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the rectangle contained by the segments of the chord, by the square

of the line intercepted between them, according as it is drawn without

or within the circle.

9. If a diameter of a circle be produced to bisect a line at right

angles, the length of which is the double of a mean proportional

between the whole line through the centre and the part without the

circle; and from any point in the double of the mean proportional

a line be drawn cutting the circle; the sum of the squares of the

segments of the double mean proportional will be equal to twice the

frectangle contained by this cutting line and the part without the

circle.

10. If from a point without a circle two straight lines be drawn,

one through the centre to the circumference, and the other perpen-

dicular to it, and on the former a mean proportional be taken

between the whole line and the part without the circle ; any other

line passing through that extremity of the mean proportional which is

within the circle, and terminated by the circumference and perpen-

dicular, will be similarly divided.

11. If a chord be drawn parallel to the diameter of a circle, and

from any point in the diameter lines be drawn to its extremitiés ; the

sum of their squares will be equal to the sum of the squares of the

segments of the diameter.

12. If through a point within or without a circle, two straight

lines be drawn at right angles to each other, and meeting the cir-

cumference; the squares of the segments of them are together equal

to the square of the diameter.

13. If from a point without a circle there be drawn two straight

lines, one of which touches the circle and the other cuts it, and

from the point of contact a perpendicular be drawn-to the diameter;

the
square

of the line which touches the circle is equal to the square

of that part of the cutting line which is intercepted by the perpen-

dicular, together with the rectangle contained by the segments of

that part of it which is within the circle.

14. A straight line drawn from the concourse of two tangents to

the concave circumference of a circle is divided harmonically by the

convex circumference and the chord which joins the points of

contact.

15. If from the extremities of any chord in a circle straight lines

be drawn to any point in the circumference meeting a diameter per-

pendicular to the chord; the rectangle contained by the distances of

their points of intersection from the centre is equal to the square

described upon the radius.

16. If from any point in the base, or base produced, of the seg-

ment of a circle, a line be drawn making therewith an angle equal to

the angle in the segment, and from the extremity of the base any line

be drawn to the former, and cutting the circumference; the rectangle

contained by this line and the part of it within the segment is always

of the same maguitude.

17. To determine the locus of the extremities of any number of

straight lines drawu from a given point, so that the rectangle con-

tained by each and a segment cut off from each by a line given in

position may be equal to a given rectangle.

18. If from a given point two straight lines be drawn containing

a given angle, and such that their rectangle may be equal to a given

rectilineal figure, and one of them be terminated by a straight line

given in position; to determine the locus of the extremity of the

other.

19. If from the vertical angle of a triangle two lines be drawn to

the base making equal angles with the adjacent sides; the squares of

those sides will be proportional to the rectangles contained by the
adjacent segments of the base.

20. If a line placed in one circle be made the diameter of a

second, the circumference of the latter passing through the centre of

the former, and any chord in the former circle be drawn through this

diameter perpendicularly; the rectangle contained by the segments

made by the circumference of the latter circle will be equal to that

contained by the whole diameter and a mean proportional between

its segments.

21. If semicircles be described on the segments of the base made

by a perpendicular drawn from the right angle of a triangle; they

will cut off from the sides, segments which will be in the triplicate

ratio of the sides.

22. If from any point in the diameter of a semicircle a perpen-
dicular be drawn, and from the extremities of the diameter lines be

24. If two circles cut each other, and from either point of inter-

section lines be drawn meeting both circumferences; the rectangles

contained by the segments of these lines are to one another in the

ratio of the perpendiculars drawn from their intersection with the

inner circumferences upon the line joining the intersections of the

circles.

25. If on opposite sides of any point in the chord of a circle, two

lines be taken, one terminating in the chord the other in the chord

produced, whose rectangle is equal to that contained by the segments

of the chord ; and the extremities of the lines so taken be joined to

those of any other chord passing through the same point; the line

joining their intersections of the circle will be parallel to the first

chord.

26. If from two points without a circle two tangents be drawn,

the sum of the squares of which is equal to the square of the line

joining those points; and from one of them a line be drawn cutting

the circle, and two lines from the other point to the intersections with

the circumference; the points in which these two lines cut the circle,

are in the same straight line with the former point.

27. If from the vertex of iangle there be drawn a line to any

point in the base, from which point lines are drawn parallel to the

sides; the sum of the rectangles of each side and its segment adja-

cent to the vertex will be equal to the square of the line drawn from

the vertex, together with the rectangle contained by the segments of

the base.

28. If on the chord of a quadrantal arc a semicircle be described ;

the area of the lune so formed will be equal to the area of the triangle

formed by the chord and terminating radii of the quadrant.

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