Plane and solid geometry . hedron is tangent to sphere 0. 14. .-. sphere 0 is inscribed in tetrahedron V-ABC. Kbasovs 5. Args. 3 & 4. 6. § 194. 7. § 617, I. 8. § 639. 9. § 688. 10. § 688. 11. § 688. 12. § 54, 1. 13. § 925. 14. §§ 926, 927. III. Discussion The discussion is left as an exercise for the student. Ex. 1501. The six planes bisecting the dihedral angles of a tetrahe-dron meet in a point which is equidistant from the four faces of thetetrahedron. Hint. Compare with § 258. Ez. 1502. Inscribe a sphere in a given cube. Ex. 1503. The volume of any tetrahedron is equal to the produc
Plane and solid geometry . hedron is tangent to sphere 0. 14. .-. sphere 0 is inscribed in tetrahedron V-ABC. Kbasovs 5. Args. 3 & 4. 6. § 194. 7. § 617, I. 8. § 639. 9. § 688. 10. § 688. 11. § 688. 12. § 54, 1. 13. § 925. 14. §§ 926, 927. III. Discussion The discussion is left as an exercise for the student. Ex. 1501. The six planes bisecting the dihedral angles of a tetrahe-dron meet in a point which is equidistant from the four faces of thetetrahedron. Hint. Compare with § 258. Ez. 1502. Inscribe a sphere in a given cube. Ex. 1503. The volume of any tetrahedron is equal to the product ofits surface and one third the radius of the inscribed See §§ 491 and 492. Ex. 1504. Find a point within a triangular pyramid such that theplanes determined by the lines joining this point to the vertices shall the pyramid into four equivalent parts. Hint. Compare with Ex. 1094. 428 SOLID GEOMETRY Propositiox VIII. Problem 931. To circumscribe a sphere about a given tetrahe-dron. V. Given tetrahedron V-ABC. To circumscribe a sphere about tetrahedron V-ABC, I. Construction 1. Through D, the mid-point of AB, construct plane D0± AB;through E, the mid-point of BC, construct plane \ andthrough F^ the mid-point of VB^ construct plane FO ± VB. 2. Join 0, the point of intersection of the three planes, to A. 3. The sphere constructed with 0 as center and OA as radiuswill be circumscribed about tetrahedron V-ABC. 11. Outline of Proof 1. Prove that the three planes OD, OF, and OF intersect eachother in three lines. 2. Prove that these three lines of intersection meet in apoint, as 0. 3. Prove that OA = OB=OC= OV. 4. .-. sphere 0 is circumscribed about tetrahedron V-ABC, 932. Cor. Four points not in the same plane determinea sphere. 933. Questions. Are the methods used in §§ 930 and 931 similarto those used in §§ 321 and 323 ? In § 930 could the lines forming theedges of the dihedral angles bisected be three lines meeting in one ver
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