SECTION II. Page 24.

1.

IF

F a straight line be drawn to touch a circle, and be parallel

to a chord; the point of contact will be the middle point of the arc

cut off by that chord.

COR. 1. Parallel lines placed in a circle cut off equal parts of the

circumference.

COR. 2. The two straight lines in a circle which join the ex-

tremities of two parallel chords are equal to each other.

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

to the concave part of the circumference, making equal angles with

the line joining the same point and the centre, the parts of these lines

which are intercepted within the circle are equal.

3. Of all straight lines which can be drawn from two given points

to meet on the convex circumference of a given circle; the sum of

those two will be the least, which make equal angles with the tangent

at the point of concourse.

4. If a circle be described on the radius of another circle; any

straight line drawn from the point where they meet to the outer cir-

cumference, is bisected by the interior one.

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

section diameters be drawn; the extremities of these diameters and

the other point of intersection shall be in the same straight line.

6. If two circles cut each other, the straight line joining their

two points of intersection is bisected at right angles by the straight

line joining their centres.

7. To draw a straight line which shall touch two given circles.

8. If a line touching two circles cut another line joining their

centres, the segments of the latter will be to each other, as the dia-

meters of the circles.

9. If a straight line touch the interior of two concentric circles,

and be placed in the outer; it will be bisected at the point of

contact.

10. If any number of equal straight lines be placed in a circle;

to determine the locus of their points of bisection.

11. If from a point in the circumference of a circle any number

of chords be drawn; the locus of their points of bisection will be a

circle.

12. If on the radius of a given semicircle, another semicircle be

described, and from the extremity of the diameters any lines be

drawn cutting the circumferences, and produced, so that the part

produced may always have a given ratio to the part intercepted

between the two circumferences; to determine the locus of the ex-

tremities of these lines.

13. If from a given point without a given circle straight lines be

drawn and terminated by the circumference; to determine the locus

of the points which divide them in a given ratio.

14. Having given the radius of a circle; to determine its centre

when the circle touches two given lines which are not parallel.

15. Through three given points which are not in the same

straight line, a circle may be described; but no other circle can pass

through the same points.

16. From two given points on the same side of a line given

in position, to draw two straight lines which shall contain a given

angle, and be terminated in that line.

17. If from the extremities of any chord in a circle perpendi-

culars be drawn, meeting a diameter; the points of intersection are

equally distant from the centre.

18. If from the extremities of the diameter of a semicircle per-

pendiculars be let fall on any line cutting the semicircle; the parts

intercepted between those perpendiculars and the circumference are

equal.

19. In a given circle to place a straight line parallel to a given

straight line, and having a given ratio to it.

20. Through a given point, either within or without a given

circle, to draw a straight line, the part of which intercepted by the

circle shall be equal to a given line, not greater than the diameter of

the circle.

21. From a given point in the diameter of a semicircle produced,

to draw a line cutting the semicircle, so that lines drawn from the

points of intersection to the extremities of the diameter, cutting each

other, may have a given ratio.

22. From the circumference of a given circle, to draw to a straight

line given in position, a line which shall be equal and parallel to a

given straight line.

23. The bases of two given circular segments being in the same

straight line; to determine a point in it such, that a line being drawn

through it making a given angle, the part intercepted between the

circumferences of the circles may be equal to a given line.

24. If two chords of a given circle intersect each other, the

angle of their inclination is equal to half the angle at the centre

which stands on an arc equal to the sum or difference of the arcs

intercepted between them, according as they meet within or without

the circle.

25. If from a point without two circles which do not meet each

other, two lines be drawn to their centres, which have the same

ratio that their radii have; the angle contained by tangents drawn

from that point towards the same parts will be equal to the angle

contained by lines drawn to the centres.

26. To determine the Arithmetic, Geometric and Harmonic

means between two given straight lines.

27. If on each side of any point in a circle any number of equal

arcs be taken, and the extremities of each pair joined the sum of

the chords so drawn will be equal to the last chord produced to

meet a line drawn from the given point through the extremity of the

first arc.

28. If the circumference of a semicircle be divided into an odd

number of equal parts, and through the points which are equally

distant from the diameter lines be drawn; the segments of these

lines intercepted between radii drawn to the extremities of the most

remote, will together be equal to a radius of the circle.

29. If from the extremities and the point of bisection of any arc

of a circle, lines be drawn to any point in the opposite circumference;

the sum of those drawn from the extremities will have to that from

the point of bisection, the same ratio that the line joining the extre-

mities has to that joining one of them and the point of bisection.

30. If two equal circles cut each other, and from either point

of intersection a circle be described cutting them; the points where

this circle cuts them, and the other point of intersection of the equal

circles are in the same straight line.

31. If two equal circles cut each other, and from either point of

intersection a line be drawn meeting the circumferences; the part of

it intercepted between the circumferences will be bisected by the

circle whose diameter is the common chord of the equal circles.

32. If two circles touch each other externally or internally; any

straight line drawn through the point of contact will cut off similar

segments.

33. If two circles touch each other externally or internally; two

straight lines drawn through the point of contact will intercept arcs,

the chords of which are parallel.

34. If two circles touch each other externally or internally; any

two straight lines drawn through the point of contact, and terminated

both ways by the circumference, will be cut proportionally by the

circumference.

35. If two circles touch each other externally, and parallel

diameters be drawn; the straight line joining the extremities of these

diameters will pass through the point of contact.

36. If two circles touch each other and also touch a straight

line; the part of the line between the points of contact is a mean

proportional between the diameters of the circles.

37. If two circles touch each other externally, and the line join-

ing their centres be produced to the circumferences; and from its

middle point as a centre with any radius whatever a circle be de-

scribed, and any line placed in it passing through the point of contact;

the parts of the line intercepted between the circumference of this

circle and each of the others will be equal.

38. If from the point of contact of two circles which touch each

other internally, any number of lines be drawn; and through the

points, where these intersect the circumferences, lines be drawn from

any other point in each circumference, and produced to meet; the

angles formed by these lines will be equal.

39. If two circles touch each other internally, and any two per-

pendiculars to their common diameter be produced to cut the cir-

cumferences; the lines joining the points of intersection and the

point of contact are proportional.

40. If three circles, whose diameters are in continued proportion,

touch each other internally, and from the extremity of the least

diameter passing through the point of contact a perpendicular be

drawn, meeting the circumferences of the other two circles; this

diameter and the lines joining the points of intersection and contact

are in continued proportion.

41. If a common tangent be drawn to any number of circles

which touch each other internally, and from any point in this tangent

as a centre, a circle be described cutting the others, and from this

centre lines be drawn through the intersections of the circles respec-

tively; the segments of them within each circle will be equal.

42. If from any point in the diameter of a circle produced, a

tangent be drawn; a perpendicular from the point of contact to the

diameter will divide it into segments which have the same ratio that

the distances of the point without the circle from each extremity of

the diameter, have to each other.

43. If from the extremity of the diameter of a given semicircle

a straight line be drawn in it, equal to the radius, and from the centre

a perpendicular let fall upon it and produced to the circumference;

it will be a mean proportional between the lines drawn from the point

of intersection with the circumference to the extremities of the

diameter.

44. If from the extremity of the diameter of a circle, two lines

be drawn, one of which cuts a perpendicular to the diameter, and

the other is drawn to the point where the perpendicular meets the

circumference; the latter of these lines is a mean proportional between

the cutting line, and that part of it which is intercepted between the

perpendicular and the extremity of the diameter.

45. In the diameter of a circle produced, to determine a point,

from which a tangent drawn to the circumference, shall be equal to

the diameter.

46. To determine a point in the perpendicular at the extremity

of the diameter of a semicircle, from which if a line be drawn to the

b