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PROPOSITION XXXV.-THEOREM.

If two straight lines within a circle cut one another, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other.

LET the two straight lines a c, bd, within the circle abcd, cut one another in the point e; the rectangle contained by a e, ec is equal to the rectangle contained by be, e d

a

b

d

If a c, bd pass each of them through the centre, so that e is the centre, it is evident that ae, ec, be, ed being all equal, the rectangle a e, e c, is likewise equal to the rectangle be, ed.

But let one of them bd pass through the centre, and cut the other ac which does not pass through the centre, at right angles, in the point e: then, if b d be bisected in f, f is the centre of the circle abcd; join af. And because bd, which passes through

d

the centre, cuts the straight line a c, which does not pass through the centre, at right angles in e; a e, ec are equal (iii. 3) to one another and because the straight line bd is cut into two equal parts in the point f, and into two unequal in the point e, the rectangle be, ed, together, with the square of ef, is equal (ii. 5) to the square of fb; that is, to the square of fa: but the squares of a e, ef, are equal (i. 47) to the square of fa: therefore the rectangle be, ed, together with the square of ef, is equal to the squares of a e, ef: take away the common square of ef, and the remaining rectangle be, ed is equal to the remaining square of a e; that is, to the rectangle ae, e c.

b

Join

Next, let bd, which passes through the centre, cut the other ac, which does not pass through the centre, in e, but not at right angles : then, as before, if bd be bisected in f, f is the centre of the circle. af, and from f draw (i. 12) fg perpendicular to ac; therefore ag is equal (iii. 3) to gc; wherefore the rectangle a e, ec, together with the square of eg, is equal (ii. 5) to the square of a g: to each of these equals

d

f

add the square of gf; therefore the rectangle ae, ec, together with the squares of eg, gf, is equal to the squares of a g, gf: but the squares of eg, gf, are equal (i. 47) to the square of ef; and the squares of ag, gf, are equal to the square of af: therefore the rectangle a e, e c, C together with the square of ef, is equal to the square of af; that is, to the square of fb: but the square of fb is equal (ii. 5) to the rectangle be, ed, together with the square of ef; therefore the rectangle a e, ec, together with the square of ef, is equal to the rectangle be, ed, together with the square of ef: take away the common square of ef, and the remaining rectangle a e, e c, is therefore equal to the remaining rectangle be, ed.

مة

b

Lastly, let neither of the straight lines ac, bd pass through the centre: take the centre f and through e, the intersection of the straight lines ac, db, draw the diameter gefh: and because the rectangle a e, ec, is equal, as has been shewn, to the rectangle ge, eh; and, for the same reason, the rectangle be, ed is equal to the same rectangle ge, eh; therefore the rectangle a e, ec is equal to the rectangle be, ed. Wherefore, if two straight lines, &c. Q. E. D.

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PROPOSITION XXXVI.-THEOREM.

If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it, the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it.

LET d be any point without the circle a bc, and d cald b two straight lines drawn from it, of which dca cuts the circle, and db touches the same: the rectangle a d. dc is equal to the square of db.

Either d ca passes through the centre, or it does not; first, let it pass

through the centre e, and join eb; therefore the angle ebd is a right (iii. 18) angle; and because the straight line ac is bisected in e, and produced to the point d, the rectangle a d, dc, together with the square of e c, is equal (ii. 6) to the square of ed, and ce is equal to eb: therefore the rectangle ad, dc, together with the square of e b, is equal to the square of ed. But the square of ed is equal (i. 47) to the squares of eb, bd, because ebd is a right angle: therefore the rectangle ad, dc, together with the square of e b, is equal to the squares of e b, bd. Take away the common square of eb; therefore the remaining rectangle a d, dc, is equal to the square of the tangent db.

d

a

e

d

But if d ca does not pass through the centre of the circle abc, take (iii. 1) the centre e, and draw ef perpendicular (i. 12) to a c, and join eb, ec, ed: and because the straight line ef, which passes through the centre, cuts the straight line a c, which does not pass through the centre, at right angles, it shall likewise bisect (iii. 3) it; therefore af is equal to fc and because the straight line ac is bisected in f, and produced to d, the rectangle ad, dc, together with the square of fc, is equal (ii. 6) to the square offd. To each of these equals add the square of fe; therefore the rectangle a d, d c, together with the squares of cf, fe, is equal to the squares of df, fe; but the square of ed is equal

f/

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(i. 47) to the squares of df, fe, because efd is a right angle: and the
square of ec is equal to the squares of cf, fe; therefore the rectangle
ad, dc, together with the square of ec, is equal to the square of ed:
and ce is equal to eb; therefore the rectangle a d, dc, together with the

square of e b, is equal to the square of ed; but the
squares of eb, bd are equal to the square (i. 47)
of ed, because ebd is a right angle; therefore the
rectangle ad, dc, together with the square of e b,
is equal to the squares of eb, bd. Take away the
common square of eb; therefore the remaining
rectangle a d, dc is equal to the square of db.
Wherefore, if from any point, &c. Q. E. D.

COR. If from any point without a circle, there
be drawn two straight lines cutting it, as a b, a c,
the rectangles contained by the whole lines, and the
c parts of them without the circle, are equal to one
another, viz. the rectangle ba, a e, to the rectangle
ca, af: for each of them is equal to the square of
the straight line a d, which touches the circle.

PROPOSITION XXXVII.-THEOREM.

If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it; and if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square of the line which meets it, the line which meets shall touch the circle.

LET any point d be taken without the circle abc, and from it let two straight lines dca and db be drawn, of which dca cuts the circle, and db meets it; if the rectangle a d, dc be equal to the square of db; db touches the circle.

Draw (iii. 17) the straight line de touching the circle a bc, find its centre f, and join fe, fb, fd; then fed is a right (iii. 18) angle and because de touches the circle abc, and dca cuts it, the rectangle a d,

point, &c. Q. E. D.

e

dc is equal (iii. 36) to the square of de: but the
rectangle ad, dc is, by hypothesis, equal to the
square of db: therefore the square of de is equal
to the square of db; and the straight line de
equal to the straight line db; and fe is equal to
fb, wherefore de, ef are equal to db, bf; and
the base fd is common to the two triangles def,
dbf; therefore the angle def is equal (i. 8) to
the angle dbf; but def is a right angle, there-
fore also db fis a right angle: and fb, if produced,
is a diameter; and the straight line which is drawn
at right angles to a diameter, from the extremity
of it, touches (iii. 16) the circle: therefore bd
touches the circle abc. Wherefore, if from a

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EXERCISES ON BOOK III.

SECT. I.-PROBLEMS.

1. Given a circle, a square, and a point without the circle, to draw from the point a straight line cutting the circle, such that the rectangle contained by the part of the line within, and the part without the circle, shall be equal to the given square.

2. Given a circle and a point within it which is not the centre, to draw through that point a chord which will be bisected in the point.

3. Given a circle with its diameter produced to a given point, to find in the part produced a point, such that if a tangent be drawn from it to the circle, the tangent shall be equal to the part of the produced diameter lying between the point and the given point.

4. Given a circle with the diameter produced and a straight line, to draw a tangent from the diameter produced to the circle, equal to the given straight line.

5. Given two circles, to draw a straight line which shall touch them. 6. Given the vertical angle, the altitude and the sum of the three sides of a triangle, to construct the triangle.

7. Given the vertical angle, the altitude and the base of a triangle, to construct the triangle.

8. Given a segment of a circle and a straight line, to describe on the line another segment of a circle which shall be similar to the given segment.

9. Given a point, a straight line, and a point in the line, to describe a circle which shall pass through the first point, and touch the straight line in the other point.

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10. The line drawn from the centre of a circle perpendicular to any chord is also perpendicular to all chords parallel to the former.

11. If two parallel chords in a circle are bisected, the bisecting line is also perpendicular to them.

12. If from any two points in the circumference of the greater of two

concentric circles, chords are drawn so as to touch the circumference of the lesser circle, these chords shall be equal.

13. If two parallel chords are drawn in a circle, the arcs intercepted between their extremities shall be equal.

14. If any point be taken without a circle, there cannot be drawn from that point more than two equal straight lines to touch the circumference of the circle.

15. If an equilateral triangle be inscribed in a circle, and from a point in the circumference straight lines be drawn to the three angles of the triangle, the greatest of these lines is equal to the sum of the other two.

16. If a quadrilateral be described about a circle, the angles subtended by any two opposite sides of the figure at the centre of the circle shall be equal to two right angles.

17. If two circles cut each other, the straight line which joins their centres will bisect the straight line which joins the points of intersection.

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