THEOREM 45. [EUCLID III. 29.]

In equal circles, chords which cut off equal arcs are equal.

Let ABC, DEF be equal circles whose centres are G and H; and let the arc BKC the arc ELF.

=

It is required to prove that the chord BC= the chord EF.

Join BG, EH.

Proof. Apply the OABC to the ODEF, so that G falls on H and GB along HE.

Then because the circles have equal radii,

.. B falls on E, and the ces coincide entirely.

And because the arc BKC = the arc ELF,

.. C falls on F.

.. the chord BC coincides with the chord EF;

.. the chord BC= the chord EF.

Q.E.D.

EXERCISES ON ANGLES IN A CIRCLE.

1. P is any point on the arc of a segment of which AB is the chord. Shew that the sum of the angles PAB, PBA is constant.

2. PQ and RS are two chords of a circle intersecting at X: prove that the triangles PXS, RXQ are equiangular to one another.

3. Two circles intersect at A and B; and through A any straight line PAQ is drawn terminated by the circumferences: shew that PQ subtends a constant angle at B.

4. Two circles intersect at A and B; and through A any two straight lines PAQ, XAY are drawn terminated by the circumferences; shew that the arcs PX, QY subtend equal angles at B.

X5. P is any point on the arc of a segment whose chord is AB: and

the angles PAB, PBA are bisected by straight lines which intersect at O. Find the locus of the point O.

6. If two chords intersect within a circle, they form an angle equal to that at the centre, subtended by half the sum of the arcs they cut off.

7. If two chords intersect without a circle, they form an angle equal to that at the centre subtended by half the difference of the arcs they cut off.

8. The sum of the arcs cut off by two chords of a circle at right angles to one another is equal to the semi-circumference.

9. If AB is a fixed chord of a circle and P any point on one of the arcs cut off by it, then the bisector of the angle APB cuts the conjugate arc in the same point for all positions of P.

10. AB, AC are any two chords of a circle; and P, Q are the middle points of the minor ares cut off by them; if PQ is joined, cutting AB in X and AC in Y, shew that AX=AY.

11. A triangle ABC is inscribed in a circle, and the bisectors of the angles meet the circumference at X, Y, Z. Shew that the angles of the triangle XYZ are respectively

12. Two circles intersect at A and B; and through these points lines are drawn from any point P on the circumference of one of the circles shew that when produced they intercept on the other circum ference an arc which is constant for all positions of P.

13. The straight lines which join the extremities of parallel chords in a circle (i) towards the same parts, (ii) towards opposite parts, are equal.

14. Through A, a point of intersection of two equal circles, two straight lines PAQ, XAY are drawn: shew that the chord PX is equal to the chord QY.

15. Through the points of intersection of two circles two parallel straight lines are drawn terminated by the circumferences: shew that the straight lines which join their extremities towards the same parts are equal.

16. Two equal circles intersect at A and B; and through A any straight line PAQ is drawn terminated by the circumferences: shew that BP BQ.

Shew

17. ABC is an isosceles triangle inscribed in a circle, and the bisectors of the base angles meet the circumference at X and Y. that the figure BXAYC must have four of its sides equal.

What relation must subsist among the angles of the triangle ABC, in order that the figure BXAYC may be equilateral?

18. ABCD is a cyclic quadrilateral, and the opposite sides AB, DC are produced to meet at P, and CB, DA to meet at Q: if the circles circumscribed about the triangles PBC, QAB intersect at R, shew that the points P, R, Q are collinear.

19. P, Q, R are the middle points of the sides of a triangle, and X is the foot of the perpendicular let fall from one vertex on the opposite side: shew that the four points P, Q, R, X are concyclic.

[See page 64, Ex. 2: also Prob. 10, p. 83.]

20.

Use the preceding exercise to shew that the middle points of the sides of a triangle and the feet of the perpendiculars let fall from the vertices on the opposite sides, are concyclic.

21. If a series of triangles are drawn standing on a fixed base, and having a given vertical angle, shew that the bisectors of the vertical angles all pass through a fixed point.

22. ABC is a triangle inscribed in a circle, and E the middle point of the arc subtended by BC on the side remote from A: if through E a diameter ED is drawn, shew that the angle DEA is half the difference of the angles at B and C.

TANGENCY.

DEFINITIONS AND FIRST PRINCIPLES.

1. A secant of a circle is a straight line of indefinite length which cuts the circumference at two points.

2. If a secant moves in such a way that the two points in which it cuts the circle continually approach one another, then in the ultimate position when these two points become one, the secant becomes a tangent to the circle, and is said to touch it at the point at which the two intersections coincide. This point is called the point of contact.

Since a secant can cut a circle at two points only, it is clear that a tangent can have only one point in common with the circumference, namely the point of contact, at which two points of section coincide. Hence we may define a tangent as follows:

3. A tangent to a circle is a straight line which meets the circumference at one point only; and though produced indefinitely does not cut the circumference.

Fig. 1.

Fig. 2.

Fig. 3.

4. Let two circles intersect (as in Fig. 1) in the points P and Q, and let one of the circles turn about the point P, which remains fixed, in such a way that Q continually approaches P. Then in the ultimate position, when Q coincides with P (as in Figs. 2 and 3), the circles are said to touch one another at P.

Since two circles cannot intersect in more than two points, two circles which touch one another cannot have more than one point in common, namely the point of contact at which the two points of section coincide. Hence circles are said to touch one another when they meet, but do not cut one another.

NOTE. When each of the circles which meet is outside the other, as in Fig. 2, they are said to touch one another externally, or to have external contact: when one of the circles is within the other, as in Fig. 3, the first is said to touch the other internally, or to have internal contact with it.

INFERENCE FROM DEFINITIONS 2 AND 4.

If in Fig. 1, TQP is a common chord of two circles one of which is made to turn about P, then when Q is brought into coincidence with P, the line TP passes through two coincident points on each circle, as in Figs. 2 and 3, and therefore becomes a tangent to each circle. Hence

Two circles which touch one another have a common tangent at their point of contact.