DEFINITIONS 776 A sphere is inscribed in a polyedron when all the faces of the polyedron are tangent to the sphere. 777 A sphere is circumscribed about a polyedron when all the vertices of the polyedron lie in the surface of the sphere. PROPOSITION VI. THEOREM 778 A sphere may be inscribed in any tetraedron. CONCLUSION. A sphere may be inscribed in D-ABC. PROOF Let the planes OAB, OAC, and OBC bisect the diedral angles at the edges AB, AC, and BC respectively. Then every point in the plane OAB is equidistant from the faces ABC and ABD; every point in the plane OAC is equidistant from the faces ABC and ADC; and every point in the plane OBC is equidistant from the faces ABC and DBC. § 570 Hence O, the common intersection of these three planes, is equidistant from the four faces of the tetraedron, and is therefore the center of an inscribed sphere. § 776 Q. E. D. 779 COROLLARY. The six planes bisecting the six diedral angles of a tetraedron intersect in the same point. PROPOSITION VII. THEOREM 780 A sphere may be circumscribed about any tetraedron. CONCLUSION. A sphere may be circumscribed about D-ABC. PROOF Let H and K be the centers of the circles circumscribed about the faces ABC and DBC respectively. Draw HFL to the plane ABC, and KG to the plane DBC. Join E the middle point of BC to H and K. Then HE and KE are each L to BC. .. the plane HEK is 1 to BC. ..the plane HEK is to the planes ABC and DBC. ... HF and KG lie in the plane HEK. ... HF and KG intersect in some point O. Since all points in HF are equidistant from A, B, C, and all points in KG are equidistant from D, B, C, O is equidistant from D, A, B, C, and is therefore the of a sphere circumscribed about D-ABC. § 188 § 523 § 563 § 565 § 532 § 532 center 8 777 Q. E. D. 781 COROLLARY 1. The four perpendiculars erected at the centers of the circles circumscribing the four faces of a tetraedron meet in a point. 782 COROLLARY 2. The six planes perpendicular to the six edges of a tetraedron at their middle points intersect in a point. SPHERICAL ANGLES. DEFINITIONS 783 The angle of two intersecting curves is the angle formed by the tangents to the curves at their point of intersection. 784 A spherical angle is the angle formed by the intersection of two arcs of great circles of a sphere. PROPOSITION VIII. THEOREM 785 A spherical angle is measured by the arc of the great circle described from its vertex as a pole and included between its sides, produced if necessary. HYPOTHESIS. PAP' and PBP' are great circles to which PS and PT are tangents at P, and AB is an arc of a great circle whose pole is P. CONCLUSION. The spherical / APB is measured by the arc AB. PROOF Draw the radii OA and OB. PS is to PO (§ 251), and OA is L to PO. $ 274 § 115 786 COROLLARY. A spherical angle is equal to the plane angle of the diedral angle formed by the planes of the two circles. SPHERICAL POLYGONS. DEFINITIONS 787 A spherical polygon is a portion of the surface of a sphere bounded by three or more arcs of great circles. The sides of a spherical polygon are the bounding arcs; the vertices are the points in which the sides intersect; the angles are the angles formed by the sides. Since the sides of a spherical polygon are arcs, they are usually measured in degrees, minutes, and seconds. 788 The planes of the sides of a spherical polygon form by their intersections a polyedral angle whose vertex is the center of the sphere, whose face angles are measured by the sides of the polygon, and whose diedral angles are equal to the angles of the polygon. From these rela A B tions of polyedral angles and spherical polygons, we have the following 789 COROLLARY. From any property of polyedral angles we may infer an analogous property of spherical polygons. 790 A diagonal of a spherical polygon is the arc of a great circle joining two vertices not consecutive. 791 A convex spherical polygon is a spherical polygon whose corresponding polyedral angle is convex. A spherical polygon is considered convex unless otherwise stated. 792 A spherical triangle is a spherical polygon of three sides. Like a plane triangle, it may be right, obtuse, acute, scalene, isosceles, equilateral, or equiangular. 793 Any side of a spherical triangle is less than the sum of the other two sides. B HYPOTHESIS. AC is the longest side of the spherical ▲ ABC. PROPOSITION X. THEOREM 794 The sum of the sides of any spherical polygon is less than the circumference of a great circle. D HYPOTHESIS. ABCD is a spherical polygon. PROOF. Draw the radii OA, OB, OC, OD. Then AOB + BOC+ COD + AOD <360°. ▲ ≤ .. AB+ BC + CD + AD < 360°. § 591 § 788 Q. E. D. |