RE: [obm-l] 2 questoes do IME

[Luís Lopes <qed_texte@xxxxxxxxxxx>]

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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