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Book IV.

THE

ELEMENTS

A

OF

EUCLID.

BOOK IV.

DEFINITIONS.

I.

Rectilineal figure is faid to be infcribed in another rectilineal figure, when all the angles of the infcribed figure are upon the fides of the figure in which it is in

scribed, each upon each.

II.

In like manner, a figure is faid to be described
about another figure, when all the fides of the

circumfcribed figure pafs thro' the angular

points of the figure about which it is defcribed, each thro' each.

III.

A rectilineal figure is faid to be inscribed in a
circle, when all the angles of the infcribed
figure are upon the circumference of the

circle.

IV.

A rectilineal figure is faid to be defcribed about a circle, when each fide of the circumfcribed figure touches

the circumference of the circle.

V.

In like manner a circle is faid to be infcribed
in a rectilineal figure, when the circumfe-
rence of the circle touches each fide of the
figure.

VI.

A circle is faid to be defcribed about a recti

lineal figure, when the circumference of the circle paffes thro' all the angular points of the figure about which it is described.

VII.

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

Book IV.

IN

PROP. I. PROB.

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 straight line, not greater than the diameter of the circle.

A

E

2. 3. I.

B

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 a to D, and from the center C, at the distance CE, defcribe the circle AEF, and join CA. therefore, because C is the center of

the circle AEF, CA is equal to D

CE; but D is equal to CE, therefore D is equal to CA. wherefore in the circle ABC a ftraight line is placed equal to the given ftraight line D, which is not greater than the diameter of the circle. Which was to be done.

i

I

PROP. II. PROB.

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

Book IV.

2. 17. 3.

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.

b

Draw a the ftraight line GAH touching the circle in the point b. 23. 1. 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 straight

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nate segment of the circle. but HAC is equal to the angle DEF, therefore also the angle ABC is equal to DEF. for the fame reafon, the angle ACB is equal to the angle DFE; therefore the red. 32. 1. maining angle BAC is equal to the remaining angle EDF. wherefore the triangle ABC is equiangular to the triangle DEF, and it is infcribed in the circle ABC. Which was to be done.

3. 23. I.

A

d

PROP. III. PROB.

BOUT a given circle to defcribe a triangle equiangular 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.

Produce EF both ways to the points G, H, and find the center K of the eircle ABC, and from it draw any straight line KB; at the point K in the straight line KB, make the angle BKA equal to the angle DEG, and the angle BKC equal to the angle DFH; and thro' the points A, B, C draw the straight lines LAM, b. 17. 3. MBN, NCL touching the circle ABC. therefore because 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 c. 18. 3. A, B, C are right angles. and because the four angles of the

c

quadrilateral figure AMBK are equal to four right angles, for it Book IV. 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

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fore the remaining M

B

N

angle AMB is equal to the remaining angle DEF. în like manner the angle LNM may be demonftrated to be equal to DFE; and therefore the remaining angle MLN is equal to the remaining e. 32. *. 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.

PROP. IV. PROB.

To infcribe a circle in a given triangle.

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

See N.

Bifect the angles ABC, BCA by the straight lines BD, CD a. 9. 1. meeting one another in the point D, from which draw b DE, DF, b. 12. 1. DG perpendiculars to AB, BC,

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A

F

to both; therefore their other fides fhall be equal; wherefore c. 26. 1.

G

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