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duced, 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.

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

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; therefore the squares of B BD, DA, are equal to the squares of BC, CD, DA, and twice the rectangle BC, CD: but the square of BA is equal* to the squares of BD, DA, * 47. 1. because the angle at D is a right angle; and the square of CA is equal* to the squares of ČD, DÀ; therefore * 47. 1. 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.

PROP. XIII. THEOR.

XII. In every triangle, the square of the side sübtending either See N.

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* AD from the op- * 12. t. posite angle: the square of AC, opposite to the angle B, shall be less than the squares of CB, BA, by twice the rectangle CB, BD.

First, let AD fall within the triangle ABC: and be cause the straight line CB is divided into two parts in

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the point D, the squares of CB, BD are equal * 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 equalt to twice the
rectangle CB, BD, and the squares of
AD, DC: 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; there-
fore the squares of CB, BA are equal to the square of
AC, and twice the rectangle CB, BD; 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, the angle ACB
is greater* than a right angle; and
therefore the square of AB is equal * to
the squares of AC, CB, and twice the
rectangle BC, CD: to each of these equals add the

square of BC; therefore the squares of AB, BC are + % Ax. equal to the square of AC, and twice 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 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 BC is the straight line between the perpendicular and the acute angle at B:

and it is manifest that the squares of AB, * 47.1. and BC, are equal* to the square of AC and

twice the square of BC: therefore in every
triangle, &c. Q. E. D.

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PROP. XIV. PROB.

See N.

To describe a square that shall be equal to a given recti

lineal

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

H

G

****** Describe* the rectangular parallelogram BCDE * 45. 1.

equal to the rectilineal figure A. Then if the sides of
it, BE, ED are equal to one
another, it is t a square, and

† 30 Def. what was required is now done: but if they are not equal, pro

BA duce one of them BE to F, and make + EF equal to ED, and

+ 3. 1. bisect + BF in G; and from the centre G, at the dis- 7 10. 1. tance GB, or GF, describe the semicircle BHF, and produce DE to H. The square described upon EH shall be equal to the given rectilineal figure A.

Join GH: and 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 square of GF: but GF * 5. 2. is equal + to GH: therefore the rectangle BE, EF, to- + 15 Def. gether with the square of EG, is equal to the square of GH: but the squares of HE, EG are equal* to the * 47. 1. square of GH: therefore the rectangle BE, EF, together with the square of EG, is equal t to the squares of + 1 Ax.

. 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 con- +3 Ax. tained 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; + Constr. therefore the rectilineal figure A is equal + to the square + 1 Ax. 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.

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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."

II.

A straight line is said to touch a cir

cle, 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 dis

tant 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 perpendi

cular falls, is said to be farther from the centre.

VI.

A segment of a circle is the figure contained

by a straight line and the circumference it cuts off.

VII.

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

VIII.

An angle in a segment is the angle con

tained by two straight lines drawn from any point in the circumference of the segment to the extremities of the straight line which is the base of the segment.

IX.

And an angle is said to insist or stand upon the cir

cumference intercepted between the straight lines that contain the angle.

X.

A sector of a circle is the figure contained

by two straight lines drawn from the centre, and the circumference between them.

XI.

Similar segments of circles are those

in which the angles are equal, or which contain equal angles.

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