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equal to four right angles, for it can be divided into two tri- Book IV. angles: and that two of them KAM, KBM are right angles, the other two AKB, L AMB are equal to two right angles :

D but the angles DEG, DEF are likewise equal to A

d 13. 1. two right angles;

K therefore the an

GE FH gles AKB, AMB are equal to the angles DEG, DEF

N. of which AKB is MB equal to DEG; wherefore the remaining angle AMB is equal to the remaining angle DEF: in like manner, the angle LNM may be demonstrated to be equal to DFE; and therefore the remaining angle MLN is equal to the remaining angle EDF:e 32. 1. 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 inscribe a circle in a given triangle.

See N

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

Bisect a the angles ABC, BCA by the straight lines BD, CD a 9.1. meeting one another in the point D, from which drawb DE, b 12. 1. DF, DG perpendiculars to AB, А BC, CA: and because the angle EBD is equal to the angle FBD, for the angle ABC is bisected by BD, and that the right angle BED is equal to the right angle BFD, the two triangles EBD, E

D FBD have two angles of the one equal to two angles of the other, and the side BD, which is opposite to one of the equal angles in each, is common to both; there

B F fore their other sides shall be

Book IV. equalc; wherefore DE is equal to DF: for the same reason,

DG is equal to DF; therefore the three straight lines DE, DF, c 26. 1. DG are equal to one another, and the circle described from the

centre D, at the distance of any of them, shall pass through the extremities of the other two, and touch the straight lines AB, BC, CA, because the angles at the points E, F, G are right an

gles, and the straight line which is drawn from the extremity of d 16. 3.

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 inscribed in the triangle ABC. Which was to be done.

PROP. V. PROB.

See N.

TO describe a circle about a given triangle.

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

circle about ABC. a 10. 1. Bisect a AB, AC in the points D, E, and from these points b 11. 1. draw DF, EF at right angles b 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 absurd: let them meet in F, and join FA; also, 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 base FB: in like manner, it may be shown, that CF is equal to FA; and therefore BF is equal to FC; and FA, FB, FC are equal to one another ;

C 4. 1.

wherefore the circle described from the centre F, at the distance Book IV. of one of them, shall pass through the extremities of the other two, and be described about the triangle ABC. Which was to be done.

Cor. And it is manifest, that when the centre of the circle falls within the triangle, each of its angles is less than a right angle, each of them being in a segment greater than a semicir. cle; but, when the centre is in one of the sides of the triangle, the angle opposite to this side, being in a semicircle, is a right angle ; and, if the centre falls without the triangle, the angle opposite to the side beyond which it is, being in a segment less than a semicircle, is greater than a right angle: wherefore, if the given triangle be acute angled, the centre of the circle falls within it; if it be a right angled triangle, the centre is in the side opposite to the right angle ; and, if it be an obtuse angled triangle, the centre falls without the triangle, beyond the side opposite to the obtuse angle.

PROP. VI. PROB.

TO inscribe a square in a given circle.

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

Draw the diameters AC, BD at right angles to one another; and join AB, BC, CD, DA ; because BE is equal to ED, for E is the centre, and that EA is common,

A and at right angles to BD; the base BA is equal a to the base AD; and, for

a 4. 1. the same reason, BC, CD are each of

E them equal to BA or AD; therefore

B

D the quadrilateral figure ABCD is equilateral. It is also rectangular; for the straight line BD, being the diameter of the circle ABCD, BAD is a semicircle ; wherefore the angle BAD is a

C right b angle; for the same reason each of the angles ABC, BCD, b 31. 3. CĎA is a right angle ; therefore the quadrilateral figure ABCD is rectangular, and it has been shown to be equilateral ; therefore it is a square; and it is inscribed in the circle ABCD. Which Was to be done.

Book IV.

PROP. VII. PROB.

TO describe a square about a given circle.

Let ABCD be the given circle; it is required to describe a square about it.

Draw two diameters AC, BD of the circle ABCD, at right ana 17.3.

gles to one another, and through the points A, B, C, D draw a FG, GH, HK, KF touching the circle ; and because FG touches

the circle ABCD, and EA is drawn from the centre E to the b 18. 3. point of contact A, the angles at A are right bangles ; for the

same reason, the angles at the points B, C, D are right angles; and because the angle AEB is a right G A

angle, as likewise is EBG, GH is pac 28. 1. rallel c to AC; for the same reason, AC is parallel to FK, and in like man

E ner GF, HK may each of them be de

B monstrated to be parallel to BED;

D therefore the figures GK, GC, AK,

FB, BK are parallelograms; and GF d 34. 1. is therefore equal d to HK, and GH to FK; and because AC is equal to BD, H с

K and that AC is equal to each of the two GH, FK ; and BD to each of the two GF, HK: GH, FK are each of them equal to GF or HK ; therefore the quadrilateral figure FGHK is equilateral. It is also rectangular; for GBEA being a parallelogram, and AEB a right angle, AGB & is likewise a right angle: in the same manner, it may be shown that the angles at H, K, F are right angles; therefore the quadrilateral figure FGHK is rect. angular, and it was demonstrated to be equilateral ; therefore it is a square ; and it is described about the circle ABCD. Which was to be done.

PROP. VIII. PROB.

TO inscribe a circle in a given square.

Let ABCD be the given square ; it is required to inscribe å

circle in ABCD. a 10.1. Bisect a each of the sides AB, AD, in the points F, E, and b 31. 1. through E drawb EH parallel to AB or DC, and through F

draw FK parallel to AD or BC; therefore each of the figures Book IV.
AK, KB, AH, HD, AG, GC, BG, GD is a parallelogram, and their
opposite sides are equals; and because AD is equal to AB, and c 34. 1.
that AE is the half of AD, and AF the half of AB, AE is equal
to AF; wherefore the sides opposite A E
to these are equal, viz. FG to GE; in
the same manner, it may be demon-
strated that GH, GK are each of them
equal to FG or GE; therefore the

F
G

K four straight lines GE, GF, GH, GK are equal to one another; and the circle described from the centre G, at the distance of one of them, shall pass through the extremities of the other B H C three, and touch the straight lines AB, BC, CD, DA: because the angles at the points E, F, H, K are right d angles, and that the straight line which is drawn from d 29. 1. the extremity of a diameter, at right angles to it, touches the circle ®; therefore each of the straight lines AB, BC, CD, DA e 16. 3. touches the circle, which therefore is inscribed in the square ABCD. Which was to be done.

PROP. IX. PROB.

TO describe a circle about a given square.

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Let ABCD be the given square; it is required to describe a circle about it.

Join AC, BD culting one another in E; and because DA is equal to AB, and AC common to the triangles DAC, BAC, the two sides DA, AC are equal to the two BA, AC; and the base DC is equal

D to the base BC; wherefore the angle

E DAC is equal a to the angle BAC, and

a 8. L. the angle DAB is bisected by the straight line AC: in the same manner, it may be demonstrated that the angles ABC, BCD, CDA are severally bisected by the straight B

C lines BD, AC; therefore, because the angle DAB is equal to the angle ABC, and that the angle EAB is the half of, DAB, and EBA the half of ABC; the angle EAB is equal to the angle EBA; wherefore the side EA is equal o to the side EB: in the same manner, it may be b 6.1.

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