Sidebilder
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which it is infcribed.

B

D

A

F

E

C

which it is defcribed.

As the Triangle DEF is infcribed in the Triangle ABC.

II. In like manner a Figure is faid to be described about a Figure, when each Side of the faid circumfcribed Figure touches each Angle of the Figure about

So the Triangle ABC is defcribed about the Triangle DEF.

III.

III. A right-lin❜d Figure is faid to be infcrib'd in a Circle, when each Angle of the infcrib'd Figure fhall touch the Periphery of the Circle.

IV. And a right-lin❜d Figure is faid to be defcrib'd about a Circle, when each of the Sides of that Figure which is circumfcrib'd, touches the Periphery of the Circle.

V. In like manner, a Circle is faid to be infcrib'd in a right-lin❜d Figure, when the Periphery of the Circle fhall touch each of the Sides of the Figure in which it is infcrib'd.

VI. A Circle is faid to be defcrib'd about a right-lin❜d Figure, when the Periphery of the Circle touches every Angle of the Figure that it circumfcribes.

A

B

VII. A right Line is faid to be apply'd in a Circle, when the Extremes of it fhall be in the Periphery of the Circle; as the right Light AB.

PROP.

PROP. I. Probl. 1.

To apply a right Line AB in a given Circle ABC, equal to a given right Line D, which is not greater than AC the Diameter of the Circle.

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About the Centre A, with the Distance AE=D defcribe a a Circle meeting the given a 3 poft. & Circle in B; then if the right Line AB be drawn, 3. 1. it fhall be "AE= ‘D. Q. E. F.

PROP. II. Probl. 2.

To defcribe a Triangle ABC, in a given Circle ABC, equiangular to a given Triangle DEF.

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15. def. conft.

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H

b

a

23.1.

Let the right Line GH touch the given a 17.3. Circle in A; make the Ang. HAC=F; and b the Ang." GAB=E, and join BC. I say the thing is done.

For

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For the Ang, BHACF; and the Ang. CGABE. whence alfo the Ang. BAC =D, and fo the Triangle BAC is infcrib'd in the Circle equiangular to the Triangle DEF. Q. E. F.

PRO P. III. Probl. 3.

To defcribe the Triangle LNM about a given Circle ABC, equiangular to a given Triangle

DEF.

D

G

a 23. 1.

C

1.

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Continue out the Side EF both ways. At the Centre I, make the Ang. AIB DEG, and the Ang. BICDFH; then let three right Lines 17. 3. LN, LM, MN touch the Circle in the Points A, B, C, and I fay the thing is done.

ax. 13.

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For that the right Lines LN, LM, MN, will meet, will thus appear; because the Angles 18. 3. LAI, LBI, are d right ones; and fo a right Line AB being drawn, fhall make the Angles LAB, LBA, lefs than two right ones. There*Sekol. 32. fore fince the Ang. AIB L2. right Ang. DEG DEF; and AIB DEG; the Ang. L fhall be = DEF. After the fame way of reafoning the Ang. M DFE, therefore likewife the Ang. ND, and fo the Triangle LNM is circumfcribed about the Cire cle, Equiangular to the given Triangle EDF. Q.E. F.

f

13. I.

• conft. 3 ax.

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h

PROP.

E

D

PROP. IV. Probl. 4.

To infcribe a Circle EFG in a given Triangle ABC.

Bife&ta the two An- a 9. I. gles B and C by the

right Lines BD, CD

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DF, DG. a Circle defcribed from the Centre D

through E fhall pass through G and F, and

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touch the three Sides of the Triangle.

d

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e

26. 1.

For the Angle DBE = DBF; and the confir. Ang. DEB = DFB; and the Side DB is com- d 12. ax. mon. Therefore DE-DF. After the fame way of reafoning DG DF, therefore a Circle defcribed from the Centre D, fhall pass through E, F, G; and fince the Angles at E, F, G, are right ones, it fhall touch all the three Sides of the Triangle. Q. E. F.

SCHOLIU M.

Hence if the three Sides of a Triangle be known, we can find their Segments made by the Points of Contact of the infcrib'd Circle. Thus,

Let AB be 12, AC 18, BC 16. Then shall ABBC 28, from whence take 18

=

AE+FC, and there remains 10

AC

BE+

BF. Therefore BE, or BF=5. whence FC
or CG11. and fo GA, or AE =
= 7.

PROP.

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