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BOOK VI.

PLANES AND SOLID ANGLES.

ON LINES AND PLANES. 428. Def. A Plane has already been defined as a surface such that the straight line joining any two points in it lies wholly in the surface.

The plane is considered to be indefinite in extent, so that however far the straight line be produced, all its points lie in the plane. A plane is usually represented by a quadrilateral supposed to lie in the plane.

429. DEF. The Foot of a line is the point in which it meets the plane.

430. DEF. A straight line is perpendicular to a plane if it be perpendicular to every straight line of the plane drawn through its foot.

In this case the plane is perpendicular to the line.

431. DEF. The Distance from a point to a plane is the perpendicular distance from the point to the plane.

432. Def. A line is parallel to a plane if all its points be equally distant from the plane.

In this case the plane is parallel to the line.

433. DEF. A line is oblique to a plane if it be neither perpendicular nor parallel to the plane.

434. DEF. Two planes are parallel if all the points of either be equally distant from the other.

435. DEF. The Projection of a point on a plane is the foot of the perpendicular from the point to the plane.

436. DEF. The projection of a line on a plane is the locus of the projections of all its points.

437. DEF. The plane embracing the perpendiculars which project the points of a straight line upon a plane is called the projecting plane of the line.

438. DEF. The angle which a line makes with a plane is the angle which it makes with its projection on the plane.

This angle is called the Inclination of the line to the plane.

439. DEF. A plane is determined by lines or points, if no other plane can embrace these lines or points without being coincident with that plane.

440. Def. The intersection of two planes is the locus of all the points common to the two planes.

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441. An infinite number of planes may embrace the same straight line.

Thus, if the plane M N em M brace the line A B it may be made to revolve about A B as an axis, and to occupy an infinite number of positions, each of which is the position of a plane embracing the line A B.

442. A plane is determined by a straight line and a point without that line.

Thus, let any plane embracing the straight line AB revolve about the line as an axis

B / until it embraces the point C.

IN Now if the plane revolve either way about the line A B as an axis, it will cease to embrace the point C.

Hence any other plane embracing the line A B and the point C must be coincident with the first plane.

§ 439 443. Three points not in a straight line determine a plane.

For, by joining any two of the points, we have a straight line and a point which determine a plane.

§ 442 444. Two intersecting straight lines determine a plane.

For, a plane embracing one of these straight lines and any point of the other line (except the point of intersection) is determined.

§ 442 445. Two parallel straight lines determine a plane.

For, a plane embracing either of these parallels and any point in the other is determined.

§ 442

PROPOSITION I. THEOREM. 446. If two planes cut one another their intersection is a straight line.

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Let M N and PQ be two plane's which cut one another.

We are to prove their intersection a straight line.
Let A and B be two points common to the two planes.

Draw the straight line A B. Since the points A and B are common to the two planes, the straight line A B lies in both planes.

$ 428 Now, no point out of this line can be in both planes ;

for, if it be possible, let C be such a point. But there can be but one plane embracing the point C and the line A B.

§ 442 .. C does not lie in both planes. ... every point in the intersection of the two planes lies in the straight line A B.

Q. E. D

PROPOSITION II. THEOREM. 447. From a point without a plane only one perpendicular can be drawn to the plane; and at a given point in a plane only one perpendicular can be erected to the plane.

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Fig. 1.

Fig. 2. Let C D (Fig. 1) be a perpendicular let fall from the

point C to the plane MN.

We are to prove that no other I can be drawn from the point C to the plane M N.

If it be possible, let C B be another I to the plane MN, and let a plane P Q pass through the lines C B and C D.

The intersection of P Q with the plane M N is a straight line B D.

§ 446 Now if C D and C B be both I to the plane, the A CBD would have two rt. 2, CBD and C D B, which is impossible.

§ 102 Let DC (Fig. 2) be a perpendicular to the plane M N at

the point D.

If it be possible, let D A be another I to the plane from the point D,

and let a plane P Q pass through the lines D C and D A. The intersection of P Q with the plane M N is a straight line.

Now if DC and D A could both be I to the plane M N at D, we should have in the plane P Q two straight lines I to the line D Q at the point D, which is impossible.

$ 61

Q. E. D. 448. COROLLARY. A perpendicular is the shortest distance from a point to a plane.

PROPOSITION III. THEOREM. - 449. If a straight line be perpendicular to each of two straight lines drawn through its foot in a plane it is perpen(licular to the plane.

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Let DC be perpendicular to each of the two lines

A C A and BC B' drawn through its foot in the
plane M N.
We are to prove DCI to the plane M N.

Take CA = C A and C B=CB'.

Join A B and A' B'.
Then A B and A' B' are symmetrical with respect to C, $ 426

(their extremities being symmetrical).
Through C draw any line HC H' in the plane M N.
Then H and H' are symmetrical,

$ 422 (being corresponding points in the symmetrical lines A B and A' B').

About C, the centre of symmetry, revolve A B, keeping AC, and BCI to C D, until it comes into coincidence with A' B'.

Then the point H will coincide with its symmetrical point H', and Z DC H will coincide with, and be equal to, ZD CH'. ..4 DCH and DCH' are rt. 4.

H and D C H' are rt. 2. § 25

..DC is I to HC H'. Now since DC is I to any line, H CH', drawn through its foot in the plane M N, it is I to every such line. .. DC is I to the plane M N.

§ 430 Q. E. D.

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