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THE

E LE MENT S

OF

EUCL I D.

воок IV.

DEFINITION S.

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I.

Right lined figure is faid to be infcribed in a right lined Book IV, figure, when every one of the angles of the infcribed fi

gure touches every one of the fides of the figure wherein it is infcribed.

II.

A right lined figure is faid to be described about a right lined figure, when every one of the fides of the circumfcribed figure touches each of the angles of the right lined figure.

III.

A right lined figure is infcribed in a circle when each of the angles of the inscribed figure touches the circumference of the circle.

IV.

A right lined figure is defcribed about a circle when each of the fides of the circumfcribed figure touches the circumference of the circle.

V.

A circle is infcribed in a right lined figure, when the circumference of the circle touches all the fides of the figure in which it is infcribed.

VI.

A circle is defcribed about a right lined figure when the cir cumference of the circle touches all the angles of the figure.

VII. A

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A right line is applied in a circle when its extremes are in the circumference of the circle.

* 3.1.

PROP. I. PRO B.

To apply a right line in a circle, equal to a given right line, not greater than the diameter of the circle.

It is required to apply a right line in the circle ABC, equal to a given right line D, not greater than the diameter of the

circle.

a

Draw the diameter BC; if equal to D, what was required is done; if not, the diameter BC is greater than D; put CE equal to D2; about the center C, with the distance CE, defcribe the circle AEF; then CA is equal to CE; but CE is equal to D; therefore CA is equal to D. Wherefore, there is drawn, &c.

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b 23, I.

€ 32. 3+

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IN Na given circle, to infcribe a triangle equiangular to a given, triangle.

It is required to inscribe a triangle, in a given circle ABC, equiangular tó a given triangle DEF: Draw the right line GAH, touching the circle in the point A; with the right line AH, at the point A, make the angle HAC equal to the angle DEF, and the angle GAB equal to DFE; join BC.

Then the angle HAC is equal to the angle ABC; but the angle HAC is equal to DEF; therefore, ABC is equal to DEF: And BAG is equal to ACB ; but BAG is equal to DFE; therefore ACB, is equal to DFE; therefore the remaining third à Cor. 32. angles BAC, EDF are equal; therefore the triangle ABC, is infcribed in the circle ABC, and equiangular to the triangle DEF, which was required. Wherefore, &c.

d

Book IV.

A

PRO P. III. PRO B.

BOUT a given circle, to defcribe a triangle equiangular
to a triangle given.

It is required to defcribe a triangle about the given circle ABC equiangular to the given triangle DEF; produce the fide EF both ways to G, H; find the center of the circle K, and draw KB any how; at the point K, with the right line KB, make the angles BKA, BKC, equal to the angles DEG, DFHa, 23. ri each to each; at the points A, B, C, draw the right lines LAM, MBN, LCN, tangents to the circle, in the points A, B, C.

b 16. 3.

Then the angles that LM, MN, LN, make with the right lines KA, KB, KC, are right angles; therefore the angles 18. 3. AKB, AMB, are equal to two right angles 4, and equal to d 32. 1. DEF, DEG; but BKA was made equal to DEG; therefore e 13. I. the remainder DEF is equal to AMB. For the fame reafon f Ax. 8. x7 DFE is equal to LNM, and the remaining angle MLN equal to EDF; wherefore the triangles LMN, DEF, are equiangular. Which was required.

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I.

It is required to inscribe a circle in the given triangle ABC. Bifect the angles ABC, ACB, by the right lines BD, DC 1, a 9. x. meeting each other in the point D; from which let fall DF, DE, DG, perpendiculars, upon the right lines AB, BC, ACb 12. x. Then, becaufe the two angles DFB, DBF, in the triangle DBF, are equal to the two angles DEB, DBE, in the triangle DBE, and the fide DB common to both; the remaining fides BE, ED, are equal to BF, FD, each to each. For the fame c 26. I, reafon DF is equal to DG; therefore D is the center, and with any of the diftances the circle EFG may be infcribed in the gi ven triangle ABC: Which was required.

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