PROP. IV.—Theorem.
If a right line (EF) be perpendicular to each of two intersecting lines (AB, CD), it will be perpendicular to any line GH, which is both coplanar and concurrent with them.
Dem.—Through any point G in GH draw a line BC intersecting AB, CD, and so as to be bisected in G; and join any point F in EF to B, G, C. Then, because EF is perpendicular to the lines EB, EC, we have
| BF2 = BE2 + EF2, and CF2 = CE2 + EF2; | |||||||||||
| ∴ BF2 + CF2 | = BE2 + CE2 + 2EF2. | ||||||||||
| Again | BF2 + CF2 | = 2BG2 + 2GF2 [II. x. Ex. 2], | |||||||||
| and | BE2 + CE2 | = 2BG2 + 2GE2; | |||||||||
| ∴ 2BG2 + 2GF2 | = 2BG2 + 2GE2 + 2EF2; | ||||||||||
| ∴ GF2 | = GE2 + EF2. |
Hence the angle GEF is right, and EF is perpendicular to EG.
Def. vi.—A line such as EF, which is perpendicular to a system of concurrent and coplanar lines, is said to be perpendicular to the plane of these lines, and is called a normal to it.
Cor. 1.—The normal is the least line that may be drawn from a given point to a given plane; and of all others that may be drawn to it, the lines of any system making equal angles with the normal are equal to each other.
Cor. 2.—A perpendicular to each of two intersecting lines is normal to their plane.
PROP. V.—Theorem.