of them, shall pass through the extremities of the other two, and touch the straight lines a b, bc, ca. Because the angles at the points e, f, g are right angles, and the straight line which is drawn from the extremity of a diameter at right angles to it, touches (iii. 13) the circle: therefore the straight lines ab, bc, ca do each of them touch the circle, and the circle efg is inscribed in the triangle abc. Which was to be done.

b

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PROPOSITION V.—PROBLEM.

To describe a circle about a given triangle.

LET the given triangle be abc; it is required to describe a circle about abc.

Bisect (i. 10) a b, a c in the points d, e, and from these points draw df, ef at right angles (i. 11) to ab, ac; df, ef, produced, meet one another. For, if they do not meet, they are parallel, wherefore a b, ac, which are at right angles to them, are parallèl; which is absurd. Let

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them meet in f, and join fa; also if the point f be not in bc, join bf, cf: then, because a d is equal to db, and df common, and at right angles

In like manner, it may

to a b, the base af is equal (i. 4) to the base fb. be shewn that cf is equal to fa; and therefore bf is equal to fc; and fa, fb, fc, are equal to one another; wherefore the circle described from the centre f, at the distance of one of them, shall pass through the extremities of the other two, and be described about the triangle abc. Which was to be done.

COR. And it is manifest, that when the centre of the circle falls within the triangle, each of its angles is less than a right angle, each of them being in a segment greater than a semicircle; but, when the centre is in one of the sides of the triangle, the angle opposite to this side, being in a semicircle, is a right angle; and, if the centre falls without the triangle, the angle opposite to the side beyond which it is, being in a segment less than a semicircle, is greater than a right angle: wherefore, if the given triangle be acute-angled, the centre of the circle falls within it; if it be a right-angled triangle, the centre is in the side opposite to the right angle; and if it be an obtuse-angled triangle, the centre falls without the triangle, beyond the side opposite to the obtuse angle.

PROPOSITION VI.-PROBLEM.

To inscribe a square in a given circle.

LET abcd be the given circle; it is required to inscribe a square in abcd.

Draw the diameters a c, bd, at right angles to one another, and join ab, bc, cd, da; because be is equal to ed, for e is the centre, and that ea is common, and at right angles to bd; the base ba is equal (i. 4) to the base ad; and, for the same reason, bc, cd are each of them equal to ba, or ad; therefore the quadrilateral figure abcd is equilateral. It is also rectangular; for the straight line bd, being the dia- b meter of the circle a bcd, bad is a semicircle; wherefore the angle bad is a right (iii. 31) angle; for the same reason, each of the angles abc, bed, cda, is a right angle; therefore the quadrilateral figure abcd is rectangular,

d

and it has been shewn to be equilateral; therefore it is a square; and it is inscribed in the circle a bcd. Which was to be done.

PROPOSITION VII.—PROBLEM.

To describe a square about a given circle.

LET abcd be the given circle; it is required to describe a square about it. Draw two diameters a c, b d of the circle abc d, at right angles to one another, and through the points a, b, c, d, draw (iii. 17) fg, gh, hk, kf, touching the circle; and because fg touches the circle abcd, and ea is drawn from the centre e to the point of contact a, the angles at

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