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the squares of BC, CA, by twice the rectangle BC, CD. Therefore, in obtuse-angled triangles, etc.

Q. E. D.

PROPOSITION XIII.

THEOR. 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, let fall the perpendicular (12.1.) 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, the squares of CB, BD are equal (7. 11.) to twice the rectangle contained by CB, BD, and the square of DC:

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To each of these equals add the square of AD: therefore the squares of CB, BD, DA are equal 3 (2 Ax.) to twice the rectangle CB, BD, and the squares of AD, DC: But the square of AB is equal (47. 1.) 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, BD; that is, square of AC alone is less than the squares of CB, BA, by twice the rectangle CB, BD.

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Secondly, let AD fall without the triangle ABC: then, because the angle at D is a right angle, the angle ACB is greater (16. 1.) than a right angle; and therefore the square of AB is equal (12. 11.) to the squares of AC, CB, and twice the rectangle BC, CD:

To these equals add the square of BC, and the squares of AB, BC are equal (2 Ax.) to the square of AC, and twice the square of BC, and twice the rectangle BC, CD:

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But because BD is divided into two parts in C, the rectangle DB, BC is equal (3. 11.) to the rectangle BC, CD and the square 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 A 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 equal (47. 1. and 2 Ax.) to the square of AC and twice the square of BC. Therefore, in every triangle, etc. Q. E. D.

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PROPOSITION XIV.

PROB. To describe a square that shall be equal to a given rectilineal

figure.

Let A be the given rectilineal figure it is required to describe a square that shall be equal to A.

Describe (45. 1.) a rectangular parallelogram BCDE equal to the rectilineal figure A.

If then the sides of it, BE, ED, are equal to one another, it is a (30 Def.) square, and what was required is now done :

But if they are not equal, produce one of them BE to F, and make (3. 1.) EF

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equal to ED, and bisect (10. 1.) BF in G; and from the centre G, at the distance GB, or GF, describe the semicircle BHF, and produce DE to H, and join GII:

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 (5. 11.) to the square of GF:

But GF is equal (15 Def.) to GH; therefore the rectangle BE, EF, together with the square of EG, is equal to the square of GH:

But the squares of HE, EG are equal (47. 1.) 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 (3 Ax.) 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 (Constr.) 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.

EXERCISES ON BOOK II.

1. (a.) The difference of the internal segments of a line, or the sum of the external, is double of the distance between the points of section and bisection.

(b.) Also, that if half the sum of two lines be increased by half their difference, the sum will be equal to the greater of them; and if diminished by half their difference, the remainder will be equal to the less of them.

2. If two lines be each of them divided into any number of parts, equal or unequal to one another, the rectangle under the two lines is equal to all the rectangles taken together, under each separate part of the one with each separate part of the other.

(a.) What will take place when all the divisions are equal?

3. Employ this to demonstrate 4 and 7, without first proving that the smaller squares are about the diagonal of the original square. Show likewise that those propositions are respectively equivalent to

these two:

(a.) The square on a line is greater than the squares described on its internal segments by twice the rectangle under those segments;

(b.) The square on a line is less than the squares on its external segments by twice the rectangle under those segments.

4. The difference of the squares on two lines are equal to the rectangle under their sum and difference.

Show that Props. 4 and 5 are cases of this property.

5. The squares on the sum and difference of two lines are equal to twice the sum of the squares on the lines themselves.

6. The square on the sum of two lines exceeds the square on their difference by four times the rectangle under the lines themselves.

7. If a straight line be drawn from the vertex of a triangle to the middle of the base, the sum of the squares on the two sides is double the square on that line, together with double the square on half the base.

8. Three times the sum of the squares on the three sides of a triangle is equal to four times the sum of the squares of the lines drawn from the angles to bisect the opposite sides.

9. The difference between the squares on two sides of a triangle is equal to the difference between the squares on the segments of the base made by a perpendicular from the opposite angle, whether the segments be internal or external.

10. The sum of the squares on the diagonals of a parallelogram is equal to the sum of the squares on its four sides.

11. Lines being drawn from any point to the angles of a rectangle : the squares on those drawn to the extremities of one diagonal are together equal to the squares on those drawn to the other extremity. 12. Let the four sides and two diagonals of any quadrilateral be bisected: then

(a.) The sum of the squares on the four sides of the quadrilateral will be equal to the sum of the squares on the two diagonals together with four times that of the line joining the middle points of the diagonals;

(b.) The sum of the squares on either pair of opposite sides, together with the squares on the diagonals, is equal to the sum of the squares on the other pair of opposite sides, together with four times the square of the line which joins their middle points.

13. If two opposite sides of a quadrilateral be parallel, the sum of the squares on its diagonals is equal to the squares on the two sides which are not parallel together with twice the rectangle under the parallel sides.

14. If a line be drawn from the vertex of an isosceles triangle to meet the base or the base produced, the square upon one side of the triangle is equal to the square upon this line, increased or diminished

respectively by the rectangle under the segments of the base, as the segments are internal or external.

15. The greatest rectangle which can be made by the segments of a given line is the square described upon half that line.

16. In the figure to I. 47, show that,

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BOOK III.

DEFINITIONS.

1. EQUAL circles are those of which the diameters are equal, 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 another, so that their centres coincide, the circles must likewise coincide, since the straight lines from the centres are equal."

2. A straight line is said to touch a circle when it meets the circle, and being produced, does not cut it.

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

4. 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.

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

6. A segment of a circle is the figure contained by a straight line and the circumference it cuts off.

7. "The angle of a segment is that which is contained by the straight line and the circumference."

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