the circle in D, shew that the angle DCB will be equal to half the angle BAC.

30. If the line AD bisect the angle A in the triangle ABC, and BD be drawn without the triangle making an angle with BC equal to half the angle BAC, shew that a circle may be described about ABCD.

31. Two equal circles intersect in A, B: PQT perpendicular to AB meets it in T and the circles in P, Q. AP, BQ meet in R; AQ, BP in S; prove that the angle RTS is bisected by TP.

32. If the angle contained by any side of a quadrilateral and the adjacent side produced be equal to the opposite angle of the quadrilateral, prove that any side of the quadrilateral will subtend equal angles at the opposite angles of the quadrilateral.

33. If DE be drawn parallel to the base BC of a triangle ABC, prove that the circles described about the triangles ABC and ADE have a common tangent at A.

34. Describe a square equal to the difference of two given squares.

35. If tangents be drawn to a circle from any point without it, and a third line be drawn between the point and the centre of the circle, touching the circle, the perimeter of the triangle formed by the three tangents will be the same for all positions of the third point of contact.

36. If on the sides of any triangle as chords, circles be described, of which the segments external to the triangle contain angles respectively equal to the angles of a given triangle, those circles will intersect in a point.

37. Prove that if ABC be a triangle inscribed in a circle, such that BA=BC, and AA' be drawn parallel to BC, meeting the circle again in A', and A'B be joined cutting AC in E, BA touches the circle described about the triangle AEA'.

38. Describe a circle, cutting the sides of a given square, so that its circumference may be divided at the points of intersection into eight equal arcs.

39. A is the extremity of the diameter of a circle, O any point in the diameter. The chord which is bisected at O subtends a greater or less angle at A than any other chord through O, according as O and A are on the same or opposite sides of the centre.

40. Shew that the square on the tangent drawn from any point in the outer of two concentric circles to the inner equals the difference of the squares on the tangents, drawn from any point, without both circles, to the circles.

41. If from a point without a circle, two tangents PT PT', at right angles to one another, be drawn to touch the circle, and if from T any chord TQ be drawn, and from T" a perpendicular TM be dropped on TQ, then T'M=QM.

42. Find the loci:

(1) Of the centres of circles passing through two given points.

(2) Of the middle points of a system of parallel chords in a circle.

(3) Of points such that the difference of the distances of each from two given straight lines is equal to a given straight line.

(4) Of the centres of circles touching a given line in a given point.

(5) of the middle points of chords in a circle that pass through a given point.

(6) Of the centres of circles of given radius which touch a given circle.

(7) Of the middle points of chords of equal length in a circle.

(8) Of the middle points of the straight lines drawn from a given point to meet the circumference of a given circle.

43. If the base and vertical angle of a triangle be given, find the locus of the vertex.

44. A straight line remains parallel to itself while one of its extremities describes a circle. What is the locus of the ther extremity ?

45. A ladder slips down between a vertical wall and a horizontal plane: what is the locus of its middle point?

46. AB is the diameter of a circle; ACD is a chord produced to D, so that AC=CD. Find the locus of the point in which BC and the line joining D to the centre intersect.

47. ABC is a line drawn from a point A, without a circle, to meet the circumference in B and C. Tangents are drawn to the circle at B and C which meet in D. What is the locus of D?

48. Two circles intersect in the points A, B; any straight line CDEF is drawn cutting the circles in C, D, E, F; prove that AC intersects BD and AE intersects BF in points which lie on a circle passing through A and B.

49. The angular points A, C of a parallelogram ABCD move on two fixed straight lines OA, OC, whose inclination is equal to the angle BCD; shew that the points B, D will move on two fixed straight lines passing through 0.

50. On the line AB is described the segment of a circle, in the circumference of which any point C is taken If AC, BC be joined, and a point P taken in AC so that CP is equal to CB, find the locus of P.

51. Find the locus of the centre of the circles circumscribing two trapeziums into which a parallelogram is divided by any line equal to one of its shorter sides.

52. If a parallelogram be described having the diameter of a given circle for one of its sides, and the intersection of its diagonals on the circumference, shew that the extremity of each of the diagonals moves on the circumference of another circle of double the diameter of the first.

53. One diagonal of a quadrilateral inscribed in a circle is fixed, and the other of constant length. Shew that the sides will meet if produced on the circumference of a fixed circle.

BOOK IV. INTRODUCTORY REMARKS. 175

BOOK IV.

INTRODUCTORY REMARKS.

EUCLID gives in this Book of the Elements a series of Problems relating to cases in which circles may be described in or about triangles, squares, and regular polygons, and of the last-mentioned he treats of three only:

the Pentagon, or figure of 5 sides,

The student will find it useful to remember the following Theorems, which are established and applied in the proofs of the Propositions in this Book.

I. The bisectors of the angles of a triangle, square, or regular polygon meet in a point, which is the centre of the inscribed circle.

II. The perpendiculars drawn from the middle points of the sides of a triangle, square, or regular polygon meet in a point, which is the centre of the circumscribed circle.

III. In the case of a square, or regular polygon the inscribed and circumscribed circles have a common centre.

IV. If the circumference of a circle be divided into any number of equal parts, the chords joining each pair of consecutive points form a regular figure inscribed in the circle, and the tangents drawn through the points form a regular figure described about the circle.

PROPOSITION I. PROBLEM.

In a given circle to place a straight line equal to a given straight line, which is not greater than the diameter of the circle.

Let ABC be the given O, and D the given line, not greater than the diameter of the

.

It is required to place in the

ABC a st. line

=

D.

Draw EC, a diameter of ABC.

Then if EC=D, what was required is done.

But if not, EC is greater than D. From EC cut off EF=D, and with centre E and radius EF describe a O AFB, cutting ABC in A and B ; and join AE.

the

Then, . E is the centre of ○ AFB,

.. EA = EF,

and.. EA=D.

Thus a st. line EA equal to D has been placed in ABC.

Q. E. F.

Ex. Draw the diameter of a circle, which shall pass at

a given distance from a given point.