72. A bridge consists of three arches, whose spans are 49 ft., 32 ft., and 49 ft. respectively: show that the point on either bank of the river at which the middle arch subtends the greatest angle is 63 feet distant from the bridge. See (477).

73. If the sum of the squares of two lines be given, their sum is a maximum when the lines are equal.

74. Of all triangles having the same base and vertical angle, the isosceles triangle has the sum of the sides a maximum.

75. Of all triangles inscribed in a circle, the equilateral triangle has the maximum perimeter.

SOLID GEOMETRY.

BOOK VI.

PLANES AND SOLID ANGLES.

DEFINITIONS.

480. A plane has been defined (11) as a surface in which the straight line joining any two of its points lies wholly in the surface.

A plane is indefinite in extent, so that, however far the straight line is produced, all its points lie in the plane. But to represent a plane in a diagram, it is necessary to take a limited portion of it, and it is usually represented by a parallelogram supposed to lie in the plane.

481. A plane is said to be determined by certain lines or points, when it is the only plane which contains these lines or points.

482. Any number of planes may be passed through any given straight line.

For, if a plane is passed through any given straight line AB, the plane may be turned about AB as an axis, and made to occupy an infinite number of positions, each of which will be a different plane passing through AB.

A

B

Hence a given straight line does not determine the position of a plane.

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

For, if the plane containing the straight line AB, turn about this line as an axis until it contains the given point C, the plane is evidently determined,

A

.C

B

If it is then turned, in either direction, about AB, it will no longer contain the point C.

484. A plane is determined by three points not in the same straight line.

For, if we join any two of the points by a straight line, this line and the third point determine a plane. (483) 485. A plane is determined by two intersecting straight lines.

For, a plane passing through AB and any point C in AC, in addition to the point of intersection A, contains the two straight lines AB, AC (480), and is determined.

A

B

(483)

486. A plane is determined by two parallel straight lines.

For, two parallel straight lines lie in the same plane (68); and since this plane contains either of these parallels and any point in the other, it is determined.

(483)

487. A straight line is perpendicular to a plane when it is perpendicular to every straight line which it meets in that plane.

Conversely, the plane in this case is said to be perpendicular to the line.

The point in which a line meets a plane is called the foot of the line.

488. A straight line is parallel to a plane when it never meets the plane, however far both may be produced.

Conversely, the plane in this case is said to be parallel to the line.

489. Two planes are parallel when they do not meet, however far they may be produced.

490. The projection of a point on a plane is the foot of the perpendicular let fall from the point to the plane,

491. The projection of a line on a plane is the locus of the projections of all the points of this line.

492. The angle which a straight line makes with a plane is defined to be the acute angle between the straight line and its projection on the plane, and is called the inclination of the line to the plane.

493. By the distance of a point from a plane is meant the shortest distance from the point to the plane.

494. The line that determines the projection of a point. on a plane is called the projecting line of that point. The plane including all the projecting lines of a straight line is called the projecting plane of the line.

LINES AND PLANES.

Proposition 1. Theorem.

495. If two planes cut each other, their common inter

section is a straight line.

Hyp. Let AB, CD be two

planes which cut each other.

To prove their common inter

section is a straight line.

Proof. Let E and H be any

two pts. common to both planes. A Join E and H by the st. line

EH.

E

B

Because E and H are in both

planes, the st. line EH lies in both planes.

(480)

Because a st. line and a pt. out of it cannot lie in two

planes,

(483)

... EH contains all the pts. common to both planes. ... EH is the common intersection of the two planes.

Q.E.D.

Proposition 2. Theorem.

496. If oblique lines are drawn from a point to a plane : (1) Two oblique lines meeting the plane at equal distances from the foot of the perpendicular are equal.

(2) Of two oblique lines meeting the plane at unequal distances from the foot of the perpendicular, the more remote is the longer.

Hyp. Let OP be to the plane

(2) Proof. On PC take PB PA, and join OB.

=

N

(Hyp.)

(104)

497. COR. 1. The perpendicular is the shortest distance from a point to a plane; therefore, by the distance of a point from a plane is meant the perpendicular distance from the point to the plane.

(493)

498. COR. 2. Equal oblique lines from a point to a plane meet the plane at equal distances from the foot of the perpendicular; and of two unequal oblique lines, the greater meets the plane at the greater distance from the foot of the perpendicular.

499. Cor. 3. The locus of the point in a plane at a given distance from a fixed point without the plane, is a circle whose centre is the foot of the perpendicular.