8. ABC is a triangle in which AB=AC; P is a point in BC, and Q in BC produced; show that AQ>AB and AP<AB.

9. Half the perimeter of a triangle is greater than any one side and less than any two sides.

10. D is the mid-point of the side BC of the ABC. If AB>AC, show that BAD<< CAD.

11. Q and R are points in BC, and P is any point within the AABC; prove that BC+CA+AB>QR+RP+PQ.

12. ABCD is a quadrilateral whose diagonals intersect in O, and P is any point within the ▲ABO; show that

PA+PB+PC+PD<AC+BD+2AB.

13. If, in the ABCD, E be a point in AB, or AB produced, such that CE=CD, show that ED bisects the AEC.

Examine the case in which E is a point in BA produced.

14. In the right-angled ▲ ABC, AB=AC, AC is produced to D, so that CD=BC, and CD to E, so that DE=BD; show that

< BEA is the eighth part of a right angle.

15. In the ▲ ABC, D is the mid-point of BC, DO is 1 BC and O is a point in DO such that AO-BO; show that BOC=2<BAC.

16. In the AABC, D is a point in AC, or AC produced, such that AD=AB; show that CBD=} ( < B ~ ≤ C).

17. ABC, ABD are adjacent supplementary angles, through P a point on the bisector of ABC, PQR is drawn || CD meeting AB in Q and the bisector of ABD in R; show that PQ=QR.

18. In ▲ ABC, AB> AC, BI, CI are the bisectors of 4s B and C, and IX BC; show that < BIX><CIX.

19. ABC is a triangle, right-angled at A, and AB-AC, from E, the mid-point of AC, EN is drawn 1 BC; show that BN=3EN.

20. ABC is a triangle, right-angled at A, AN is drawn 1 BC and produced to P, so that AP=AB. If ACP be a right angle, show that PC-AN.

21. The side BC of the ▲ ABC is produced to D, and the bisectors of the 8 ABC, ACD meet in J; show that BJC=} < BAC. (This is the principle of Hadley's Sextant.)

22. If, in a quadrilateral ABCD, ▲ A= ≤B and ≤ C=D; show that AB DC and AD=BC,

23. Prove that every quadrilateral figure, not a square or rectangle, must have at least one acute and at least one obtuse angle.

24. Show that no convex polygon can have more than three interior angles acute.

25. Show that an equilateral triangle, a regular decagon and a regular fifteen-sided figure may be placed so as to have one vertex in common, and exactly fill up the space round a point.

26. Show that the number of diagonals of a polygon of ʼn sides is } n (n-3).

27. In the ▲ ABC, BC is produced to X, from BX, BD is cut off equal to AB, and from CX, CE is cut off equal to AC; show that (1) DE=BC+CA-AB,

(2) < AED=}<C,

(3) ▲ ADE=rt. <+} <B,

(4) < DAE=} < A.

28. In the ▲ ABC, AB> AC, AK is the bisector of the A, from AB, AG is cut off equal to AC, and GH parallel to AK meets BC in H, show that

29. In the ▲ ABC, BQ and CR are the altitudes from B and C ; show that the straight lines drawn at right angles to BR, RQ, QC from their mid-points are concurrent.

30. ABC is a triangle, right-angled at A, D is the mid-point of BC, and the bisector of ‹ A meets the perpendicular drawn to BC through D in L; show that DL=AD.

31. The straight line joining the right angle of a right-angled triangle to a point in the hypotenuse, other than the mid-point, is greater than one of the segments of the hypotenuse base and less than the other.

32. If, in the ▲ ABC, BC be produced to X and AK be the bisector of the A; show that ≤ ABX+ <ACX=22AKX.

33. In the ▲ ABC the bisector of the 48 B and C meet in I, and PIQ BC meets AB and AC in P and Q, from P and Q lines are drawn 1 BI and CI meeting BC in R and S; show that

BC=PQ+RS.

34. ABC is a triangle having AB-AC, and AB<BC; from BA, BC equal parts BD, BE are cut off and ED produced meets CA produced in F. Show that 3 <ADE + <AFD=4 right angles. (Compare § 13, Ex. 14.)

35. ABC is an isosceles triangle, the A being a right angle; D is any point in AC. From C and D, CF and DF are drawn at right angles to BC and BD respectively. Show that DF=DB.

S$ 15-21. EUCLID I.

Theorems.

36. ABCD is a parallelogram, and equilateral as ABE, CDF are described on AB, CD towards the same parts; show that AEFD, BEFC, are parallelograms.

37. In the ABCD, AB is produced to E and CD to F, so that AE=CF; show that BEDF is a parallelogram.

38. If, through the vertices of a parallelogram, straight lines be drawn making equal angles, towards the same parts, with the sides of the parallelogram; show that the figure contained by these four straight lines, when produced to meet, is also a parallelogram.

39. P is any point in the side CD of the ABCD. From C and D, CQ and DR are drawn parallel to PB and PA, meeting AB produced in Q and R. Show that the length of QR is independent of the position of P.

40. ABCD is a trapezium and AD | BC; E, F, G, H are the midpoints of AB, BD, AC, CD; if AF and DG meet in a point on BC, show that BC=2AD, and EF=FG=GH.

41. ABC, DEF are two triangles, AB, BC are equal, parallel and drawn in the same direction as DE, EF; if AD, BE, CF be joined, show that three parallelograms are formed, one of which is equal to the sum of the other two.

42. In the ▲ ABC, E and F are the mid-points of AC and AB, BE and CF meet in G; show that

▲ ABC 3 ▲ CGB = 44 CEF = 12 ▲ EFG.

43. In the ▲ ABC, D and E are the mid-points of BC and CA, and K is the mid-point of AD; show that

A ABC=8 AKBE.

44. OAB and OCD are two straight lines which meet in O; show that ▲ BCD > A ACD.

45. The parallelogram formed by joining the mid-points of the adjacent sides of a quadrilateral is a rhombus or a rectangle according as the quadrilateral is a rectangle or a rhombus.

46. The area of a trapezium is determined when its altitude and the sum of its parallel sides are given.

47. ABCD is a rectangle; P, Q are points in AD, BC such that 8AP=AD, 8BQ=5BC; show that 8 area APQB=3 ▲ BCD.

48. In the ▲ ABC, D, E, F are the mid-points of BC, CA, AB; if the FBEH be completed, show that HC=AD.

49. The distances of the mid-point of the base of a triangle from the foot of the perpendicular drawn from either extremity of the base to the bisector of the vertical angle is equal to half the difference between the sides.

50. If in a trapezium one of the parallel sides be twice the other, the point of intersection of the diagonals shall be a point of trisection of each diagonal.

51. O is the point of intersection of the diagonals of the trapezium ABCD, in which AD || BC. Through O, POQ is drawn || AD, meeting AB and CD in P and Q; show that PO=OQ.

52. Three parallel straight lines AD, BE, CF are drawn through the vertices of the ▲ ABC meeting the opposite sides (produced when necessary) in D, E, F ; show that ▲ DEF=2 ▲ ABC.

53. In the isosceles ▲ ABC, AB=AC, and points P and Q are taken in AB, AC such that AP=CQ; show that ▲ ACP = ^ BCQ.

54. If the straight line which joins the mid-points of the diagonals of a quadrilateral be parallel to one side, it shall also be parallel to the opposite side.

55. D, E, F are the mid-points of BC, CA, AB, the sides of the ▲ ABC; AL, BM, CN are drawn perpendicular to EF, FD, DE; show that AL, BM, CN are concurrent.

56. A, B and C are three points in order on a straight line, such that BC=2 AB. Through these points parallel straight lines AP, BQ and CR are drawn meeting a given straight line DE in P, Q and R; show that 2 AP + CR=3 BQ according as A and C are on the same or on opposite sides of DE.

57. ABCD is a , BE parallel to AC and CE parallel to DB meet in E; AE, DE meet BD, AC in H, K; show that BC=2 HK.

58. O is any point within the ▲ ABC and s BOCD, COAE, AOBF are completed; show that ▲ DEF=A ABC.

BOC

59. OA, OB and OC are three straight lines such that = ≤ COA= 2 AOB. From any point P perpendiculars are drawn to OA, OB and OC, produced if necessary; show that one of these perpendiculars is equal to the sum of the other two.

60. A, B, C are three points, AB is bisected at D and DC is divided at G, so that GC=2 DG; through A, B, C, G parallel straight lines AA', BB', CC, GG', are drawn, meeting a straight line PQ in A', B', C', G'; show that AA'±BB'±CC'=±3 GG', the negative signs being taken when B, C, G are on the opposite side of PQ from A.

61. A straight line XY is drawn through G, the point of intersection of the medians of the ▲ ABC; through A, B, C parallel straight lines AA', BB', CC' are drawn, meeting XY in A', B', C'; show that AA'±BB'±CC'=0, the negative signs being taken when B, C are on the opposite side of XY from A.

62. ABCD is a quadrilateral, E and F are the mid-points of AC and BD, and G is the mid-point of EF; through A, B, C, D parallels AA', BB', CC', DD' are drawn, meeting XY a straight line through G in A', B', C', D'; show that

AA'+BB'+CC'+DD' =0,

the negative signs being taken when B, C, D are on the opposite side of XY from A.

63. A1, A2, A, are three equidistant points on one straight line, B1, B2, B3 are three equidistant points on another straight line; three parallel straight lines A,P1, A2P2, A ̧P ̧ meet three other parallel straight lines B1P1, B2P2, B3P3, in P1, P2, P3; show that P1, P2, P3 are collinear.

64. ABCD is a parallelogram, P is a point in AD, and PQ is drawn parallel to AB to meet BC in Q, R is a point in AB, and RS is drawn parallel to AP to meet CD in S; show that 2AAQS-BD-PR.

65. P is any point on the diagonal AC of the ABCD, or on AC produced; show that ▲ PABA PAD. Conversely, if the ▲ PAB=▲ PAD, show that P must be a point on the diagonal of the parallelogram which has AB, AD as two adjacent sides.

66. ABCD is a parallelogram and P is any point, through P are drawn GH || BC and EF || AB, meeting the sides of ABCD in E, F, G, H; show that EG, HF, and one of the diagonals of ABCD are concurrent.

67. ABCD is a quadrilateral; AB, CD meet at E; AD, BC meet at F; P, Q, R are the mid-points of AC, BD, EF; show that P, Q, R are collinear.

68. ABC, CDE are triangles of equal area, so placed that AC, CE and also BC, CD form straight lines; show that the straight line joining the mid-points of AD and BE passes through C.

69. The diagonals of a quadrilateral intersect at right angles. Show that it is possible in this case to turn over the four corners