alizes theorems or relieves them from dependence upon the particularities of a diagram. Thus the two figures of Art. 100° differ in that in the first figure the secants intersect without the circle, and in the second figure they intersect within, while the theorem applies with equal generality to both. The Principle of Continuity may be stated as follows :— When a figure, which involves or illustrates some geometric property, can undergo change, however small, in any of its parts or in their relations without violating the conditions upon which the property depends, then the property is continuous while the figure undergoes any amount of change of the same kind within the range of possibility.

105°. Let AB be a chord dividing the O into unequal arcs, and let P and Q be any points upon

the major and minor arcs respectively.
(102°, Def.)

Let O be the centre.

I. The radii OA and OB form two

angles at the centre, a major angle A denoted by a and a minor angle de

noted by ẞ. These together make up a circumangle.

C

B

2. The chords PA, PB, and QA, QB form two angles at the circle, of which APB is the minor angle and AQB is the major angle.

3. The minor angle at the circle, APB, and the minor angle at the centre, ß, stand upon the minor arc, AQB, as a base. Similarly the major angles stand upon the major arc as base.

4. Moreover the ▲APB is said to be in the arc APB, so that the minor angle at the circle is in the major arc, and the major angle at the circle is in the minor arc.

5. When B moves towards B' all the minor elements increase and all the major elements decrease, and when B comes to B' the minor elements become respectively equal to the major, and there is neither major nor minor.

When B, moving in the same direction, passes B', the elements change name, those which were formerly the minor becoming the major and vice versa.

106°. Theorem.-An angle at the circle is one-half the corresponding angle at the centre, major corresponding to major and minor to minor.

The theorem is thus proved for the minor angles. But since the limiting conditions require only an angle at the circle and an angle at the centre, the theorem remains true

while B moves along the circle. And when B passes B' the angle APB becomes the major angle at the circle, and the angle AOB minor becomes the major angle at the

centre.

.'.

the theorem is true for the major angles.

Cor. 1. The angle in a given arc is constant.

(105°, 4)

Cor. 2. Since

and

LAPBLAOB minor,
LAQB LAOB major,

(37°)

LAPB+LAQB = a straight angle.

and LAOB minor + LAOB major=4 right angles

...

And APBQ is a concyclic quadrangle.

Hence a concyclic quadrangle has its opposite internal

angles supplementary.

Cor. 3. D being on AQ produced,

(40°, Def. 1)

LBQD is supplementary to LAQB.

But LAPB is supplementary to LAQB,

...

LAPB=4BQD.

Hence, if one side of a concyclic quadrangle be produced, the external angle is equal to A the opposite internal angle.

Cor. 4. Let B come to B'.

Then

..

(Fig. of 106°)

LAOB' is a straight angle,

LAPB' is a right angle.

But the arc APB' is a semicircle,

B

(102°, Def.)

Therefore the angle in a semicircle is a right angle.

107°. Theorem.—A quadrangle which has its opposite angles supplementary has its vertices concyclic.

(Converse of 106°, Cor. 2)

ABCD is a quadrangle whereof the LADC

is supplementary to LABC; then a circle can pass through A, B, C, and D.

A

Proof. If possible let the through A,

B, and C cut AD in some point P.

Join P and C.

B

Then LAPC is supplementary to LABC, (106°, Cor. 2)

LADC is supplementary to LABC,

.. the cannot cut AD in any point other than D, Hence A, B, C, and D are concyclic.

(hyp.)

(60°)

q.e.d.

Cor. 1. The hypothenuse of a right-angled triangle is the diameter of its circumcircle.

(88°, 3, Def.; 97°, Def.)

Cor. 2. When P moves along the ○ the ▲APC (last figure) has its base AC constant and its vertical angle APC constant. Therefore the locus of the vertex of a triangle which has a constant base and a constant vertical angle is an arc of a circle passing through the end-points of the base.

This property is employed in the trammel which is used to describe an arc of a given circle

E

It consists of two rules (16°) L and M joined at a

108°. Theorem.-The angle between two intersecting secants is the sum of those angles in the circle which stand

1. If a six-sided rectilinear figure has its vertices concyclic, the three alternate internal angles are together equal

to a circumangle.

2. In Fig. 105°, when B comes to Q, BQ vanishes; what is the direction of BQ just as it vanishes?

3. Two chords at right angles determine four arcs of which a pair of opposite ones are together equal to a semicircle.

4. A, B, C, D are the vertices of a square, and A, E, F of an

equilateral triangle inscribed in the same circle. What is the angle between the lines BE and DF? between BF and ED?

SPECIAL SECANTS--TANGENT.

109°. Let P be a fixed point on the OS and Q a variable

one.

The position of the secant L, cutting the circle in P and Q, depends upon the position of Q.

T

T'

As Q moves along the the secant rotates about P as pole. While Q makes one complete revolution along the the secant L passes through two special positions. The first of these is when Q is farthest distant from P, as at Q', and the secant L becomes a centreline. The second is when Q comes into coincidence with P, and the secant takes the position TT' and becomes a tangent.

Def. 1.- -A tangent to a circle is a secant in its limiting position when its points of intersection with the circle become coincident.

That the tangent cannot cut or cross the O is evident. For if it cuts the O at P it must cut it again at some other point. And since P represents two points we would have the absurdity of a line cutting a circle in three points.

(94°)

Def. 2.-The point where P and Q meet is called the point of contact. Being formed by the union of two points it represents both, and is therefore a double point.

From Defs. I and 2 we conclude

I. A point of contact is a double point.

2. As a line can cut a O only twice it can touch a O only once.

3. A line which touches a O cannot cut it.

4. A is determined by two points if one of them is a given point of contact on a given line; or, only one circle can pass through a given point and touch a given line at a given point. (Compare 97°.)