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Book II.

PROP. XII. THEOR.

IN obtuse angled triangles, if a perpendicular be drawn from any of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle is greater than the squares of the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle between the perpendicular and the obtuse angle.

a

Let ABC be an obtuse angled triangle, having the obtuse angle ACB, and from the point A let AD be drawna 12. 1. perpendicular to BC produced: The square of AB is greater than the squares of AC, CB, by twice the rectangle BC, CD.

A

b 4.2.

Because the straight line BD is divided into two parts in the point C, the square of BD is equal to the squares of BC, CD, and twice the rectangle BC, CD: To each of these equals add the square of DA; and the squares of BD, DA, are equal to the squares of BC, CD, DA, and twice the rectangle BC, CD: But the square of BA is equal B to the squares of BD, DA, be

cause the angle at D is a right angle; and the square of CA is equal to the squares of CD, DA: Therefore the square of BA is equal to the squares of BC, CA, and twice the rectangle BC, CD; that is, the square of BA is greater than the squares of BC, CA, by twice the rectangle BC, CD. Therefore, in obtuse angled triangles, &c. Q. E. D.

• 47. 1.

Book II.

PROP. XIII. THEOR.

See N. IN every triangle, the square of the side subtending any of the acute angles, is less than the squares of the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle.

Let ABC be any triangle, and the angle at B one of its acute angles, and upon BC, one of the sides containing it, a 12. 1. let fall the perpendiculara AD from the opposite angle: The square of AC, opposite to the angle B, is less than the squares of CB, BA, by twice the rectangle CB, BD.

First, Let AD fall within the triangle ABC; and because the straight line CB is divided

into two parts in the point D,
7. 2. the squares of CB, BD are equalb
to twice the rectangle contained
by CB, BD, and the square of
DC: To each of these equals
add the square of AD; therefore
the squares of CB, BD, DA, are
equal to twice the rectangle CB, B
BD, and the squares of AD, DC:

A

47. 1. But the square of AB is equal to the squares of BD, DA, because the angle BDA is a right angle; and the square of AC is equal to the squares of AD, DC: Therefore the squares of CB, BA are equal to the square of AC, and twice the rectangle CB, BI), that is, the square of AC alone is less than the squares of CB, BA by twice the rectangle CB, BD.

Secondly, Let AD fall without the triangle ABC: Then, because the angle at D is a right angle, 16. 1. the angle ACB is greater than a right angle; and therefore the 12. 2. square of AB is equale to the squares of AC, CB, and twice the rectangleBC,CD: To these equals add the square of BC, and the B

C

A

squares of AB, BC are equal to the square of AC, and twice Book II. the square of BC, and twice the rectangle BC, CD: But because BD is divided into two parts in C, the rectangle DB, BC is equal to the rectangle BC, CD and the square 3.2. of BC: And the doubles of these are equal: Therefore the squares of AB, BC are equal to the square of AC, and twice the rectangle DB, BC: Therefore the square of AC alone is less than the squares of AB, BC, by twice the rectangle DB, BC.

Lastly, let the side AC be perpendicular to BC; then is BC the straight line between the perpendicular and the acute angle at B; and it is manifest, that the squares of AB, BC, are equals to the square of AC and twice the square of BC: Therefore, in every triangle, &c. Q. E.D.

A

47. 1.

PROP. XIV. PROB.

To describe a square that shall be equal to a given See N. rectilineal figure.

Let A be the given rectilineal figure; it is requi⚫ed to

describe a square

that shall be equal to A.

a

Describe the rectangular parallelogram BCDE equal 45. 1. to the rectilineal figure A. If then the sides of it, BE, ED,

[blocks in formation]

EF equal to ED, and bisect BF in G; and from the ceutre G, at the distance GB, or GF, describe the semicircle BHF, and produce DE to H, and join GH: Therefore because the straight line BF is divided into two equal parts in the point G, and into two unequal at E, the rectangle BE, EF, together with the square of EG, is equal to the 5.2. square of GF: But GF is equal to GH: Therefore the rectangle BE, EF, together with the square of EG, is equal

< 47. 1.

Book II. to the square of GH: But the squares of HE, EG are equale to the square of GH: Therefore the rectangle BE, EF, together with the square of EG, is equal to the squares of HE, EG: Take away the square of EG, which is common to both; and the remaining rectangle BE, EF is equal to the square of EH: But the rectangle contained by BE, EF is the parallelogram BD, because EF is equal to ED; therefore BD is equal to the square of EH; but BD is equal to the rectilineal figure A; therefore the rectilineal figure A is equal to the square of EH. Wherefore a square has been made equal to the given rectilineal figure A, viz. the square described upon EH. Which was to be done.

THE

39

ELEMENTS

OF

EUCLI D.

BOOK III.

DEFINITIONS.

I.

EQUAL circles are those of which the diameters are equal, Book III. or from the centres of which the straight lines to the circumferences are equal.

"This is not a definition, but a theorem, the truth of 'which is evident; for, if the circles be applied to one an' other, so that their centres coincide, the circles must like'wise coincide, since the straight lines from the centres are equal.'

6

II.

A straight line is said to

touch a circle, when it meets the circle, and being produced does not cut it.

III.

Circles are said to touch one another, which meet but do not cut one another. IV.

Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal. V.

And the straight line on which the greater perpendicular falls, is said to be further from the centre.

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