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XXIX.); .. in As AEC, BED, the S ACE, CAE EBD, BDE, each to each, and, adjacent to each, AC = BD; .'. AE = ED, and BE = EC (Prop. XXVI.): therefore the diams. AD, BC, mutually bisect each other.

2. But, let the diams. AD, BC bisect each other in E; then, since, in the ▲ AEC, BED, the two AE, EC = the two DE, EB, each to each, and the included vertical / AEC included vertical ▲ BED (Prop. XV.) ; ... the base AC = base BD, and the ACE = ZEBD (Prop. IV.): .. AC is I to BD. In like manner, AB is to CD; .. ABCD is am.

Wherefore the diameter &c.-Q. E. D.

PROP. FF. PROB.

GEN. ENUN.-To bisect a parallelogram by a straight line drawn from a given point in one of its sides.

PART. ENUN.-Let ABCD be a m, E a gn. pt. in the side AB; then it is required to bisect ABCD by a line drawn through E.

CONST.-Draw the diam. BC, and bisect it in F (Prop. X.) ; join EF, and produce it to G (Post. 1 and 2); then the ABCD is bisected by EG.

m

DEMONST.-Because AB is c

A

Ε

B

F

D

to CD, and BC meets them; .. the alternate EBF = alternate FCG (Prop. XXIX.), and the vertical 4 EBF = vertical / GCF (Prop. XV.), also BF = FC (Const.); .. the ▲ EFB = ▲ GCF. (Prop. XXVI.) 2. But ▲ ABC = ▲ BCD (Prop. XXXIV.); .., by subtraction, trapezium AEFC = trapezium BFGD (Ax. 3).

3. Hence, by addition, the trapezium AEGC = trapezium EBDG (AX. 1).

Wherefore them ABCD has been bisected by the line EG drawn through the gn. pt. E.-Q. E. F.

PROP. GG. THEOR.

GEN. ENUN.-The diameters of rectangular parallelograms are equal to one another; and in any other parallelogram the diameter which joins the acute angles is greater than that which joins the obtuse.

PART. ENUN.-Case 1. Let ABCD be a rectangular ",

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(Def. 10), .. the base AD = the base CB. (Prop. IV.) PART. ENUN.-Case 2. Let ABCD not be rectangular, as a rhombus or rhomboid, of which the opposite / ABD,

ACD are acute, and CAB, CDB, obtuse; then BC is > than AD. DEMONST.

For, as before,

the two AB, BD

BA, AC, each

to each; but the

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the base AD. (Prop. XXIV.)

Wherefore the diameters &c.-Q. E. D.

COR. Since the diams. of a □m bisect each other (Prop. EE), if the hypothenuse of a rt. Zd▲ be bisected, the st. line drawn from the rt. to the pt. of bisection is half the hypothenuse.

PROP. XXXV. THEOR.

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GEN. ENUN.-Parallelograms upon the same base, and between the same parallels, are equal to one another.

PART. ENUN.- Let the ms ABCD, EBCF (Fig. 2 and 3) be upon the same base BC, and between the same | AF, BC; then the

S

m

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S

DEMONST.-1. If the sides AD, DF of the ABCD, DBCF (Fig. 1), opposite to the base BC, terminate in the same pt. D, it is plain that each of them, being bisected by its diameter (Prop. XXXIV.), = 2 DBC; ▲ ... the m ABCD = DBCF (Ax. 6).

m

2. But if the sides AD, EF, opposite to the base BC (Figs. 2 and 3), be not terminated in the same pt.; then, by the property of a AD = BC = EF. (Prop. XXXIV.)

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m

3. To or from each of these equals add or subtract DE; .'. AD + DE (or AE) = EF + DE (or DF). Also AB = `DC.

4... in the two ▲ EAB, FDC, the two EA, AB = FD, DC, each to each, and the interior EAB = exterior FDC (Prop. XXIX.); .'. ▲ EAB = ▲ FDC. (Prop. IV.)

5. Take each of these s from the trapezium ABCF, and the remainders will be = (Ax. 3): i. e. the □m ABCD = m EBCF.

Wherefore

Q. E. D.

8

upon the same base, &c.—

The equality of ms is that of their areas; and it appears from this Proposition that any obliquem is equal to a rectangular, or rectangle, upon the same base and between the same |, i. e. of the same base and altitude. Now it is known that the area of a rectangle the base × its altitude; and .. generally the area of any m = the base the altitude. The converse of the Proposition is also true.

PROP. HH. THEOR.

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GEN. ENUN.-Equal parallelograms, upon the same base, are between the same parallels.

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m

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ms

B

G

ABCD, EBCG are upon the same, they are = to one

3. But the " ABCD =
EBCF, or the < = = >, which is impossible.
4. .. EG is not in the same st. line with AD.

m EBCF; .. them EBCG =

5. In like manner, it may be proved that no other line but EF is in the same st. line with AD; .. AD, EF are in the same st. line, and .. | with BC. (Prop. XXXIV.) Wherefore ms &c.-Q. E. D.

PROP. XXXVI. THEOR.

GEN. ENUN.-Parallelograms upon equal bases, and between the same parallels, are equal to one another.

A

D

E

PART. ENUN.-Let ABCD, EFGH be upon equal bases BC, FG, and between the same ||S AH, BG; then the m ABCD

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DEMONST.-1. Because BC = FG, and FG = EH, .'. BC = EH (Ax. 1).

2. But BC, Eн are ||, and joined towards the same parts by BE, CH; .. BE and CH are also and (Prop. XXXI.); and .'. EBCH is a m. (Prop. XXXIV. Def.)

3. Now, because the □ms ABCD, EBCH are upon the same base BC, and between the same | AH, BC, ... □m ABCD = (Prop. XXXV.)

S

4. For the same reason, EBCH;.'. □ ABCD = □

m

m

EBCH.

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EFGH (Ax. 1).

Whereforems &c.-Q. E. D.

The converse of this Proposition may be demonstrated in the same manner as that of the foregoing; and it is left to the student to shew, that equalms, upon = bases in the same straight line, are between the same || ". (See also Prop. XL.) The following are deductions :

PROP. KK. THEOR.

GEN. ENUN.-If two sides of a trapezium are parallel, its area is equal to half that of a parallelogram, between the same parallels, whose base is equal to the two parallel sides taken together.

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