The First Six Books of the Elements of Euclid

13. The rectangle contained by the perpendiculars from any point in a circle, on the diagonals of an inscribed quadrilateral, is equal to the rectangle contained by the perpendiculars from the same point on either pair of opposite sides.

14. The rectangle contained by the sides of a triangle is greater than the square on the internal bisector of the vertical angle, by the rectangle contained by the segments of the base."

15. If through A, one of the points" of 'intersection of two circles, we draw any line ABC, cutting the circles again in B and C, the tangents at B and C intersect at a given angle.

16. If a chord of a given circle pass through a given point, the locus of the intersection of tangents at its extremities is a right line.

17. The rectangle contained by the distances of the point where the internal bisector of the vertical angle meets the base, and the point where the perpendicular from the vertex meets it from the middle point of the base, is equal to the square on half the difference of the sides.

18. State and prove the Proposition analogous to 17 for the external bisector of the vertical angle.

19. The square on the external diagonal of a cyclic quadrilateral is equal to the sum of the squares on the tangents from its extremities to the circumscribed circle.

20. If a variable circle touch a given circle and a given line, the chord of contact passes through a given point.

21. If A, B, C be three points in the circumference of a circle, and D, E the middle points of the arcs AB, AC; the if then line DE intersect the chords AB, AC in the points F, G, AF is equal to AG.

22. Given two circles, O, O'; then if any secant cut O in the points B, C, and O' in the points B', C', and another secant cuts them in the points D, E; D', E' respectively; the four chords BD, CE, B'D', C'E' form a cyclic quadrilateral.

23. If a cyclic quadrilateral be such that a circle can be inscribed in it, the lines joining the points of contact are perpendicular to each other.

24. If through the point of intersection of the diagonals of a cyclic quadrilateral the minimum chord be drawn, that point will bisect the part of the chord between the opposite sides of the quadrilateral.

25. Given the base of a triangle, the vertical angle, and either the internal or the external bisector of the vertical angle; construct it.

26. If through the middle point A of a given arc BAC we draw any chord AD, cutting BC in E, the rectangle AD. AE is constant.

27. The four circles circumscribing the four triangles formed by any four lines pass through a common point.

28. If X, Y, Z be any three points on the three sides of a triangle ABC, the three circles about the triangles YAZ, ZBX, XCY pass through a common point.

29. If the position of the common point in the last question be given, the three angles of the triangle XYZ are given, and conversely.

30. Place a given triangle so that its three sides shall pass through three given points.

31. Place a given triangle so that its three vertices shall lie on three given lines.

32. Construct the greatest triangle equiangular to a given one whose sides shall pass through three given points.

33. Construct the least triangle equiangular to a given one whose vertices shall lie on three given lines.

34. Construct the greatest triangle equiangular to a given one whose sides shall touch three given circles.

35. If two sides of a given triangle pass through fixed points, the third touches a fixed circle.

36. If two sides of a given triangle touch fixed circles, the third touches a fixed circle.

37. Construct an equilateral triangle having its vertex at a given point, and the extremities of its base on a given circle.

38. Construct an equilateral triangle having its vertex at a given point, and the extremities of its base on two given circles. 39. Place a given triangle so that its three sides shall touch three given circles.

40. Circumscribe a square about a given quadrilateral.

41. Inscribe a square in a given quadrilateral.

42. Describe circles-(1) orthogonal (cutting at right angles) to a given circle and passing through two given points; (2) orthogonal to two others, and passing through a given point; (3) orthogonal to three others.

43. If from the extremities of a diameter AB of a semicircle two chords AD, BE be drawn, meeting in C, AC . AD + BC .BE AB2.

44. If ABCD be a cyclic quadrilateral, and if we describe any circle passing through the points A and B, another through B and C, a third through C and D, and a fourth through D and A; these circle sintersect successively in four other points E, F, G, H, forming another cyclic quadrilateral.

45. If ABC be an equilateral triangle, what is the locus of the point M, if MA = MB + MC?

46. In a triangle, given the sum or the difference of two sides and the angle formed by these sides both in magnitude and position, the locus of the centre of the circumscribed circle is a right line.

47. Describe a circle-(1) through two given points which shall bisect the circumference of a given circle; (2) through one given point which shall bisect the circumference of two given circles.

48. Find the locus of the centre of a circle which bisects the circumferences of two given circles.

49. Describe a circle which shall bisect the circumferences of three given circles.

50. AB is a diameter of a circle; AC, AD are two chords meeting the tangent at B in the points E, F respectively prove that the points C, D, E, F are concyclic.

51. CD is a perpendicular from any point C in a semicircle on the diameter AB; EFG is a circle touching DB in E, CD in F, and the semicircle in G; prove-(1) that the points A, F, G are collinear; (2) that AC = AE.

52. Being given an obtuse-angled triangle, draw from the obtuse angle to the opposite side a line whose square shall be equal to the rectangle contained by the segments into which it divides the opposite side.

53. O is a point outside a circle whose centre is E; two perpendicular lines passing through O intercept chords AB, CD on the circle; then AB2+ CD2 + 40E2 = 8R2.

54. The sum of the squares on the sides of a triangle is equal to twice the sum of the rectangles contained by each perpendicular and the portion of it comprised between the corresponding vertex and the orthocentre; also equal to 12R2 minus the sum of the squares of the distances of the orthocentre from the vertices.

55. If two circles touch in C, and if D be any point outside the circles at which their radii through C subtend equal angles, if DE, DF be tangent from D, DE. DF = DC2.

BOOK IV.

INSCRIPTION AND CIRCUMSCRIPTION

TRIANGLES AND OF REGULAR POLYGONS IN AND ABOUT CIRCLES.

DEFINITIONS.

1. If two rectilineal figures be so related that the angular points of one lie on the sides of the other1, the former is said to be inscribed in the latter; 2, the latter is said to be described about the former.

II. A rectilineal figure is said to be inscribed in a circle when its angular points are on the circumference. Reciprocally, a rectilineal figure is said to be circumscribed to a circle when each side touches the circle.

III. A circle is said to be inscribed in a rectilineal figure when it touches each side of the figure. Reciprocally, a circle is said to be circumscribed to a rectilineal figure when it passes through each angular point of the figure.

IV. A rectilineal figure which is both equilateral and equiangular is said to be regular.

Observation.-The following summary of the contents of the Fourth Book will assist the student in remembering it :

1. It contains sixteen Propositions, of which four relate to triangles, four to squares, and four to pentagons, and four miscellaneous Propositions.

2. Of the four Propositions occupied with triangles

(a) One is to inscribe a triangle in a circle.

(B) Its reciprocal, to describe a triangle about a circle.
(2) To inscribe a circle in a triangle.

(8) Its reciprocal, to describe a circle about a triangle.

3. If we substitute in (a), (B), (7), (8) squares for triangles, and pentagons for triangles, we have the problems for squares and pentagons respectively.

4. Every Proposition in the Fourth Book is a problem.

PROP. I.-PROBLEM.

In a given circle (ABC) to place a chord equal to a given line (D) not greater than the diameter.

Sol.-Draw any diameter AC of the circle; then, if

AC be equal to D,

:]; D

the thing required is done; if not, from AC cut off the part AE equal to D [I. III. and with A as centre and AE as radius, describe the circle EBF, cutting the circle ABC