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The following Theorems, depending for their demonstration upon those already demonstrated, are introduced as exercises for the pupil. In some of them references are made to the propositions upon which the demonstration depends. They are not connected with the propositions in the following books, and can be omitted if thought best.
38. The square on the sum A C of two straight lines A B, B C is equivalent to the squares on A B and BC, together with twice the rectangle AB. BC.
Or, algebraically, if a = A B, and b
39. Corollary. The square on a line is four times the square on half of the line.
40. The square on the difference AC of two straight lines A B, BC is equivalent to the squares on AB and BC, diminished by twice the rectangle AB. BC.
Or, algebraically, if a = A B, and b = BC, (a - b)2= a2 2ab+b2
41. The rectangle contained by the sum and difference of two lines AB, B C is equivalent to
42. Parallelograms are to each other as the products of their bases and altitudes. (10.)
43. Parallelograms having equal bases are to each other as their altitudes; those having equal altitudes are as their bases.
44. Where must a line from the vertex be drawn to bisect a triangle? (13.)
45. Two or more lines parallel to the base of a triangle divide the other sides, or the other sides produced, proportionally.
46. Lines joining the middle points of the adjacent sides of a quadrilateral form a parallelogram; and the perimeter of this parallelogram is equal to the sum of the diagonals of the quadrilateral. Draw the diagonals. (18.)
47. Lines drawn from the vertex of a triangle divide the opposite side and a parallel to it proportionally.
48. State and prove the converse of 47.
49. ABCD is a parallelogram; E and F the middle points of AB and CD. BF and DE trisect the diagonal A C.
50. If two triangles have two sides of the one equal respectively to two sides of the other, and the included angles supplementary, the triangles are equivalent.
51. The diagonals divide a parallelogram into four equivalent triangles. Two triangles standing on opposite sides are equal.
52. If the middle points of the sides of a triangle are joined, the area of the triangle thus formed is one fourth the area of the original triangle.
53. Every line passing through the intersection of the diagonals of a parallelogram bisects the parallelogram.
54. If a point within a parallelogram is joined to the vertices, the two triangles formed by the joining lines and two opposite sides are together equivalent to half the parallelogram.
Through the point draw lines parallel to the sides of the parallelogram.
55. State and prove the proposition if the point named in 54 is without the parallelogram.
56. The area of a trapezoid is equal to twice the area of the triangle formed by joining the extremities of one non-parallel side to the middle point of the other.
57. Two triangles are similar if two angles of the one are equal respectively to two angles of the other.
58. Two triangles are similar if their homologous sides are proportional.
59. Definition. When a point is taken on a given line, or a given line produced, the distances of the point from the extremities of the line are called the segments. If the point is within the given line, the sum of the segments, if in the line produced, the difference of the segments, is equal to the line.
60. The line bisecting any angle, interior or exterior, divides the opposite side into segments which are proportional to the adjacent sides. Let B be the bisected angle of a triangle ABC. Through C draw a line parallel to the bisecting line and meeting AB. If the interior angle at B is bisected, AB must be produced; if the exterior angle, A C. In the latter case, if E is the point where the bisecting line meets AC produced, the segments of the base (59) are A E and CE. (I. 17.) (I. 45.) (16.)
61. Two triangles having an angle of the one equal to an angle in the other are to each other as the rectangles of the sides containing the equal angles; or
ABC: ADE=ABX AC: ADX AE
62. Prove Theorem XII., first
drawing G C and BF; then proving the triangles A G C and ABF equal.
Turn the triangle ABF on the point A in its own plane till A B coincides with A G; where will F be? (7, 11.)
63. Prove that if GH, KI, and LB, in the figure above, are produced, they will meet in the same point.
64. Prove Theorem XII., first producing FA to GH, and producing GH, KI, and LB till they meet.
65. Prove Theorem XII., first constructing the squares on opposite sides of AB and BC from that on which they are drawn in the figure in Art. 62; moving the square A GHB on AB, a distance
equal to BC in the direction BA; then proving that these squares are divided into parts that can be made to coincide with the parts of the square on A C.
66. If A is an acute angle of the triangle ABC, and BD is the perpendicular from B to A C, then BC2 = A B2 + AC2 — 2 AC × AD
67. If A is an obtuse angle of the triangle B ABC, and BD is the perpendicular from B to AC, then
B C2 = A B2 + A C2 + 2 A C × AD
68. Show that if the angle A becomes a right angle, both 66 and 67 reduce to the same as 27; and if C becomes a right angle, both reduce to the same as the second equation in 28.
69. If a line is drawn from the vertex of any angle of a triangle to the middle of the opposite side, the sum of the squares of the other two sides is equivalent to twice the square of the bisecting line together with twice the square of a segment of the bisected side.
Draw a perpendicular from the same vertex to the opposite side. (66, 67.)
70. The sum of the squares of the four sides of a parallelogram is equivalent to the sum of the squares of the diagonals. (69.) (39.)
71. In the figure in Art. 62 draw HI, KE, FG. The triangle HIB is equal, and the triangles CKE, GAF are equivalent to ABC.
72. The squares of the sides of a right triangle are as the segments of the hypothenuse made by a perpendicular from the vertex of the right angle.
73. The square of the hypothenuse is to the square of either side as the hypothenuse is to the segment adjacent to this side made by a perpendicular from the vertex of the right angle.
74. The side of a square is to its diagonal as 1:2; or the square described on the diagonal of a square is double the square itself.
75. (Converse of 30.) Two polygons composed of the same number of similar triangles similarly situated are similar.
1. A Circle is a plane figure bounded by a curved line called the circumference, every point of which is equally distant from a point within called the centre; as ABDE.
2. The Radius of a circle is a line drawn from the centre to the circumference; as CD.
3. The Diameter of a circle is a line drawn through the centre and terminating at both ends in the circumference; as A D.
4. Corollary. The radii of a cir
cle, or of equal circles, are equal; also the diameters are equal, and each is equal to double the radius.
5. An Arc is any part of the circumference; as A FB.
6. A Chord is the straight line joining the ends of an arc; as A B.
7. A Segment of a circle is the part of the circle cut off by a chord; as the space included by the arc A FB and the chord A B.
8. A Sector is the part of a circle included by two radii and the intercepted arc; as the space B C D.
9. A Tangent (in geometry) is a line which touches, but does not, though produced, cut the circumference; as G D.