PROPOSITION VIII. THEOREM

473. If, from a point without a plane, oblique lines be drawn to the plane:

(1) Those meeting the plane at equal distances from the foot of the perpendicular are equal.

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

Hyp. EA is to plane MN. Oblique lines EB, EC, and ED are drawn so that AB= AC and AD> AC.

To prove (1)

(2)

HINT.

EBEC.

ED> EC.

1. Prove by means of equal triangles

2. ED2 = EA2 + AD2, EC2 = EA2 + AC2.

474. COR. 1. Conversely, equal oblique lines drawn from a point to a plane meet the plane at equal distances from the foot of the perpendicular from the point to the plane, and of two unequal oblique lines, the greater one meets the plane at a greater distance from the foot.

475. COR. 2. The locus of a point in space that is equidistant from all the points in the circumference of a circle is a

straight line perpendicular to the plane of the circle, and passing through the center of the circle.

476. COR. 3. The perpendicular to a plane is the shortest line that can be drawn to the plane from a point without.

477. DEF. The distance of a point from a plane is the length of the perpendicular to that plane.

PROPOSITION IX. THEOREM

478. The intersections of two parallel planes with a third plane are parallel.

Hyp. The parallel planes MN and PQ are intersected by a third plane RS in AB and CD, respectively.

To prove

Proof.

AB CD.

AB and CD cannot meet, for otherwise planes MN and PQ would meet.

AB and CD lie in the same plane.

Hence

AB is to CD.

Q.E.D.

479. COR. 1. Parallel lines included between parallel planes

are equal.

480. COR. 2. Two parallel planes are everywhere equally distant.

Ex. 1038. What is the locus of a point having a given distance from a given plane?

PROPOSITION X. THEOREM

481. Any plane containing one, and only one, of two parallel lines is parallel to the other line.

Hyp. AB is to CD, and plane MN contains CD.

Proof. AB and CD determine a plane, intersecting MN in CD.

Hence, if AB meets MN, it must meet MN in CD.
But, since AB | CD, this is impossible.

Hence

plane MN is to AB.

Q.E.D.

482. COR. 1. Through a given straight line a plane can be passed parallel to any other given straight line; and if the lines are not parallel, only one such plane can be drawn.

483. COR. 2. Through a given point a plane can be passed parallel to any two given straight lines in space; and if the lines are not parallel, only one such plane can be drawn.

PROPOSITION XI. THEOREM

484. A line parallel to a plane is parallel to the intersection of this plane with any plane passing through the line.

B

C

Hyp.

M

Plane MN is to AB.

To prove any plane AC containing AB intersects MN in CD so that CD is to AB.

Proof. AB and CD cannot meet, for otherwise AB would meet the plane MN.

AB and CD are in the same plane.

486. COR. 2. If two angles ABC and A'B'C' have their sides respectively parallel, their planes are parallel.

B.

B

c'

Since AB is to A'B', plane A'B'C' is to AB, and, similarly, is to BC.

Hence planes ABC and A'B'C' are parallel.

(Cor. 1.)

Ex. 1039. If a line is parallel to a plane and parallel to another line, the second line is parallel to the plane.

Ex. 1040. Find the locus of a point equidistant from three given points.

Ex. 1041. A line parallel to two intersecting planes is parallel to the intersection of these planes.

Ex. 1042. What is the locus of a point equidistant from two parallel planes?

Ex. 1043. To construct a line parallel to two intersecting planes.

Ex. 1044. The midpoints of two opposite sides of a quadrilateral in space and the midpoints of the two diagonals determine the vertices of a parallelogram.

Ex. 1045. The lines joining the midpoints of the opposite sides of any quadrilateral in space and the line joining the midpoints of the diagonals meet in a point.