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