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1. If 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

extremities 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

circumference, 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

diameters 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.

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

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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 eir-

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