(2) Required, to draw the transverse pair of common tangents.

Cons. and proof. With centre A and radius the sum of the radii of the two Os, describe a O, and complete the construction, and proof, as in (1).

E

EXERCISES.

THEOREMS.

1. If a straight line cut two concentric circles, the parts of it intercepted between the two circumferences are equal.

2. If one circle touch another internally at P, prove that the straight line joining the extremities of two parallel diameters of the circles, towards the same parts, passes through P.

3. In Ex. 2, if a chord AB of the larger circle touches the smaller one at C, prove that PC bisects the angle APB.

4. If two circles touch externally at P, prove that the straight line joining the extremities of two parallel diameters, towards opposite parts, passes through P.

5. Two circles with centres A and B touch each other externally, and both of them touch another circle with centre O internally show that the perimeter of the triangle AOB is equal to the diameter of the third circle.

6. In two concentric circles any chord of the outer circle which touches the inner, is bisected at the point of contact. 7. If three circles touch one another externally in P, Q, R, and the chords PQ, PR of two of the circles be produced to meet the third circle again in S, T, prove that ST is a diameter.

8. Points P, Q, R on a circle, whose centre is 0, are joined; OM, ON are drawn perpendicular to PQ, PR respectively join MN, and show that if the angle OMN

is greater than ONM, then the angle PRQ is greater than PQR.

9. A circle is described on the radius of another circle as diameter, and two chords of the larger circle are drawn, one through the centre of the less at right angles to the common diameter, and the other at right angles to the first through the point where it cuts the less circle. Show that these two chords have their greater segments equal to each other and their less segments equal to each other.

10. O is the centre of a circle, P is any point in its circumference, PN a perpendicular on a fixed diameter: show that the straight line which bisects the angle OPN always passes through one or the other of two fixed points on the circumference.

11. Two tangents are drawn to a circle at the opposite extremities of a diameter, and intercept from a third tangent a portion AB: if C be the centre of the circle show that ACB is a right angle.

12. A straight line touches a circle at A, and from any point P, in the tangent, PB is drawn meeting the circle at B so that PB is equal to PA: prove that PB touches the circle.

13. OC is drawn from the centre O of a circle perpendicular to a chord AB: prove that the tangents at A, B intersect in OC produced.

14. TA, TB are tangents to a circle, whose centre is 0; from a point P on the circumference a tangent is drawn cutting TA, TB, or those produced, in C, D: prove that the angle COD is half the angle AOB.

15. AB is the diameter and C the centre of a semicircle: show that O, the centre of any circle inscribed in the semicircle, is equidistant from C and from the tangent to the semicircle parallel to AB.

16. If from any point without a circle straight lines be drawn touching it, the angle contained by the tangents is double the angle contained by the straight line joining the

points of contact and the diameter drawn through one of them.

17. C is the centre of a given circle, CA a radius, B a point on a radius at right angles to CA; join AB and produce it to meet the circle again at D, and let the tangent at D meet CB produced at E: show that BDE is an isosceles triangle.

18. Let the diameter BA of a circle be produced to P, so that AP equals the radius; through A draw the tangent AED, and from P draw PEC touching the circle at C and meeting the former tangent at E; join BC and produce it to meet AED at D: then will the triangle DEC be equilateral.

Let O be the centre of the given . Produce OC to a pt. F so that CF = compare as PCO, PCF, etc.

CO:

19. APB is a fixed chord passing through P, a point of intersection of two circles AQP, PBR; and QPR is any other chord of the circles passing through P: show that AQ and RB when produced meet at a constant angle.

20. Two circles whose centres are A and B touch externally at C; the common tangent at C meets another common tangent DE at F: prove that (1) CF, DF, FE are equal; (2) each of the angles AFB, DCE is a right angle; (3) DE touches the circle described on AB as diameter.

21. The diagonals AC, BD of a quadrilateral ABCD inscribed in a circle intersect at right angles at P: prove that the straight line drawn from P to the middle point of one of the sides of the quadrilateral is perpendicular to the opposite sides.

Bisect AB in E, produce EP to meet CD in F, etc.

22. If a side of a quadrilateral inscribed in a circle be produced, the exterior angle is equal to the interior and opposite angle: and conversely, if the exterior angle of a quadrilateral made by any side and the adjacent side produced be equal to the interior and opposite angle, a circle can be described about the quadrilateral.

23. ABCD is a quadrilateral inscribed in a circle with centre O. If the angles BAD, BOD are together equal to two right angles, prove that the angle BCD is two-thirds of two right angles.

24. If two pairs of opposite sides of a hexagon inscribed in a circle are parallel, the third pair of opposite sides are parallel.

See (242).

25. A circle is described passing through the ends of the base of a given triangle: prove that the straight line joining the points, in which it cuts the sides or sides produced, is parallel to a fixed straight line.

26. In the semicircle ABCDE, the chord BD which is parallel to the diameter AE bisects the radius OC at right angles: prove that the arc BC is double the arc AB.

27. The straight lines which bisect the vertical angles of all triangles on the same base, on the same side of it, and with the same vertical angle, are concurrent.

28. AB, AC are chords of a circle: show that the straight line, which joins the middle points of the arcs AB, AC, cuts off equal portions of the chords.

Let D, E be the mid, pts. of arcs AB, AC; take D, E on opp. sides of diam. through A, etc.

29. BAC, BA'C are two angles in the same segment of a circle; AP, A'P' are drawn making the angles BAP, BA'P' equal to the angles BCA, BCA' respectively: prove that AP is parallel to A'P'.

30. CD is a chord of a circle at right angles to the diameter AB; E is any point in the arc BC; AE cuts CD in F: prove that the angles DFE, ACE are equal.

31. Prove that two of the straight lines which join the ends of two equal chords are parallel, and that the other two are equal.

Let the equal chds. AB, CD be joined towards the same parts by BC, AD; towards opp. parts by AC, BD; ... chd. AB = chd. CD, etc.

32. ABCD is a quadrilateral inscribed in a circle. Find the relation between the sides in order that AC the angle BAD, and BD bisect the angle ABC.

ZDAC CAB,... arc DC = arc CB, ... CB =CD, etc.

may bisect

33. Two circles touch internally at O, and a line is drawn cutting one of the circles in P, P', the other in Q, Q': show that PQ, P'Q' subtend equal angles at O.

Draw tang. OB, B being towards P, Q, etc.

34. Two circles touch externally at A; the tangent at B to one of them cuts the other in C, D: prove that BC and BD subtend supplementary angles at A.

Draw common tang. AE meeting BC in E, EAB = 2 EBA, etc.

35. If the opposite pairs of sides of a quadrilateral in a circle be produced to meet, and the angles so formed be bisected, the bisectors are at right angles to each other.

Let ABCD be a cyclic quadl.; let AD, BC meet in E, and AB, DC in F, etc.

36. If from any point on the circumference of the circle circumscribing a triangle, perpendiculars be drawn to the sides, show that the feet of these perpendiculars are collinear (164).

Let P be any pt. in the O ce of the circumscribing the ▲ ABC. From P draw PL to BC, PN 1 to AB, etc.

37. In any triangle ABC, if O be the orthocentre (171), and L, M, N the feet of the perpendiculars, the circle described through L, M, N will (1) bisect OA, OB, OC, and (2) will also pass through the middle points D, E, F of the sides of the triangle.

LOCI.

38. Find the locus of a point at a given radial distance. from the circumference of a given circle.

See (157).

39. On a given base as hypotenuse right triangles are described: find the locus of their vertices.

See (240).