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THEOREM XLIII.

The two diagonals of any parallelogram bisect each other; and the sum of their squares is equivalent to the sum of the squares of the four sides of the parallelogram.

Let ABCD be any parallelogram, and AC and BD its diagonals.

We are now to prove,

1st. That AE = EC, and DE

EB.

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2d. That AC2 + BD2 = AB2 + BC2 + CD2 + AD2. 1. The two triangles ABE and CDE are equal, because AB CD, the angle ABE the alternate angle CDE, and the vertical angles at E are equal; therefore, AE, the side opposite the angle ABE, is equal to CE, the side opposite the equal angle CDE; also EB, the remaining side of the one ▲, is equal to ED, the remaining side of the other triangle.

2. As ACD is a triangle whose base, AC, is bisected in E, we have, by (Th. 42),

2AE2 + 2ED2
2ED2 = AD2 + DO2 (1)

And as ACB is a triangle whose base, AC is bisected in E, we have

2AE2 + 2EB2 = AB2 + BC2 (2)

By adding equations (1) and (2), and observing that EB' ED', we have

2

=

4AE2 + 4ED2 = AD2 + DC2 + AB2 + BO2

But, four times the square of the half of a line is equiv alent to the square of the whole line, (Th. 36, Corollary); therefore 4AE = AC, and 4ED DB'; and by substituting these values, we have

=

AC2 + BD2 = AB2 + BC2 + DC2 + AD2,

which equation conforms to the enunciation of th theorem.

THEOREM XLIV.

If a line be bisected and produced, the rectangle contained by the whole line and the part produced, together with the square of one half the bisected line, will be equivalent to the square on a line made up of the part produced and one half the bisected line.

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ed. Also, complete the construction of the rectangle ADEK.

Then we are to prove that the rectangle, AE, and the square, GL, are together equivalent to the square, CDFG.

AE, will be

The two complementary rectangles, CL and LF, are equal, (Th. 31). But CL-AH, the line AB being bisected at C; therefore AL is equal to the sum of the two complementary rectangles of the square CF. To AL add the square BE, and the whole rectangle, equal to the two rectangles CE and EM. To each of these equals add HM, or the square on HL or its equal CB, and we have rectangle AE+ square HM CD but rectangle AE= AD × BD, and square HM CB3. Hence the theorem, etc.

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SCHOLIUM, If we represent AB by 2a, and BD by x, then AD= 2ax, and AD × BD = 2ax + x2. But CB a2; adding this equation to the preceding, member to member, we get AD × BD+ CB2: a2+2ax + x2= a+x2. But CD=a+x; hence this equation is equivalent to the equation AD × DB + CB2 = CD3, which is the algebraic proof of the theorem.

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THEOREM XLV.

If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the two unequal parts together with the square of the line between the points of division, will be equivalent to the square on one half the line.

Let AB be a line bisected in C, and divided into two unequal parts in D.

We are to prove

that AD × DB +

B

A

CD2 = AC2, or CB3.

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We see by inspection that AD AC+ CD, and BD AC-CD; therefore by (Th. 38), we have

AD × BD = AC2 — CD2,

By adding CD to each of these equals, we obtain

AD x BD + CD2 = AC2

Hence the theorem.

BOOK II.

PROPORTION.

DEFINITIONS AND EXPLANATIONS.

THE word Proportion, in its common meaning, denotes that general relation or symmetry existing between the different parts of an object which renders it agreeable to our taste, and conformable to our ideas of beauty or utility; but in a mathematical sense,

1. Proportion is the numerical relation which one quantity bears to another of the same kind.

As the magnitudes compared must be of the same kind, proportion in geometry can be only that of a line to a line, a surface to a surface, an angle to an angle, or a volume to a volume.

2. Ratio is a term by which the number which measures the proportion between two magnitudes is designated, and is the quotient obtained by dividing the one

B

by the other. Thus, the ratio of A to B is or A: B,

Α'

in which A is called the antecedent, and B the consequent. If, therefore, the magnitude A be assumed as the unit or standard, this quotient is the numerical value of B expressed in terms of this unit.

It is to be remarked that this principle lies at the foundation of the method of representing quantities by numbers. For example, when we say that a body weighs twenty-five pounds, it is implied that the weight of this body has been compared, directly or indirectly, with that of the standard, one pound. And so of geometrica

magnitudes; when a line, a surface, or a volume is said to be fifteen linear, superficial, or cubical feet, it is understood that it has been referred to its particular unit, and found to contain it fifteen times; that is, fifteen is the ratio of the unit to the magnitude.

When two magnitudes are referred to the same unit, the ratio of the numbers expressing them will be the ratio of the magnitudes themselves.

Thus, if A and B have a common unit, a, which is contained in A, m times, and in B, n times, then A = ma

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3. A Proportion is a formal statement of the equality of two ratios.

Thus, if we have the four magnitudes A, B, C and D, B D

uch that

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this relation is expressed by the pro

portion A: B:: C: D, or A: BC: D, the first of which is read, A is to B as Cis to D; and the second, the ratio of A to B is equal to that of C to D.

4. The Terms of a proportion are the magnitudes, or ore properly the representatives of the magnitudes compared.

5. The Extremes of a proport on are its first and fourth

Lerms.

6. The Means of a proportion are its second and third

Lerms.

7. A Couplet consists of the two terms of a ratio. The

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