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By adding a2 to both members, and transposing c', we have

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If in any triangle a line be drawn from any angle to the middle of the opposite side, twice the square of this line, together with twice the square of half the side bisected, will be equal to the sum of the squares of the other two sides.

Let ABC be a triangle, its base

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=a+x, and DB-a-x. Put AM=m.

Now by (th. 36) we have the two following equations:

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The two diagonals of any parallelogram bisect each other; and the sum of their squares is equal to the sum of the squares of all the

four sides of the parallelogram.

Let ABCD be any parallelogram, and D

draw its diagonals AC and BD.

We are now to show, 1st. That AE

=EC, DE=EB. 2d. That AC2+BD2 =AB2+BC2+DC2+AD2.

C

E

B

1. The two triangles ABE and DEC are equal, because AB -DC, the angle ABE the alternate angle EDC, and the vertical angles at E are equal; therefore, AE, the side opposite the angle ABE, is equal to EC, the side opposite the equal angle EDC: also EB, the remaining side of the one A is equal to ED, the remaining side of the other triangle.

2. As ADC is a triangle whose base AC is bisected in E, we have, by (th. 39),

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As ABC is a triangle whose Lase, AC, is bisected in E, we have

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By adding equations (1) and (2), and observing that

EB2=ED2, we have

4AE2+4ED2=AD2+DC2+AB2+BC2

But four times the square of the half of a line is equal to the square of the whole (scholium to th. 33); therefore 4AE2=AC2, and 4ED2=DB2; and by making the substitutions we have

AC+DB2=AD2+DC2+AB2+BC2. Q. E. D.

BOOK II.

PROPORTION.

THE word Proportion has different shades of meaning, accord ing to the subject to which it is applied: thus, when we say that a person, a building, or a vessel is well proportioned, we mean nothing more than that the different parts of the person or thing bear that general relation to each other which corresponds to our taste and ideas of beauty or utility, but in a more concise and geometrical

sense,

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

DEFINITIONS AND EXPLANATIONS.

In Geometry, the quantitities between which proportion can exist, are of three kinds, only. 1st. A line to a line. 2d. A surface to a surface. 3d. A solid to a solid.

To find the numerical relation which one quantity bears to another, we must refer them both to the same standard of measure. If a quantity, as A, be contained exactly

1

A

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B

C

D 1

-1

E

F

a certain number of times in another quantity, B, the quantity A is said to measure the quantity B; and if the same quantity, A, be contained exactly a certain number of times in another quantity, C, A is also said to be a measure of the quantity C, and it is called a common measure of the quantities B and C; and the quantities B and C will, evidently, bear the same relation to each other that the numbers do which represent the multiple that each quantity is of the common measure A.

Thus, if B contain A three times, and C contain A also three times, B and C being equimultiples of the quantity A, will be

equal to each other; and if B contain A three times, and C contain A four times, the proportion between B and C will be the same as the proportion between the numbers 3 and 4.

Again, if a quantity, D, be contained as often in another quantity, E, as A is contained in B, and as often in another quantity, F, as A is contained in C, the ratio of E to F, or the proportion between them, will be the same as the proportion between B and C; and in that case, the quantities B, C, E, and F, are said to be proportional quantities; a relation which is commonly expressed thus, B: C::E: F

To find the numerical relation that any quantity, as A, has to any other quantity of the same kind as B, we simply divide B by A, and the quotient may appear in the form of a fraction, thus: Now this fraction, or the value of this quotient, is always a numeral, whatever quantities may be expressed by A and B.

B

A

D

To find the numerical relation between D and E, we simply divide E by D, or write which denotes the division; and if we find the same quotient as when we divided B by A, then we may write

E'

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If B contains A three times, and D contains E three times, as we have just supposed, equation (1) is nothing more than saying

that

3=3

When we divide one quantity by another to find their numerical relation, the quotient thus obtained is called the ratio.

When the ratio between two quantities is the same as the ratio between two other quantities, the four quantities constitute a proportion.

N. B. On this single definition rests the whole subject of geometrical proportion.

On this definition, if we suppose that B is any number of times A, and D the same number of times E, then

Or more concisely :

A is to B as E is to D;

A: B=E:D. The signs:: meaning equal ratio.

Now it is manifest, that if E is greater than A, D will be greater than B. If A=E, then_B=D, &c., &c.; and whatever relation or ratio A is of E, the same ratio B will be of D; and whatever relation B is of A, the same relation D will be of E. This shows that the means may be changed, or made to change places.

Or,

A:E=B:D, which is the former pro

portion with the middle terms or means changed.

The first and third of four magnitudes are called the antecedents; the second and fourth, the consequents.

A simple relation or ratio exists between any two magnitudes of the same kind; but a proportion, in the full sense of the term, must consist of four quantities.

When the two middle quantities are equal, as,

A: B=B: C

then the three quantities, A, B, and C, are said to be continued proportionals; and B is said to be the mean proportional between A and C; and C is said to be the third proportional to A and B. In the proportion A: B-C: D, the last D is said to be the fourth proportional to A, B, and C.

By the same rule of expression, A may be called the first proportional, B the second, and C the third; for either one can be found when the other three are given, as we shall subsequently explain.

When quantities have the same constant ratio from one to the other, they are said to be in continued proportion,

Thus the numbers 1, 2, 4, 8, 16, &c., are in continued portion; the constant ratio from term to term being 2.

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If there be two magnitudes which have a common measure, x, so that the first magnitude may be expressed by mx, the second by nx; and two other magnitudes which have a common measure, y, so that the first may be expressed by my, the second by ny; that is, the two common measures x and y having the same equimultiples, m and n, to make up the magnitudes; then the four magnitudes will be in geometrical proportion.

Or

mx: nx=my: ny

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