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VI.

A circle is faid to be described about a rectilineal figure, when the circumference of the circle paffes through all the angular points of the figure about which it is defcribed.

Book IV.

VII.

A ftraight line is faid to be placed in a circle, when the extre mities of it are in the circumference of the circle.

PROP. I. PROB.

IN a given circle to place a ftraight 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 ftraight line, not greater than the diameter of the circle.

a

Draw BC the diameter of the circle ABC; then, if BC is equal to D, the thing required is done; for in the circle ABC a ftraight line BC is placed equal to D: But, if it is not, BC is greater than D; make CE equal to D, and from the center C, at the distance CE, defcribe the circle AEF, and join CA: Therefore, because Ċ is the center of the circle AEF, CA is equal to CE; but Dis

D

C

A

a 3. 1.

E

B

F

equal to CE; therefore D is equal to CA: Wherefore in the circle ABC 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.

PROP. II. PRO B.

IN a given circle to inscribe a triangle equiangular to a given triangle.

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Book IV.

a 17 3.

b 23. I.

C 32. 3.

d 32. I.

Let ABC be the given circle, and DEF the given triangle; it is required to infcribe in the circle ABC a triangle equiangular to the triangle DEF.

a

Draw the ftraight line GAH touching the circle in the point A, and at the point A, in the ftraight line AH, make the angle HAC equal to the angle DEF; and at the point A, in the ftraight line

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G

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the circle ABC, and E
AC is drawn from
the point of contact,
the angle HAC is
equal to the angle

H

C

ABC in the alternate fegment of the circle: But HAC is equal to the angle DEF; therefore alfo the angle ABC is equal to DEF: For the fame reafon, the angle ACB is equal to the angle DFE; therefore the remaining angle BAC is equal to the remaining angle EDF: Wherefore the triangle ABC is equiangular to the triangle DEF, and it is infçribed in the circle ABC. Which was to be done.

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a 23. I.

b 17.3.

c. 18. 3.

A

BOUT a given circle to defcribe a triangle equian, gular to a given triangle.

Let ABC be the given circle, and DEF the given triangle; it is required to defcribe a triangle about the circle ABC equiangular to the triangle DEF.

2

Produce EF both ways to the points G, H, and find the center K of the circle ABC, and from it draw any straight line KB at the point K in the ftraight line KB, make 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 ftraight lines LAM, MBN, NCL touching the circle ABC: Therefore, becaufe LM, MN, NL touch the circle ABC in the points A, B, C, to which from the center are drawn KA, KB, KC, the angles at the points A, B, C, are right c angles: And becaufe the four angles of the quadrilateral fi

b

gure

gure AMBK are equal to four right angles, for it can be divided in- Book IV. to two triangles; and that two of them KAM, KBM are right an

gles, the other two

AKB, AMB are

L

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DEF, of which AKB is equal to DEG; wherefore the remaining angle AMB is equal to the remaining angle DEF: In like manner, the angle LNM may be demonftrated to be equal to DFE; and therefore the remaining angle MLN is equal to the e 32, 1. remaining angle EDF: Wherefore the triangle LMN is equiangular to the triangle DEF: And it is described about the circle ABC. Which was to be done.

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Let the given triangle be ABC; it is required to inscribe a circle in ABC.

A

See N.

Bifect the angles ABC, BCA by the ftraight lines BD, CD a. 9. x. meeting one another in the point D, from which drawb DE, DF, b 12. 1. DG perpendiculars to AB, BC, CA: And because the angle EBD is equal to the angle FBD, for the angle ABC is bifected 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 fide BD, which is oppofite to one of the equal angles in each, is common to both; therefore their other B F fides fhall be equal; where

E

D

fore

c 26. &.

Book IV. fore DE is equal to DF: For the fame reafon, DG is equal te DF; therefore the three ftraight lines DE, DF, DG are equal to one another, and the circle defcribed from the center D, at the distance of any of them, fhall pafs through the extremities of the other two, and touch the ftraight lines AB, BC, CA, because the angles at the points E, F, G are right angles, and the ftraight line which is drawn from the extremity of a diameter at right angles to it, touches d the circle: Therefore the straight lines AB, BC, CA do each of them touch the circle, and the circle EFG is infcribed in the triangle ABC. Which was to be done.

d 16.3.

PROP. V. PROB.

See N.

2 10. 1.

b II. I.

To defcribe a circle about a given triangle.

Let the given triangle be ABC; it is required to describe a circle about ABC.

Bifecta AB, AC in the points D, E, and from these points draw DF, EF at right angles to AB, AC; DF, EF produced

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meet one another: For, if they do not meet, they are parallel, wherefore AB, AC, which are at right angles to them, are parallel; which is abfurd: Let them meet in F, and join FA ; alfo, if the point F be not in BC, join BF, CF: Then, because AD is equal to DB, and DF common, and at right angles to AB, the base AF is equal to the bafe FB: In like manner, it may be fhewn that CF is equal to FA; and therefore BF is equal to FC: and FA, FB, FC are equal to one an

other.

the

other; wherefore the circle described from the center F, at the Book IV.
diftance of one of them, fhall pafs through the extremities of
the other two; and be described about the triangle ABC, which
was to be done.

COR. And it is manifeft, that, when the center of the circle
falls within the triangle, each of its angles is lefs than a right
angle, each of them being in a fegment greater than a femicir-
cle; but, when the center is in one of the fides of the triangle,
the angle oppofite to this fide, being in a femicircle, is a right
angle; and, if the center falls without the triangle, the angle
oppofite to the fide beyond which it is, being in a fegment lefs
than a femicircle, is greater than a right angle: Wherefore, if
the given triangle be acute angled, the center of the circle falls
within it; if it be a right angled triangle, the center is in the
fide oppofite to the right angle; and, if it be an obtufe angled
triangle, the center falls without the triangle, beyond the fide
oppolite to the obtufe angle.

ΤΟ

PRO P. VI. PRO B.

To inscribe a square in a given circle.

Let ABCD be the given circle; it is required to inscribe a fquare in ABCD.

A

Draw the diameters AC, BD at right angles to one an-
other; and join AB, BC, CD, DA; becaufe BE is equal to
ED, for E is the center, and that EA
is common, and at right angles to
BD; the base BA is equal to the
bafe AD; and, for the fame reason,
BC, CD are each of them equal to
BA or AD; therefore the quadri- B
lateral figure ABCD is equilateral.
It is alfo rectangular; for the straight
line BD, being the diameter of the
circle ABCD, BAD is a femicircle;
wherefore the angle BAD is a right

E

D

C

8 4.

bangle; for the fame reafon each of the angles ABC, BCD, b 31. 3.
CDA is a right angle; therefore the quadrilateral figure ABCD
is rectangular, and it has been fhewn to be equilateral; there-
fore it is a fquare; and it is infcribed in the circle ABCD.
Which was to be done.

PROP.

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