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concurrent sets of planes

Source: Romania IMO TST 1991 p2

February 19, 2020
concurrent planesconcurrentplanestetrahedron

Problem Statement

Let A1A2A3A4A_1A_2A_3A_4 be a tetrahedron. For any permutation (i,j,k,h)(i, j,k,h) of 1,2,3,41,2,3,4 denote: - PiP_i – the orthogonal projection of AiA_i on AjAkAhA_jA_kA_h; - BijB_{ij} – the midpoint of the edge AiAjA_iAj, - CijC_{ij} – the midpoint of segment PiPjP_iP_j - βij\beta_{ij}– the plane BijPhPkB_{ij}P_hP_k - δij\delta_{ij} – the plane BijPiPjB_{ij}P_iP_j - αij\alpha_{ij} – the plane through CijC_{ij} orthogonal to AkAhA_kA_h - γij\gamma_{ij} – the plane through CijC_{ij} orthogonal to AiAjA_iA_j. Prove that if the points PiP_i are not in a plane, then the following sets of planes are concurrent: (a) αij\alpha_{ij}, (b) βij\beta_{ij}, (c) γij\gamma_{ij}, (d) δij\delta_{ij}.
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