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Under which conditions there is an intersection point for all bisectors of a pyramid?

A bisector of a pyramid initiating from a vertex is a line which has the same angle with all edges who are neighbor of that vertex.(A natural generalization of bisectors in triangle).

Mikasa
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Ali Taghavi
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  • @ResidentDementor No I have no idea. But it would be interesting(and surprising) if they always have a common point. – Ali Taghavi Feb 11 '18 at 07:52
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    Since "bisect" literally means "cut into two parts," it seems strange to have a "bisector" defined by more than two equal things. I'm not sure what alternative term to suggest, however. – David K Feb 13 '18 at 13:46

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These lines meet at the same point iff there is a sphere tangent to all six edges of the tetrahedron (for each vertex, such line clearly contains all possible centres of spheres tangent to the three edges adjacent to that vertex), which is not always the case. Tetrahedra with this property has several equivalent descriptions, see Theorem 1 in http://forumgeom.fau.edu/FG2007volume7/FG200703.pdf .

Vladimir Dotsenko
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  • Thank you and +1 for your answer. I am not sure that I understood (and am convinced) that the intersection point of bisectors is the center of spher tangent to all face. I should try more. But what about the following question: Is there a common point for all medieans of a pyramid? A medean is a line starting from a vertex and ending at the median of the front face(the face front to the vertex). – Ali Taghavi Feb 15 '18 at 23:48
  • @AliTaghavi it's tangent to all edges, not to all faces. Your other question does not make sense, the sentence "A medean is a line starting from a vertex and ending at the median of the front face(the face front to the vertex)" does not define a particular line. If you mean that you connect each vertex to the centroid of the opposite face, then a rather trivial computation with vectors shows that those lines are concurrent. – Vladimir Dotsenko Feb 16 '18 at 07:50
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    Yes I meant the centroid of the front face. It was my typos. Thank for your answer. – Ali Taghavi Feb 16 '18 at 13:48