SOLID GEOMETRY.

BOOK VI.

LINES AND PLANES IN SPACE. — DIEDRALS.POLYEDRALS.

394. DEF. A plane is said to be determined by certain lines or points when one plane, and only one, can be drawn through these lines or points.

PROPOSITION I. THEOREM.

395. A plane is determined

I. By a straight line and a point without the line. II. By three points not in the same straight line. III. By two intersecting straight lines.

IV. By two parallel straight lines.

I. Let C be a point without the straight line AB. To prove that a plane is determined (§ 394) by AB and C. If any plane, as MN, be drawn through AB, it may be revolved about AB as an axis until it contains the point C. Hence, one plane, and only one, can be drawn through AB and C.

II. Let A, B, and C be three points not in the same straight line.

To prove that a plane is determined by A, B, and C.

Draw AB.

By I., one plane, and only one, can be drawn through the line AB and the point C.

Hence, one plane, and only one, can be drawn through A, B, and C.

III. Let AB and BC be two intersecting straight lines. To prove that a plane is determined by AB and BC.

By I., one plane, and only one, can be drawn through AB and any point C of BC.

But since this plane contains the points B and C, it must contain the line BC.

[A plane is a surface such that the straight line joining any two of its points lies entirely in the surface.] ($ 8.) Hence, one plane, and only one, can be drawn through AB and BC.

IV. Let AB and CD be two parallel lines.

To prove that a plane is determined by AB and CD. The parallels AB and CD lie in the same plane (§ 52). And by I., but one plane can be drawn through AB and any point C of CD.

Hence, one plane, and only one, can be drawn through AB and CD.

PROPOSITION II. THEOREM.

396. The intersection of two planes is a straight line.

Let the line AB be the intersection of the planes MN and PQ.

To prove AB a straight line.

Let a straight line be drawn between the points 1 and B. This line must lie in MN, and also in PQ.

[A plane is a surface such that the straight line joining any two of its points lies entirely in the surface.]

Then it must be the intersection of MN and PQ.
Whence, AB is a straight line.

(§ 8.)

397. DEF. If a straight line meets a plane, the point of intersection is called the foot of the line.

A straight line is said to be perpendicular to a plane when it is perpendicular to every straight line drawn in the plane through its foot.

A straight line is said to be parallel to a plane when it cannot meet the plane however far they may be produced. Two planes are said to be parallel to each other when they cannot meet however far they may be produced.

398. SCH. The following form of the second definition of § 397 is given for convenience of reference :

A perpendicular to a plane is perpendicular to every straight line drawn in the plane through its foot.

399. At a given point in a plane, one perpendicular to the plane can be drawn, and but one.

Let P be the given point in the plane MN.

To prove that a perpendicular can be drawn to MN at P, and but one.

At any point A of the straight line AB draw the lines AC and AD perpendicular to AB.

Let RS be the plane determined by AC and AD.

Let AE be any other straight line drawn through the point A in the plane RS; and draw the line CED intersecting AC, AE, and AD in C, E, and D.

Produce BA to B', making AB' = AB.

Draw BC, BE, BD, B'C, B'E, and B'D.

In the triangles BCD and B'CD, the side CD is common. And since AC and AD are perpendicular to BB' at its middle point,

BC= B'C, and BD

[If a perpendicular be erected at the middle point of a straight line, any point in the perpendicular is equally distant from the extremities of the line.]

(§ 40, I.)

[Two triangles are equal when the three sides of one are equal respectively to the three sides of the other.]

(§ 69.)

Now revolve triangle BCD about CD as an axis until it coincides with triangle B'CD.

Then B will fall at B', and the line BE will coincide with B'E; that is, BE = B'E.

Hence, since the points A and E are each equally distant from B and B′, AE is perpendicular to BB'.

[Two points, each equally distant from the extremities of a straight line, determine a perpendicular at its middle point.]

(§ 43.)

But AE is any straight line drawn through A in RS. Then, AB is perpendicular to every straight line drawn through its foot in the plane RS.

Whence, AB is perpendicular to RS.

[A straight line is said to be perpendicular to a plane when it is perpendicular to every straight line drawn in the plane through its foot.]

(§ 397.)

Now apply the plane RS to the plane MN so that the point A shall fall at P; and let AB take the position PQ. Then, PQ will be perpendicular to MN.

Hence, a perpendicular can be drawn to MN at P.

If possible, let PT be another perpendicular to MN at P; and let the plane determined by PQ and PT intersect MN in the line HK.

Then, both PQ and PT are perpendicular to HK.

[A perpendicular to a plane is perpendicular to every straight line drawn in the plane through its foot.]

(§ 398.) But in the plane HKT, only one perpendicular can be drawn to HK at P.

[At a given point in a straight line, but one perpendicular to the line can be drawn.] (§ 28.)

Hence, but one perpendicular can be drawn to MN at P.

400. COR. I. A straight line perpendicular to each of two straight lines at their point of intersection is perpendicular to their plane.

401. COR. II. From a given point without a plane, one perpendicular to the plane can be drawn, and but one.