RECAPITULATION.

Theorems and Problems of Plane Geometry.

On angles, perpendiculars, and obliques.

1. Only one perpendicular can be drawn through a given point to a given straight line (§ 36).

2. The adjacent angles which one straight line makes with another are either two right-angles or are together equal to two right-angles (§ 20).

3. When two straight lines cut one another, the vertically opposite angles are equal (§ 24).

4. Every point in a perpendicular raised from the middle of a straight line is equally distant from the two extremities (§ 28).

5. Every oblique line drawn from a given point to a given straight line, is greater than the perpendicular from the same point to the straight line (§ 35).

6. Of two oblique lines from the same point, that which has the greater departure is the greater (§ 39).

7. Of two oblique lines from the same point, that which is greater has the greater departure (§ 39).

Relation between the sides and angles of a triangle.

8. The angles at the base of an isosceles triangle are equal (§ 73).

9 If two angles of a triangle are equal, the opposite sides are equal (§ 74).

10. When in a triangle two angles are unequal, the side oppo site the greater angle is greater than the side opposite the less (§ 75).

II. When in a triangle two sides are unequal, the angle opposite the greater is greater than the angle opposite the less (§ 76).

On the equality of triangles.

12. When two triangles have two sides and the included angle of the one respectively equal to two sides and the included angle of the other, the triangles are equal in all respects (§ 88 a).

13. When two triangles have two angles of the one respectively equal to two angles of the other, and a side equal to a

side, namely, either the side common to the two angles in each, or opposite one of them, the triangles are equal in all respects. (§ 886).

14. When two triangles have two sides of the one respectively equal to two sides of the other, but the angle contained by the two sides of the one greater than the angle contained by the two sides of the other, the base of that which has the greater angle is greater than the base of the other (§ 88c).

15. When two triangles have the three sides of the one respectively equal to the three sides of the other, the triangles are equal in all respects (§ 88d).

Properties of parallels.

16. Two straight lines perpendicular to a third are parallel (§ 92).

17. Any straight line perpendicular to one of two parallels is also perpendicular to the other (§ 97).

18. When one straight line falls on two parallel straight lines it makes,

1. The alternate angles equal,

2. The corresponding angles equal,

3. The interior angles on the same side equal to two right-angles (§ 98).

19. When one straight line falling on two other straight lines makes,

1. The alternate angles equal, or

2. The corresponding angles equal, or

3. The interior angles on the same side equal to two rightangles,

these two straight lines are parallel (§ 100).

Consequences of the above properties.

20. The exterior angle of a triangle is equal to the sum of the interior and opposite angles; and the three angles of a triangle are together equal to two right-angles (§ 126).

21. The opposite sides and angles of a parallelogram are equal, and a diagonal bisects it (§ 119).

22. The diagonals of a parallelogram bisect each other (§ 121). 23. The straight lines which join the corresponding extremities of equal and parallel straight lines are themselves equal and parallel (§ 107).

On the equivalence of triangles and parallelograms.

24. Parallelograms upon the same or equal bases and of the sameheight are equivalent (§ 141-143).

25. Every triangle is half a parallelogram having the same base and height (§ 144).

26. Triangles upon the same or equal bases and of the same height are equivalent (§ 145).

27. To construct a rectangle equal to any rectilinear figure (§ 147).

28. The square on the hypothenuse of a right-angled triangle is equivalent to the sum of the squares on the other sides (§ 148, 149).

29. To describe a square equivalent to a given parallelogram (§ 156).

The relation between the rectangles formed by the segments

of a line (p. 90).

30. The difference of the squares on two lines is equal to the rectangle whose sides are respectively the sum and difference of these lines (§ 155).

31. If a line be divided into two parts, the square on the whole is equal to the sum of the squares on each part, together with twice the rectangle contained by the parts (§ 155).

32. If a straight line be divided into two parts, the square on one part is equal to the sum of the squares on the whole line and the other part, less twice the rectangle contained by the whole line and this part (§ 155).

Proportion of Lines.

33. When parallel lines cut two transversals, the parts of one transversal are proportional to the parts of the other (§ 173).

34. The bisector of the vertical angle of a triangle divides the base into parts proportional to the sides (§ 175).

35. In parallel lines the segments intercepted by two transversals which intersect, are proportional to the segments of these straight lines between the meeting point and the parallels (§ 180).

36. The segments of parallel lines intercepted by transversals from the same point are proportional (§ 182).

37. To divide a straight line in mean and extreme ratio (§ 184).

The circle.

38. The diameter is the greatest chord in a circle (§ 56).

39. The diameter divides the circumference into equal parts ($ 59).

40. The perpendicular from the centre on a chord bisects the chord and the arc it subtends (§ 194).

41. Three given points determine a circle (§ 195).

42. An angle at the circumference is half the angle at the centre on the same arc (§ 206).

43. The angle inscribed in a segment of a circle is a rightangle, an acute angle, or an obtuse angle, according as the segment is equal to, greater than, or less than a semicircle (§ 207). 44. A diameter divides a circle into two equal parts.

45. The following propositions on equal circles may be proved by superposing the circles, i.e. by placing one on the other, so that their centres coincide and so that a given point in the circumference of one coincides with the corresponding point on the circumference of the other.

Ist. In equal circles, equal angles at the centre stand upon equal arcs, and also equal angles at the circumference stand upon equal arcs.

2nd. In equal circles, the angles at the centre or at the cir cumference on equal arcs, are equal.

3rd. In equal circles, equal arcs are subtended by equal chords.

4th. In equal circles, equal chords subtend equal arcs.

46. Every straight line perpendicular to the extremity of a diameter is a tangent to the circumference (§ 217).

47. If at a point in a circumference we draw a tangent and a chord, the angle between them is jual to any angle inscribed in the opposite segment (§ 228).

48. When two secants start from the same point, the segments of one of them form the extremes of a proportion of which the segments of the other form the means (§ 211).

Or,

If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.

49. If from a point without a circle a tangent and a secant be drawn, the tangent is a mean proportional between the whole secant and the part without the circle ($230).

Or,

If from any point without a circle two straight lines be drawn, one of which cuts the circle and the other touches it, the rectangle contained by the whole line which cuts the circle and the part of it without the circle is equal to the square of the tangent.

50. The line joining the centres of two circumferences is perpendicular to their common chord, and bisects it (§ 243).

51. The point of contact of two circles is situated on the line joining their centres (§ 245).

52. To describe a circle of known radius which may be touched by two given circles,—

Ist. Externally.

2nd. Internally.

3rd. Internally by one circle, and externally by the other.

53. To describe a circle of known radius,—

Ist. Touching another circle in a given point.

2nd. Touching another circle, and passing through a given point without the circumference.

3rd. Touching a given circle and a given straight line.

54. To describe a circle,

Ist. Touching a circle in a given point, and passing through a given point.

2nd. Touching a given circle and passing through two given points.

3rd. Touching a given circle and a given straight line, and passing through a given point.

55. To draw a circle, touching at the same time a given circle and two straight lines. (All the above, 52-55, are in § 247-256.)

Similar figures.

56. Two triangles are similar when the angles of one are equal to those of the other, each to each (§ 266).

57. Two triangles which have their sides parallel or perpendicular, each to each, are similar (§ 267).

58. Two triangles which have their sides proportional are similar (§ 268).

59. Two triangles which have two sides of the one proportional to two sides of the other, and the included angles equal, are similar (§ 269).

60. The perpendicular from the vertex of a right-angled triangle upon the hypothenuse, divides the triangle into two others similar to the whole and to one another (§ 271).

61. The perimeters of similar polygons have the same ratio as their homologous sides (§ 341).

62. The surfaces of similar polygons have the same ratio as the squares of the homologous sides (§ 342).