Sauda,c~oes, Oi Sergio,
i) IME 1986/1987 (9a questao) Sejam duas retas ortogonais r e r' nao coplanares. Considere sobre r dois pontos fixos A e B e sobre r' dois pontos variaveis M e M', tais que a projecao de M' sobre o plano que contem o triangulo MAB eh o ortocentro H deste triangulo. Determine o lugar geometrico dos centros das esferas circunscritas ao tetraedro ABMM'.
A solução que segue eu não entendi. Precisaria de algumas aulas de geometria espacial e bons desenhos para entendê-la. Espero que lhe seja útil. []'s L.
Dear Luís Lopes > Let r and r' be two orthogonal lines not belonging to > the same plane. Take two fixed points A and B over r > and two variable points M and M' over r' such that > the projection of M' over the plane that contains > MAB is the orthocenter H of this triangle. > Determine the locus of the centers of the spheres > that circumscribe the tetrahedre ABMM'. It is easy if we know some properties of the orthocentric tetrahedrons. If V is the common point of r' with the plane passing through r and orthogonal to r', the condition means that the tetrahedron is orthocentric with orthocenter the orthocenter H of ABV. As the centroid G of the tetrahedron moves on a line parallel to r', the center of the circumsphere, which is the reflection of H in G, will move too on a line parallel to r'. Friendly. Jean-Pierre
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