776. THEOREM. Any point in the arc of a great circle that bisects

a spherical angle is equally distant from the sides of the angle.

Given Spherical Z BAC; arc AT bisect

:

ing it; any point P, of arc AT; PD and PE, arcs of great to AB and AC, respectively.

To Prove: Arc PD arc PE.

Proof: The arcs PD and PE, if prolonged, will pass through R and S, respectively, the poles of AB and AC (?) (729). Draw arcs AR and AS.

Now,

Also RA

=

DAR LEAS. [Each is a right ; (727).]
DAP = EAP (Hyp.). Subtracting,
RAP SAP (Ax. 2).

SA (?) (724), and AP = AP.

..▲ RAP is symmetrical to ▲ SAP (?) (765, I).

Q.E.D.

.. RP = SP (?).

But RD SE. (Each is a quadrant.) .. PD = PE (Ax. 2).

777. THEOREM.

Any point on the surface of a sphere and equally distant from the sides of a spherical angle is in the arc of a great circle that bisects the angle. (The proof is similar to the proof of 776.)

ORIGINAL EXERCISES

1. Vertical spherical angles are equal.

2. If two spherical triangles, on the same or equal spheres, are mutually equilateral, their polar triangles are mutually equiangular.

3. The polar triangle of an isosceles spherical triangle is isosceles. 4. The polar triangle of a birectangular spherical triangle is birectangular.

5. If two dihedral angles of a trihedral angle are equal, the opposite face angles also are equal.

Proof: Construct a sphere having the vertex as center, etc.

6. If two face angles of a trihedral angle are equal, the opposite dihedral angles also are equal.

7. A trirectangular spherical triangle is its own polar triangle.

8. Two symmetrical spherical polygons are equivalent.

9. Any side of a spherical polygon is less than the sum of the other sides. [Draw diagonals from a vertex.]

10. If the three face angles of a trihedral angle are equal, the three dihedral angles also are equal.

11. State and prove the converse of No. 10.

12. A straight line cannot meet a spherical surface in more than two points.

13. If two dihedral angles of a trihedral angle are unequal, the opposite face angles are unequal, and the greater face angle is opposite the greater dihedral angle.

14. State and prove the converse of No. 13.

15. Two lines tangent to a sphere at a point determine a plane tangent to a sphere at the same point.

16. All the tangent lines drawn to a sphere from an external point are equal.

17. The volume of any tetrahedron is equal to one third the product of its total surface by the radius of the inscribed sphere.

18. Every point in the circumference of a great circle that is perpendicular to an arc at its midpoint is equally distant from the ends of the

arc.

19. The points of contact of all lines tangent to a sphere from an external point lie in the circumference of a circle.

20. The arcs of great circles perpendicular to the sides of a spherical triangle at their midpoints meet in a point equally distant from the vertices.

21. If the opposite sides of a spherical quadrilateral are equal, the opposite angles are equal.

22. If the opposite sides of a spherical quadrilateral are equal, the diagonals bisect each other.

23. If the diagonals of a spherical quadrilateral bisect each other, the opposite sides are equal.

24. The exterior angle of a spherical triangle is less than the sum of the opposite interior angles.

25. The sum of the angles of a spherical quadrilateral is more than four right angles and less than eight right angles.

26. If two spheres are tangent to each other, the straight line joining their centers passes through the point of contact.

27. The sum of the angles of a spherical polygon is more than 2 n right angles and less than 2 n right angles.

4

28. The arcs of great circles bisecting the angles of a spherical triangle meet in a point.

29. A circle may be inscribed in any spherical triangle.

30. If a tangent line and a secant be drawn to a sphere from an external point, the tangent is a mean proportional between the whole secant and the external segment.

31. The product of any secant that can be drawn to a sphere from an external point, by its external segment, is constant for all secants drawn through the same point.

32. If two spherical surfaces intersect and a plane be passed containing their intersection, tangents from any point in this plane to the two spherical surfaces are equal.

33. Find the distance from the center of a sphere whose radius is 15 to the plane of a small circle whose radius is 8.

34. The polar distance of a small circle is 60° and the radius of the sphere is 12 in. Find the radius of the circle.

35. The total surface of a tetrahedron is 90 sq. m., and the radius of the inscribed sphere is 4 m. Find the volume of the tetrahedron.

36. Find the radius of the sphere inscribed in a tetrahedron whose volume is 250 and total surface is 150.

37. Find the total surface of a tetrahedron whose volume is 320, if the radius of the inscribed sphere is 8.

38. Find the radius of the sphere inscribed in a regular tetrahedron whose edges are each 10 in.

39. Find the radius of the sphere circumscribed about a regular tetrahedron whose edges are each 18 in.

40. Find the radii of the spheres inscribed in and circumscribed about a cube whose edges are each 10 in.

41. The sides of a spherical triangle are 60°, 80°, 110°. Find the angles of its polar triangle.

42. The angles of a spherical triangle are 74°, 119°, 87°. Find the sides of its polar triangle.

43. The chord of the polar distance of the circle of a sphere is 12, and the radius of the sphere is 9. Find the radius of the circle.

44. The polar distance of a circle is 60° and the diameter of the circle is 8. Find the diameter of the sphere.

[Denote by R, each side of an equilateral triangle whose altitude is 4.]

45. The radii of two spherical surfaces are 11 in. and 13 in., and their centers are 20 in. apart. Find the radius of the circle of their intersection. Find also the distances from the centers of the spheres to the center of this circle.

46. The radii of two spherical surfaces are 20 m. and 37 m., and the distance between their centers is 19 m. What is the length of the diame ter of their intersection?

47 To bisect an arc of a great circle.

48. To draw an arc of a great circle perpendicular to a given arc of a great circle through a given point in the arc.

49. To bisect a spherical angle.

50. To bisect an arc of a small circle.

51. To circumscribe a circle about a given spherical triangle.

52. To construct a spherical angle equal to a given spherical angle at a given point on the same sphere.

53. To construct a spherical triangle having the three sides given. 54. To construct a spherical triangle having the three angles given. 55. To construct a plane tangent to a sphere at a given point on the surface.

56. To construct a spherical surface having the radius given and containing three given points.

57. To construct a spherical surface that shall have a given radius, touch a given plane, and contain two given points.

58. To construct a spherical surface that shall have a given radius, shall be tangent to a given sphere, and contain two given points.

59. To construct a spherical surface that shall contain four given points.

60. To construct a plane that shall contain a given line and be tangent to a given sphere.

61. To construct a plane tangent to a given sphere and parallel to a given plane.

62. What is the locus of points on the surface of a sphere: (a) Equally distant from two given points on the surface?

(b) Equally distant from two given points not on the surface?

(c) Equally distant from two given intersecting arcs of great circles?

ROBBINS' SOLID GEOM.-25

AREAS AND VOLUMES

778. A lune is a portion of the surface of a sphere bounded by two semicircumferences of great circles.

The points of intersection of the sides of a lune are the vertices of the lune.

The angles made at the vertices by the sides are the angles of the lune.

779. A zone is a portion of the surface of a sphere bounded by the circumferences of two circles whose planes are parallel.

The bases of a zone are the circumferences bounding it. The altitude of a zone is the perpendicular distance between the planes of its bases.

If one of the planes is tangent to the sphere the zone is a zone of one base.

780. A spherical degree is the one seven-hundred-and twentieth part of the surface of a sphere. If the surface of a sphere is divided into 720 equal parts, each part is a spherical degree.

The size of a spherical degree depends on the size of the sphere. It may be easily conceived to be half a lune whose angle is a degree, that is, a birectangular spherical triangle whose third angle is 1°.

How many spherical degrees in a trirectangular spherical triangle?