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11.10

Part of I Soros Olympiad 1994-95 (Rus + Ukr)

Problems(1)

property of points whose projections on edges of tetrahedron are coplanar

Source: I Soros Olympiad 1994-95 Round 1 11.10

8/1/2021
Given a tetrahedron A1A2A3A4A_1A_2A_3A_4 (not necessarily regulart). We shall call a point NN in space Serve point, if it's six projection points on the six edges of the tetrahedron lie on one plane. This plane we denote it by a(N)a (N) and call the Serve plane of the point NN. By BijB_{ij} denote, respectively, the midpoint of the edges A1AjA_1A_j, 1≤i<j≤41\le i <j \le 4. For each point MM, denote by MijM_{ij} the points symmetric to MM with respect to Bij,B_{ij}, 1≤i<j≤41\le i <j \le 4. Prove that if all points MijM_{ij} are Serve points, then the point MM belongs to all Serve planes a(Mij)a (M_{ij}), 1≤i<j≤41\le i <j \le 4.
geometry3D geometrytetrahedronprojections