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By Bak A. (ed.)

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Additional resources for Algebraic K-Theory, Number Theory, Geometry and Analysis: Proceedings of July 26-30, 1982

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32. Prove that the bisector plane of a dihedral angle at an edge of a tetrahedron divides the opposite edge into parts proportional to areas of the faces that confine this angle. 33. In tetrahedron ABCD the areas of faces ABC and ABD are equal to p and q and the angle between them is equal to α. Find the area of the section passing through edge AB and the center of the ball inscribed in the tetrahedron. 34. Prove that if x1 , x2 , x3 , x4 are distances from an arbitrary point inside a tetrahedron to its faces and h1 , h2 , h3 , h4 are the corresponding heights of the tetrahedron, then x2 x3 x4 x1 + + + = 1.

10. Let O be the center of the given sphere, r its radius; a and b the lengths of tangents drawn from points A and B; let M be the intersection point of the tangents drawn from A and B; letx be the length of the tangent drawn from M . Then AM 2 = (a ± x)2 , BM 2 = (b ± x)2 and OM 2 = r2 + x2 . , so that α + β + γ = 0 and ±2αa ± 2βb = 0. We see that point M satisfies either the relation bAM 2 + aBM 2 − (a + b)OM 2 = d1 or the relation bAM 2 − aBM 2 + (a − b)OM 2 = d2 . Each of these relations determines a plane, cf.

Prove that on the surface of the 20-hedron there are two points the distance between which is greater than 21. 33. The length of a cube’s edge is equal to a. Find the areas of the parts into which the planes of the cube’s faces split the sphere circumscribed about the cube. 42 CHAPTER 4. 34. 1) all the plane angles of which are equal to 60◦ . The surface of the ball situated inside the angle consists of two curvilinear quadrilaterals. Find their areas. 35. Given a regular tetrahedron with edge 1, three of its edges coming out of one vertex and a sphere tangent to these edges at their endpoints.

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