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Def. 2.- A line which divides a figure into two parts such that when one part is revolved about the line it may be made to fall on and coincide with the other part is an axis of symmetry of the figure.

102°. Theorem.--A centre-line is an axis of symmetry of the circle.

Proof.—Let AB and CD be equal chords meeting at P, and let PHOG be a centre line.

Let the part of the figure which lies upon the F side of PG be revolved about PG until it comes to the plane on the E side of PG.

GPA=4GPC,

G

A

B

E

P

H

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D

D

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... PC coincides with PA.

Then

And

PB PD

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one arc.

And the arc HCG, coinciding in three points with the arc HAG, is equal to it, and the two arcs become virtually but (96°, Cor. 1) Therefore PG is an axis of symmetry of the O, and divides it into two equal arcs.

q.e.d.

Def.---Each of the arcs into which a centre-line divides the circle is a semicircle.

Any chord, not a centre-line, divides the circle into unequal arcs, the greater of which is called the major arc, and the other the minor arc.

Cor. 1. By the superposition of the theorem we see that arc AB=arc CD, arc HB=arc HD, arc GA=arc GC, arc BDCA=arc DBAC.

(1st Fig.)

But the arcs BDCA and AB are the major and minor arcs

to the chord AB, and the arcs DBAC and CD are major and minor arcs to the chord CD.

Therefore equal chords determine equal arcs, major being equal to major and minor to minor.

Cor. 2. Equal arcs subtend equal angles at the centre.

103°. Theorem.-Parallel secants intercept equal arcs on a

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F

AB is || to CD,

arc AC=arc DB.

Proof-Let EF be the centre-line

EF is to CD also.

to AB. (72°)

When EBDF is revolved about EF,

B comes to coincidence with A, and D with C, and the arc

BD with the arc AC,

arc AC arc DB.

Cor. Since the chord AC chord BD,

q.e.d.

Therefore parallel chords have the chords joining their end-points equal.

EXERCISES.

I. Any plane closed figure is cut an even number of times by an indefinite line.

2. In the figure of Art. 96°, if A, B, and C shift their relative positions so as to tend to come into line, what becomes of the point O?

3. In the same figure, if ABC is a right angle where is the point O?

4. Given a circle or a part of a circle, show how to find its

centre.

5. Three equal segments cannot be drawn to a circle from a point without it.

6. The vertices of a rectangle are concyclic.

7. If equal chords intersect, the segments of one between the

point of intersection and the circle are respectively equal to the corresponding segments of the other. 8. Two equal chords which have one end-point in common lie upon opposite sides of the centre.

9. If AB and CD be parallel chords, AD and BC, as also AC and BD, meet upon the right bisector of AB or CD. Io. Two secants which make equal angles with a centre-line

make equal chords in the circle if they cut the circle.

(Converse of 100°) II. What is the axis of symmetry of (a) a square, (b) a

rectangle, (c) an isosceles triangle, (d) an equilateral triangle? Give all the axes where there are more than

one.

12. When a rectilinear figure has more than one axis of symmetry, what relation in direction do they hold to one another?

13. The vertices of an equilateral triangle trisect its circumcircle.

14. A centre-line perpendicular to a chord bisects the arcs determined by the chord.

15. Show how to divide a circle (a) into 6 equal parts, (b) into 8 equal parts.

16. If equal chords be in a circle, one pair of the connecters of their end-points are parallel chords.

(Converse of 103°, Cor.)

THE PRINCIPLE OF CONTINUITY.

104°. The principle of continuity is one of the most prolific in the whole range of Mathematics.

Illustrations of its meaning and application in Geometry will occur frequently in the sequel, but the following are given by way of introduction.

1. A magnitude is continuous throughout its extent.

Thus a line extends from any one point to another without

breaks or interruptions; or, a generating point in passing from one position to another must pass through every intermediate position.

2. In Art. 53° we have the theorem-Every point on the right bisector of a segment is equidistant from the end-points of the segment.

In this theorem the limiting condition in the hypothesis is that the point must be on the right bisector of the segment.

Now, if P be any point on the right bisector, and we move P along the right bisector, the limiting condition is not at any time violated during this motion, so that P remains continuously equidistant from the end-points of the segment during its motion.

We say then that the property expressed in the theorem is continuous while P moves along the right bisector.

3. In Art. 97° we have the theorem-The sum of the internal angles of a quadrangle is four right angles.

The limiting condition is that the figure shall be a quadrangle, and that it shall have internal angles.

A

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rangle ABCD becomes the inverted quadrangle ABCD2, and the theorem remains true. Or, the theorem is continuously true while the vertex D moves anywhere in the plane, so long as the figure remains a quadrangle and retains four internal angles.

Future considerations in which a wider meaning is given to the word "angle" will show that the theorem is still true even when D, in its motion, crosses one of the sides AB or BC, and thus produces the crossed quadrangle.

The Principle of Continuity avoids the necessity of proving theorems for different cases brought about by variations in the disposition of the parts of a diagram, and it thus gener

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 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 B. These together make up
a circumangle.

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C

P

B

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 LAPB 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.

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