3. If a quadril. ABCD be circumscribed about a D. circle EFGH, any two of its opp. sides AB+CD, or

AD+BC= half its perimeter, i. e.

AB+BC+CD+AD.

2

= AH, BE BF,
CF CG, and DGDH;

.. AE BE+ CF + DG=
AH + BF + CG + DH;

C

H

B

i. e. AB+ CD AD + BC, half the perimeter.

PROP. 8.-PROB.

To inscribe a circle in a given square.

CON.

10, I. 31, I. Through a given point to draw a parallel to a given st. line.

DEM. Def. 30, I. Cor. 2. 46, I. 34, I. Def. 5, IV.

Ax. 7. Things which are halves of the same are equal to one another.
29, I. If a line fall upon two par.
to one another; and the ext.
side; and likewise the two int.
two rt. s.

st. lines, it makes the alt. s equal
int. and opp. upon the same
s upon the same side together

=

Cor. 1. 16, III. If a st. line be drawn at rt. Zs to any diam. of a from its extremity, it shall touch the at the extremity; and a st. line touching the O at one point shall touch it at no other point.

D.1 C. 2.

2 Def. 30.

Each of the figs. GD, DH, GC, CF, GE,

EK, HE, EF is a □];

and each contains an

of the sq. GHKF; 3 Cor. 2. 46, I... each of those figures is rectangular; and .. of each, the opp. sides are equal.

4 34, I.

5 Def. 30. E 1. Now GF = GH, and GA =

GF,

.. the from E, with rad. EA, passes through

6 Ax. 7. D. 3.

.. GA =

GB, and BE

7 Sim.

8 Conc.

929, I. 10 C.

11 Cor. 16, III. 12 Def. 5, IV.

And thes at A, B, C, D, are rt. ≤s;
and the st. lines GF, FK, KH, HG are at
the ends of diams. BD, AC;

.. each of those st. lines is a tang to the ;
. the ABCD is inscribed in the GHKF.

Q. E. F.

N.B.-The diagram will illustrate the Cor. to Pr. 7, IV.

SCH.-Euclid confines himself in this book to the inscription and circumscription of circles and regular rt. lined figures,-but circles may be inscribed in segments and sectors; for example,

To inscribe a circle in a given quadrant ABC.

816, III.

9 11, III.

..the touches AC and AB in H and F ;

10 C. 3 and D. 6. and

and.. AG, joining the centres A and G, passes through D; LD or KD, a perp. to AD at D, is a tang. to arc CDB and to the DFH;

.. / GDF = / DFG, and GF = GD = HG;
HFD passes through the points, D, F, H.

Also the s at H and F are rt. /s;

11' 11, III.

.. the

DFH touches the arc CDB.

Q. E. F.

PROP. 9.-PROB.

To circumscribe a circle about a given square.

CON. Pst. 1 and 3.

DEM. Def. 30, I. Axs. 11 & 7.

Def. 6, IV. 8, I. If the As have two

sides of the one equal to two sides of the other, each to each, and have

likewise their bases equal, the

of the one, shall be equal to the

to them of the other.

which is contained by the two sides contained by the two sides equal

6, I. If two ▲s of a ▲ be equal to one another, the sides opp. to the equals shall be equal to one another.

Join AC, BD cutting in G; F
and with GA desc. a ;
that O ABCD is the
required.

. in As ABC, ADC, DA =
and AC com.;

K

B

AB, CB CD,

.. 4 DAC = BAC, i. e. Z BAD is bisd.
by AC;

So 4s ABC, BCD and CDA are bisected by
BD and AC.

Now DAB = ABC;

=

and GAB DAB, and GBA = ABC;

6 Ax. 7 & 6. I... Z GAB =

GBA, and GA = GB.

So GA = GB, and GC

7 Sim.

8 Ax. 1. 99, III.

10 D. 8.

11 Def. 6, IV.

.. GA GB

= GB = GD;

- GC GD.

Hence a from G, with rad. GA, will pass through B, C, and D.

And the ABC passes through the angular •s of the sq. ABCD,

.. the ABC is c. scrd. about the ABCD.

Q. E. F.

N.B. In the diagram a circle is also inscribed in the square ABCD.

PROP. 10.-PROB.

To construct an isosceles triangle, having each of the angles at the base double of the third, or vertical angle.

CON, Pst. 3. 1. IV. Pst. 1. 5, IV. 11, II. To divide a given line into two parts, so that the rect. contained by the whole and one of the parts shall equal the sq. of the other part.

DEM. Ax. J. 32, III. 6, I. 37, III. If from a

without a there be drawn two lines, one of a which cuts the O, and the other meets it; if the rect. contained by the whole line which cuts the O, and the part of it without the circle, be equal to the sq. of the line which meets it, the line which meets shall touch the .

Ax. 2. If equals be added to equals the wholes are equal.
32, I. If a side of a ▲ be produced, the ext.

= the two int. and opp.
Zs, and the three int. s of every ▲ are together equal to two rt. /s.
Def. 25, I. An Isosceles ▲ is that which has two equal sides or legs.
5, I. The s at the base of an isosc. ▲ are equal to each other; and if
the eq. sides be produced, the s on the other side of the base shall
be equal.

31, IV. Pst. 1. in

4 Sol.

5 Pst. 1. 5., IV.

D

BDE place BD = AC diam. of OBDE, and join AD;

then in A ABD, ▲ ABD = ▲ BDA = 2 / BAD.

Join DC, and about ▲ ADC desc. © ACD.

out of

ACD, are drawn

BCA, BD, one cutting the O in C, the other meeting it in D;

3 D. 1. 37, III. and. AB. BC= BD2, .. BD touches the

O ACD in D.

D.4 D. 3. & C.

5 32, III.

6 Add. Ax. 3.

Again BD touches the ACD, and DC
a st. line from the of contact D,
that ;

.. ▲ BDC =

Add

7 32, I. Ax. 1. But

DAC in the alt. seg. ;

CDA to each;

cuts

▲ BDA = / CDA + ≤ DAC;
ext. 4 BCDs CDA, DAC,
BDA = ▲ BCD;

USE AND APP.-The following are some of the various problems which bear a close relation to the 10th :

1o. The side AC inscribed in the smaller

ACD equals the side of a

regular pentagon in that circle, and also equals the side of a regular decagon in the larger BDE.

Prel. 1 2 Cor. 15, I. . the s formed by lines from a central point

2 3

and. in a regular polygon the centrals are all equal;
.. the central of a pent. 4-5ths of a rt. ;
and the central

of a decagon =4-10ths

= 2-5ths

the side of a reg. BEF.

F

I