drawn through the centre of its inscribed circle, and a perpendicular

be drawn to this line from one of the other angles; the point of

intersection of the perpendicular, and the two points of contact of

the inscribed circle which are adjacent to the remaining angle, are in

the same straight line.

37. If from the three angles of any triangle three straight lines

be drawn to the points where the inscribed circle touches the sides ;

these lines shall intersect each other in the same point.

38. If three circles touch each other, two of which are equal;

the vertical angle of the triangle formed by joining the points of con-

tact, is equal to either of the angles at the base of the triangle, which

is formed by joining their centres.

39. If three equal circles touch each other; to compare the area

of the triangle formed by joining their centres with the area of the

triangle formed by joining the points of contact.

40. If four straight lines intersect each other, and form four

triangles; the circles which circumscribe them will pass through one

and the same point.

41. Having given the base and vertical angle of a triangle; to

determine the locus of the extremity of the line, which always bisects

the vertical angle, and is equal to half the sum of the sides contain-

ing that angle.

42. If from the extremities of the base of a triangle inscribed in

a circle, perpendiculars be drawn to the opposite sides, intersecting

a diameter which is perpendicular to the base; the segments of the

diameter intercepted between these points and a point in it, whose

distance from the base is equal to the lesser segment of the diameter

made by the base, will be to one another in the ratio of the sides of

the triangle.

43. If the exterior angle of a triangle be bisected by a straight

line which cuts the base produced; the square of the bisecting line

is equal to the difference of the rectangles of the segments of the base

and of the sides of the triangle.

SECTION VIII. Page 263.

1. If from the centre of a circle a line be drawn to any point in

the chord of an arc; the square of that line together with the rectangle

contained by the segments of the chord will be equal to the square

described on the radius.

2. If two straight lines in a circle cut each other at right angles ;

the sums of the squares of the two lines joining their extremities will

be equal.

3. If two points be taken in the diameter of a circle, equidistant

'from the centre; the sum of the squares of the two lines drawn from

these points to any point in the circumference will be always the

same.

4. If from any point in the diameter of a semicircle there be

drawn two straight lines to the circumference, one to its point of

bisection, and the other at right angles to the diameter; the squares

of these two lines are together double of the square of the semi-

diameter.

5. If a straight line be drawn at right angles to the diameter of

a circle, and be cut by any other line; the rectangle contained by the

segments of this cutting line, together with the square of that part of

the perpendicular line which is intercepted between it and the

diameter, is always of the same magnitude.

6. A straight line being drawn from the centre of a quadrant,

bisecting the arc and meeting a tangent drawn from one extremity :

if from any point in the bounding radius a line be drawn parallel to

the tangent; the sum of the squares of the segments of it, cut off by

the aforesaid line and by the circumference will be equal to the

square of the radius.

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

lines, one of which is perpendicular to a diameter, and the other

cuts the circle; the square of the perpendicular is equal to the

rectangle contained by the whole cutting line and the part without

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

the diameter.

8. If any straight line be drawn perpendicular to the diameter of

a given circle, and produced to cut any chord; the rectangle con-
tained by the segments of the diameter will be less or greater than

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

drawn to any point in the circumference, and meeting the perpen-

dicular; the rectangle contained by the segments which they cut off

from the perpendicular, will be equal to the rectangle contained by

the segments of the diameter.

23. If from the point of bisection and any other point in a given

arc of a circle, two parallel lines be drawn, the former terminated

by the circumference, the latter by the chord of the arc; the rect-

angle contained by these two lines will be equal to that contained by

the lines which join the latter point with each extremity of the given

arc.

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.

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