M

A

lines. Join BB'. Then, since AB and AB' are equal lines drawn from A to BB', they cannot be perpendicular to BB' (I., Proposition XVI.), and consequently they are longer than the perpendicular AC from A to BB', by I., Proposition XVII., which is contrary to the hypothesis that they were shorter than any other lines that could be drawn from A to MN. There is therefore one,

B

and but one, minimum line from A to the plane. Let AP be that minimum line; then AP is perpendicular to any straight line EF drawn in the plane through its foot P. For, in the plane of the lines AP and EF, AP is the shortest line that can be drawn from A to any point in EF, since it is the shortest line that can be drawn from A to any point in the plane MN; therefore AP is perpendicular to EF (I., Proposition XVII.). Thus AP is perpendicular to any, that is, to every, straight line drawn in the plane through its foot, and is therefore perpendicular to the plane.

There can be no other perpendicular from A to the plane MN; for, if there were, both lines would be perpendicular to the line joining the points where they met the plane, and we should have two perpendiculars from a point to a line, which is contrary to I., Proposition XVI.

8. COROLLARY. At a given point in a plane, one perpendicular can be erected to the plane, and but one.

from A' to this plane. Suppose the plane M'N' to be applied

to the plane MN with the point P' upon P, and let AP be the position then occupied by the perpendicular A'P'. We then have one perpendicular, AP, to the plane MN, erected at P. There can be no other: for let PB be another perpendicular at P. Then AP and PB are both perpendicular to PC, the line of intersection of MN with the plane determined by the two lines AP and BP, at the same point, and lie in the same plane with PC, and this is contrary to I., Proposition I.

9. Scholium. By the distance of a point from a plane is meant the shortest distance; hence it is the perpendicular distance from the point to the plane.

EXERCISES.

1. Theorem. Oblique lines drawn from a point to a plane, and meeting the plane at equal distances from the foot of the perpendicular, are equal; and of two oblique lines meeting the plane at unequal distances from the foot of the perpendicular the more remote is the greater.

PROPOSITION IV.-THEOREM.

10. If a straight line is perpendicular to each of two straight lines at their point of intersection, it is perpendicular to the plane of those lines.

Let AP be perpendicular to PB and PC, at their intersection P; then AP is perpendicular to the plane MN which contains those lines.

For, let PD be any other straight line drawn through P in the plane MN. Draw any straight line BDC intersecting PB, PC, PD, in B, C, D; produce AP to A', making PA' PA, and join A and A' to each of the points B, C, D.

=

=

M

N

Since BP is perpendicular to AA', at its middle point, we have BA BA' (I., Proposition XVIII.), and for a like reason CA CA'; therefore the triangles ABC, A'BC, are equal (I., Proposition IX.), and the angle ABD is equal to the angle A'BD. The triangles ABD and A'BD are equal (I., Proposition VI.), and ADA'D. Hence the triangles APD and A'PD are equal (I., Proposition IX.). Therefore the adjacent angles APD and A'PD are equal, and PD is perpendicular to AP. AP, then, is perpendicular to any, that is, to every, line passing through its foot in the plane MN, and is consequently perpendicular to the plane.

11. COROLLARY I. At a given point of a straight line one plane can be drawn perpendicular to the line, and but one.

Let AP be the line, and P the point. Through AP pass two planes, and in each of these planes draw through P a line perpendicular to AP. The plane determined by these

two lines is perpendicular to AP at P, by Proposition

IV.

No other perpendicular plane can be drawn through P, for, if it could, a plane containing AP would intersect the two perpendicular planes in lines which would lie in the same plane with AP, and be perpendicular to AP at the same point, which is contrary to I., Proposition I.

B

12. COROLLARY II. Through a given point without a straight line one plane can be drawn perpendicular to the line, and but

one.

In the plane determined by the point and the line draw a perpendicular from the point to the line, and through the foot of this perpendicular draw, in any second plane passing through the given line, a second perpendicular to the line. The plane of these two perpendiculars is obviously a plane passing through the given point and perpendicular to the given line.

No second perpendicular plane can be drawn through the given point, for the plane determined by the line and the point would cut the two perpendicular planes in lines which would be two perpendicular lines from the given point to the given line, which is contrary to I., Proposition XVI.

EXERCISES.

1. Theorem. All the perpendiculars that can be drawn to a straight line at the same point lie in a plane perpendicular to the line at the point.

2. Theorem-If from the foot of perpendicular to a plane a straight line is drawn at right angles to any line of the plane, and its intersection with that line is joined to any point of the perpendicular, this last line will be perpendicular to the line of the plane.

M

C

D

B

PARALLEL STRAIGHT LINES AND PLANES. 13. Definitions. A straight line is parallel to a plane when it cannot meet the plane, though both be indefinitely produced. In the same case, the plane is said to be parallel to the line. Two planes are parallel when they do not meet, both being indefinite in extent.

PROPOSITION V.-THEOREM.

14. Two lines in space having the same direction are parallel.

Α

E

B

Let AB and CD be two lines having the same direction. Through AB and any point E of CD pass a plane, and in this plane draw through E a line parallel to AB. This line will have the same direction as AB (I., Axiom II.), and consequently the same direction as CD, and must therefore coincide with CD, by I., Postulate II. Hence AB and CD are parallel.

15. COROLLARY. Two lines parallel to the same line are parallel to each other. For they have the same direction.

PROPOSITION VI.-THEOREM.

16. If two straight lines are parallel, every plane passed through one of them and not coincident with the plane of the parallels is parallel to the other.