Hence HE and IF must meet in some point, as 0. Since O is in the perpendicular EH, it must be equidistant from A, D, and C. (473) And since it is in FI it is equidistant from B, C, and D. (473) Hence O is equidistant from A, B, C, and D, and a sphere described from 0 as a center with a radius equal to AO will be circumscribed about the tetraedron. 702. Cor. Four points not in the same plane determine a sphere. PROPOSITION VI. THEOREM 703. A sphere may be inscribed in any given tetraedron. B A Hyp. ABCD is a tetraedron. To prove a sphere may be inscribed in ABCD. Proof. Bisect any three diedral angles as AD, DC, and AC by the planes OAD, ODC, and OAC. Then 0 is equidistant from the four faces of the tetraedron. [To be completed by the student.] 704. Cor. The six planes bisecting the diedral angles of a tetraedron meet in a point. PROPOSITION VII. PROBLEM 705. To construct the diameter of a material sphere. Given. A material sphere ABCE. Construction. From any point P as center, with any [opening of compasses as] radius, draw circumference ABC. Measure (with compasses] the three chords AB, BC, and CA, and in any plane construct A A'B'C", having its sides respectively equal to AB, BC, and CA. Construct D'A' the radius of the circumscribed circle A'B'C'. Draw right triangle P'A"D", having the hypotenuse P"A" PA, and one arm D"A" D'A'. At A" draw A"E" I to P"A", meeting P''D' produced in E". Then PE" is the required diameter. [The proof is left to the student.] 706. DEF. The angle between two intersecting curves is the angle formed by the tangents at the point of contact. 707. DEF. A spherical angle is the angle between two intersecting great circles. Ex. 1171. The volume of any tetraedron equivalent to its surface multiplied by one-third the radius of the inscribed sphere. Ex. 1172. If in the diagram for Prop. VII, A'B C' is an equilateral triangle, find the angles between chord A' B' and arc A B'. 708. A spherical angle is measured by the arc of a great circle described from its vertex as a pole, and included between its sides, produced if necessary. Hyp. DPE is the spherical angle formed by the great circles PAC and PBC. AB is an arc of a great circle having P for its pole. To prove Z DPE is measured by are AB. AO is I to OP. (192) and AO and DP are in the plane of O PAO. Hence AO is li to DP. (84) Similarly BO is II to EP. (468) But Z AOB is measured by arc AB. Hence Z DPE is measured by arc AB. Q.E.D. 709. Cor. An angle formed by two great circles is equal to the plane angle of the diedral angle formed by their planes. SPHERICAL POLYGONS 710. DEF. A spherical polygon is a portion of the surface of a sphere bounded by three or more arcs of great circles. The arcs are the sides, their points of intersection are the vertices, and the spherical angles formed by the sides are the angles of the polygon. 711. DEF. A diagonal of a spherical polygon is an arc joining any two non-adjacent vertices. Thus, ABCD is a spherical polygon, AB, BC, etc., its sides, A, B, C, etc., its vertices, and & ABC, BCD, etc., its angles. 712. DEF. A spherical triangle is a spherical polygon of three sides. It is called isosceles, equilateral, etc., in the same cases in which a plane triangle would be so called. The planes of the sides of a spherical polygon form at the center a polyedral angle (0 - ABCD) which is said to correspond with the spherical polygon. The sides of the spherical polygon are measured by the sides of the corresponding polyedral angle, its angles are equal to the diedral angles of the corresponding polyedral angle. 713. REMARK. — By means of the relations between the parts of a spherical polygon and those of its corresponding polyedral angle, we can deduce from any theorem of polyedral angles an analogous one of spherical polygons. 714. A spherical polygon is convex if its corresponding polyedral angle is convex. All spherical polygons are supposed to be convex polygons unless stated otherwise. 715. Spherical polygons are symmetrical if their corresponding polyedral angles are symmetrical. Evidently, their parts must be respectively equal, but follow in reverse order. In general, two symmetrical spherical polygons cannot be made to coincide. The sides of a spherical polygon are usually measured in degrees. PROPOSITION IX. THEOREM 716. The sum of two sides of a spherical triangle is greater than the third side. Proof. Draw radii OA, OB, and OC. ZAOB + Z BOC> Z AOC. (522) But the central angle is measured by the intercepted arc. |