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ELEMENTS

ΘΕ

GEOMETRY.

A

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RECTILINEAL figure is faid to be infcribed in another Book IV. rectilineal figure, when all the angles of the inscribed

figure are upon the fides of the figure

in which it is infcribed, each upon

each.

II.

In like manner, a figure is said to be defcribed about another figure, when all the fides of the circumfcribed figure

pass through the angular points of the figure about which it is defcribed, each through each.

III.

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

IV.

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a 3. I.

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

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

PROP. I. PROB.

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N a given. circle to place a ftraight line, equal to a given ftraight 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.

Draw BC the diameter of

the circle ABC; then, if BC
is equal to D, the thing re-
quired is done; for in the cir-
cle ABC a straight line BC is
placed equal to D: But, if it
is not, BC is greater than D;
make CE equal a to to D, and
from the centre C, at the di-

ftance CE, defcribe the circle

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AEF, and join CA: Therefore, because C is the centre of

the

circle AEF, CA is equal to 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 straight line D, which is not greater that the diameter of the circle. Which was to be done.

Book IV.

PROP. II. PROB.

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

IN

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.

Draw a the ftraight line GAH touching the circle in the a 17. 3. point A, and at the point A, in the ftraight line AH, make b b 23. 1. the angle HAC equal to the angle DEF; and at the point A,

in the ftraight line
AG, make the angle
GAB equal to the
angle DFE, and join
BC. Therefore, be-
caufe HAG touches
the circle ABC, and
AC is drawn from
the point of contact, E

the angle HAC is e

C

qual to the angle

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ABC in the alternate fegment of the circle: But HAC is equal to the angle DEF; therefore also the angle ABC is equal to DEF: for the fame reason, the angle ACB is equal to the angle DFE; therefore the remaining angle BAC is equal d to the remaining angle EDF: Wherefore the triangle d 32. 1. ABC is equiangular to the triangle DEF, and it is infcribed in the circle ABC. Which was to be done.

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Book IV.

A

PROP. III. PRO B.

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

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

a

Produce EF both ways to the points G, H, and find the centre K of the circle ABC, and from it draw any straight a 23. 1. 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 through the points A, B, C, b 17. 3. draw the ftraight lines LAM, MBN, NCL touching b the circle ABC: Therefore, because LM, MN, NL touch the circle ABC 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 18. 3. right angles. And because the four angles of the quadrila. teral figure AMBK are equal to four right angles, for it can be divided into two triangles; and because 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 like-

d 13. 1. wife equal

e 32. I.

d

to

two right angles;
therefore the an-
gles AKB, AMB M

A

Ι.

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are equal to the angles DEG, 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 remaining angle EDF: Wherefore the triangle LMN is equiangular to the triangle DEF: And it is defcribed about the circle ABC. Which was to be done.

e

PROP.

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