80. To trisect a triangle by straight lines drawn from a point within to the vertices.

81. Parallel to the base of a triangle to draw a line equal to the sum of the lower segments of the two sides.

82. Parallel to the base of a triangle to draw a line equal to the difference of the lower segments of the two sides.

83. To inscribe in a given triangle a quadrilateral similar to a given quadrilateral.

84. To divide a given line so that the sum of the squares of the parts shall be equivalent to a given square.

85. To construct a parallelogram when there are given,

1st. Two adjacent sides and a diagonal.

2d. A side and two diagonals.

3d. The two diagonals and the angle between them.
4th. The perimeter, a side, and an angle.

86. To construct a square when the diagonal is given.

87. To construct a parallelogram equivalent to a given triangle and having a given angle.

88. To draw a quadrilateral, the order and magnitude of all the sides and one angle given.

Show that sometimes there may be two different polygons satisfying the conditions.

89. To draw a quadrilateral, the order and magnitude of three sides and two angles given.

1st. The given angles included by the given sides.

2d. The two angles adjacent, and one adjacent to the unknown side. 3d. The two angles being opposite each other.

4th. The two angles being both adjacent to the unknown side.

In any of these cases can more than one quadrilateral be drawn?

90. To draw a quadrilateral, the order and magnitude of two sides and three angles given.

1st. The given sides being adjacent.

2d. The given sides not being adjacent.

91. In a given circle to inscribe a triangle similar to a given triangle.

92. Through a given point to draw to a given circle a secant such that the part within the circle may be equal to a given line.

93. With a given radius to draw a circumference,

1st. Through two given points.

2d. Through a given point and tangent to a given line.

3d. Through a given point and tangent to a given circumference. 4th. Tangent to two given straight lines.

5th. Tangent to a given straight line and to a given circumference. 6th. Tangent to two given circumferences.

State in each of these cases how many circles can be drawn, and when the construction is impossible.

94. To draw a circumference,

1st. Through two given points and with its centre in a given line. 2d. Through a given point and tangent to a given line at a given point..

3d. Tangent to a given line at a given point, and also tangent to a second given line.

4th. Tangent to three given lines.

5th. Through two given points and tangent to a given line.

6th. Through a given point and tangent to two given lines.

95. To draw a tangent to two circumferences.

There can be drawn,

1st. When the circles are external to each other, four tangents. 2d. When the circles touch externally, three.

3d. When the circles cut, two.

4th. When the circles touch internally, one.

5th. When one circle is within the other, none.

BOOK VI.

GEOMETRY OF SPACE.

PLANES AND THEIR ANGLES.

DEFINITION.

1. Geometry of Space, or geometry of three dimensions, treats of figures whose elements are not all in the same plane. For the definition of a plane see I. 11.

THEOREM I.

2. A plane is determined,

1st. By a straight line and a point without that line;

2d. By three points not in the same straight line; 3d. By two intersecting straight lines.

1st. Let the plane M N, pass- M ing through the line AB, turn upon this line as an axis until it contains the point C; the position of the plane is evidently de

termined; for if it is turned in

A

B

N

either direction it will no longer contain the point C.

2d. If three points, A, B, C, not in the same straight line, are given, any two of them, as A and B, may be joined by a straight line; then this is the same as the 1st case.

3d. If two intersecting lines A B, A C, are given, any point, C, out of the line A B can be taken in the line A C; then the plane passing through the line A B and the point C contains the two lines A B and A C, and is determined by them.

3. Cor. 1. The intersection of two planes is a straight line; for the intersection cannot contain three points not in the same straight line, since only one plane can contain three such points.

4. Cor. 2. Through a straight line an infinite number of planes can pass. For the plane MN, revolving on A B as an axis, occupies an infinite number of positions.

DEFINITIONS.

5. A straight line is perpendicular to a plane when it is perpendicular to every straight line of the plane which it meets. Conversely, the plane, in this case, is perpendicular to the

line.

The foot of the perpendicular is the point in which it meets. the plane.

THEOREM II.

6. There can be but one perpendicular from a point to a plane.

If there could be two, they would be in the same plane (2); and the intersection of this plane with the given plane would be a straight line (3), and then there would be two perpendiculars from a point to a straight line, (the three lines in the same plane,) which is impossible (I. 57).

THEOREM III.

7. Oblique lines from a point to a plane equally distant from the perpendicular are equal; and of two oblique lines unequally distant from the perpendicular, the more remote is the greater.

Let A C, A D be oblique lines drawn to the plane M N at equal distances from the perpendicular AB:

1st. A CAD; for the triangles ABC, ABD are equal (I. 80).

2d. Let A F be more remote. From BF cut off BE — BD

=

and draw A E; then AFA E

M

Α

E

B

D

N

(I. 90); and A E AD AC; therefore A F> A D or A C.

8. Cor. 1. Conversely, equal oblique lines from a point to a plane are equally distant from the perpendicular; therefore they meet the plane in the circumference of a circle whose centre is the foot of the perpendicular. Of two unequal lines the greater is more remote from the perpendicular.

9. Cor. 2. The perpendicular is the shortest distance from a point to a plane.

THEOREM IV.

10. A line perpendicular to each of two lines at their point of intersection is perpendicular to the plane of these lines.

Let A B be perpendicular to BC, BD at their point of intersection B; then A B is perpendicular to the plane M N, in which the lines B C, BD are. Let BE be any other line through B in the plane M N. Draw a line intersecting BC, BE, BD in CED; produce A B so as to make B F=AB; join A C, A E, AD, FC, FE, FD.

As BC and BD are perpendicular

M

Α

F

N

to AF at its middle point, the triangles AC D, D E F have (I. 94) AC CF and AD=DF; and CD is common ; therefore (I. 88) the triangle ACD DCF, and the equal angles at the base D C are adjacent; hence lines drawn from the corresponding vertices A and F to corresponding points of their bases must be equal; that is, A E=EF. Hence E must be a point in a perpendicular passing through B the middle of A F (converse of I. 94), that is, A B is perpendicular to BE. Therefore A B is perpendicular to the plane M N (5).

11. Corollary. Hence to pass a plane through any point D perpendicular to a given line AF, draw a perpendicular from the point D to the line A F, and at B, the point of intersection with the given line, and not in the plane of the given line