523. THEOREM. Through a given point one plane can be passed parallel to a given plane, and only one.

X

.. only one plane can contain P and be | to AB.

524. THEOREM. Parallel lines included

Q.E.D,

R

Q.E.D.

525. THEOREM. The plane perpendicular to a line at its midpoint is the locus of points in space, equally distant from the extremities

P'M is not to AB (?) (502). .. P', any point outside of plane RS, is not equally distant from A and B (?) (68).

Hence, plane RS is the locus of points in space equally distant from A and B (?) (179).

Q.E.D.

526. THEOREM. The locus of points in space equally distant from all the points in the circumference of a circle is the line perpendicular to the plane of the circle at its center.

Given: (?). To Prove: (?).

Proof: I. Any point in AC is equally distant from all the points in the circumference of the circle (?) (520, II).

II. Any point equally distant from all points of the circumference of the circle is in AC (?) (520, III).

.. AC is the required locus (179).

M

Q.E.D.

Ex. 1. What is the locus of points equally distant from two given points?

Ex. 2. What is the locus of points equally distant from three given points?

527. The distance from a point to a plane is the length of the perpendicular from the point to the plane.

Thus, the word "distance," referring to the shortest line from a point to a plane, implies the perpendicular.

The inclination of a line to a plane is the angle between the line and its projection upon the plane.

ORIGINAL EXERCISES

1. Through one straight line a plane can be passed parallel to any other straight line in space, and only one.

Through a point of the first line draw a line || to the second.

2. Two parallel planes are everywhere equally distant.

3. If a line and a plane are parallel, another line parallel to the given line and through any point in the given plane lies wholly in the given plane.

Through the given line and the point P pass a plane cutting the given plane in PX. Use 499.

4. A straight line parallel to the intersection of two planes, but in neither, is parallel to both planes.

5. If two straight lines are parallel and two intersecting planes are passed, each containing one of the lines, the intersection of these planes is parallel to each of the given lines.

6. If three straight lines through a point meet the same straight line, these four lines all lie in the same plane.

7. If a straight line meets two parallel planes, its inclinations to the planes are equal.

8. Two parallel planes can be passed, each containing one of two 522 given lines in space. Is this ever impossible?

9. If each of three straight lines intersects the other two, the three lines all lie in a plane.

10. The projections of two parallel lines on a plane are parallel.

A

B

M

11. If two lines in space are equal and parallel, their projections on a plane are equal and parallel.

12. If a plane is parallel to one of two parallel lines, it is parallel to the other.

13. If a straight line and a plane are perpendicular to the same straight line, they are parallel.

14. Equal oblique lines drawn to a plane from one point have equal inclinations with the plane.

15. If a line and a plane are both parallel to the same line, they are parallel to each other.

16. Four points in space, A, B, C, D, are joined, and these four lines are bisected. Prove that the four lines joining (in order) the four midpoints of the first lines form a parallelogram.

Proof: Pass plane DP through points A, D, B, and plane DX through points B, C, D, - these planes intersecting in BD. ST is to BD and BD (?); etc.

=

P

17. If a plane is passed containing a diagonal of a parallelogram and perpendiculars be drawn to the plane from the other vertices of the parallelogram, they are equal. To Prove: AE = CF. Proof: Draw diagonal AC. Draw EO and OF in plane MN. EO, OF, and EOF are projections; etc.

18. If from the foot of a perpendicular to a plane, a line be drawn at right angles to any line in the plane, the line connecting this point of intersection with any point in the perpendicular is perpendicular to the line in the plane.

Given: AB to plane RS; BC 1 to DE in the plane; PC drawn from C to P, in AB.

To Prove: PC is 1 to DE.

Proof: Take CD = CE, draw PD, PE, BD, BE. its midpoint (?). .. BD = BE (?). PD=PE (?) .. PC is 1 to DE (?) (70).

M

E

R

P

E

S

BC is 1 to DE at (520, II).

19. A line PB is perpendicular to a plane at B, and a line is drawn from B meeting any line DE, of the plane, at C. If PC is perpendicular to DE, BC is perpendicular to DE.

20. Are two planes that are parallel to the same straight line necessarily parallel?

21. If each of two parallel lines is parallel to a plane, is the plane of these lines also parallel to the given plane?

22. Is a three-legged chair always stable on the floor? Why? four-legged chair always stable? Why?

Is a

23. What is the locus in space of points equally distant from two parallel planes? From two parallel lines?

24. What is the locus of points in space at a given distance from a given plane?

25. What is the locus of points in a plane at a given distance from an external point?

26. What is the locus of points in space equally distant from two points and equally distant from two parallel planes?

27. What is the locus of points in space, equally distant from the vertices of a given triangle?

28. What is the locus of all straight lines perpendicular to a given straight line at a given point?

29. What is the locus of all lines parallel to a given plane and * drawn through a given point?

30. If the points in a line satisfy one condition and the points in a plane satisfy another condition, what will be true of their intersection? What will be true if they do not intersect?

31. If the points in one plane satisfy one condition and the points in another plane satisfy another condition, what is true of their intersection? What is true if the planes are parallel?

32. To construct a plane perpendicular to a given line at a given point in the line.

33. To construct a plane perpendicular to a given line through a given external point.

34. To construct a line perpendicular to a given plane, through a given point in the plane.

35. To construct a line perpendicular to a given plane, through a given external point.

36. To construct a plane parallel to a given plane, through a given point.

37. To construct a number of equal oblique lines to a plane from a given external point.

38. To construct a line through a given point parallel to a given plane.

39. To construct a line through a given point and parallel to each of two given intersecting planes.

40. To construct a plane containing one given line and parallel to another.

41. To construct a plane through a given point parallel to any two given lines in space.