Take P any point in the arc APB, and join OP.

Take OQ a radius on the other side of AOB, and making with
OB the angle BOQ equal to BOP.
I. Prob. 5.

Then, if APB be turned about AB till it lies on the same side of AB with AQB, and in the same plane with it,

OP will fall along OQ, since the angle BOP is equal to the angle BOQ,

and P will fall on Q, since OP is equal to OQ.

Hence every point in the arc APB will coincide with some point in the arc AQB.

In like manner every point in the arc AQB may be made to coincide with some point in the arc APB.

Therefore the figure APB may be made to coincide with the figure AQB, and therefore is identically equal to it.

Q.E.D.

COR. Any two diameters at right angles to one another divide the circle into four identically equal parts, called quadrants.

Any line which divides a figure into two parts which are identically equal, so that if one part be turned about the line it will come to coincide with the other, is said to divide the figure symmetrically, and the line is said to be an axis of symmetry.

Hence Theor. 2 may be thus expressed:

A circle is symmetrical with respect to any one of its diameters; or, Every diameter of a circle is an axis of symmetry.

Ex. 4. Shew that the bisector of the vertical angle of an isosceles triangle is an axis of symmetry of the triangle.

Ex. 5. Shew that a rhombus is symmetrical with respect to each of its diagonals.

Ex. 6. If a parallelogram is symmetrical with respect to one of its diagonals, it must be a rhombus.

Ex. 7. If a quadrilateral is symmetrical with respect to each of its diagonals, it must be a rhombus.

*Ex. 8. Prove that if a curve is symmetrical with respect to every axis through a given point, it is a circle whose centre is that point.

THEOR. 3. Circles of equal radii are identically equal.

Let DEF, HKL be two circles of equal radii :

then shall the circle DEF be identically equal to the circle HKL.

Let A, B be the centres of the circles DEF, HKL, respectively.

Place the circle DEF upon the circle HKL, so that the centre A may coincide with the centre B.

Then any point on the circumference of DEF will fall on the circumference of HKL, since its distance from the centre of HKL is a radius of the circle DEF, and therefore equal to the radius of HKL. III. 1, Cor In like manner, any point on the circumference of HKL will fall on the circumference of DEF.

Hence the two circumferences coincide,

and therefore the circles coincide,

and therefore are identically equal.

Q.E.D.

COR. I. If two circles coincide, and one of them is turned through any angle about their common centre, they will continue to coincide.

COR. 2. Concentric circles of unequal radii cannot meet

COR. 3. Two circles whose circumferences meet one another cannot be concentric.

EXERCISES.

*9. If from any point within a circle two straight lines be drawn making equal angles with the diameter through the point and on opposite sides of the diameter, the two straight lines are equal.

το. Construct an equilateral triangle having two of its vertices on a given circle, and the third at a given point within the circle.

II. Construct an isosceles triangle of given vertical angle, having its vertex at a given point within the circle and the extremities of its base on the circumference.

12. The lines joining any fixed point to any number of points on the circumference of a given circle are bisected. Shew that the middle points all lie on the circumference of another fixed circle.

13. If lines are drawn to meet a fixed circle from an external point, prove that the vertices of equilateral triangles described upon them as bases will be on two fixed circles.

Enunciate and prove a similar theorem as regards squares.

14. A point is taken outside a fixed circle at a distance from the centre not greater than three times the radius. Shew that a straight line may be drawn through the point to cut the circle in such a way that the part of the line between the point and the circle is equal to the part intercepted by the circle.

15. If two straight lines be drawn through any point on a diagonal of a square parallel to the sides of the square, the points where these lines meet the sides will lie on the circumference of a circle whose centre is at the intersection of the diagonals

SECTION II.

ANGLES AT THE CENTRE AND SECTORS.

DEF. 4 An arc is a part of a circumference. Two arcs, which together make the whole circumference, are said to be conjugate. The greater of the two is called the major conjugate arc and the smaller the minor conjugate arc.

DEF. 5. The conjugate angles formed at the centre of a circle by two radii are said to stand upon the coniugate arcs opposite them intercepted by the radii, the major angle upon the major arc, and the minor angle upon the minor arc.

DEF. 6. A sector is a figure contained by an arc and the radii drawn to its extremities. The angle of the sector is the angle

at the centre which stands upon the arc of the sector.

THEOR. 4. In the same circle, or in equal circles, equal angles at the centre stand on equal arcs, and of two unequal angles at the centre the greater angle stands on the greater

arc.

In the equal circles DEF, HKL, of which A and B are the respective centres, let the angle DAE be equal to the angle HBK: