RE: [obm-l] 2 questoes do IME

[Luís Lopes <qed_texte@xxxxxxxxxxx>]

Sauda,c~oes,

Caro Sergio,

Foram muitas as respostas. Esta esclarece um pouco mais.

[]'s
L.

>Dear Luis,
>
>The answer to
>
> > 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'.
>
>is a line parallel to r'.
>
>The condition that H is proj. of M' implies that ABMM' is an orthocentric
>tetrahedron
>(such that its 4 altitudes concur). Its orthocenter H* lies on the common
>perpendicular
>of r and r'. As MM' varies, the point H* remains fixed. (This can be shown
>using the fact
>that MD*MH = const, where MD is an altitude of MAB.)
>In an orthocentric tetrahedron, the centroid is the midpoint of OH* (O the
>circumcenter).
>Obviously, the locus g of G is the image of r' under (1/2) dilation wrt the
>midpoint of AB.
>So the locus of O is the image of g under dilation with factor 2 wrt H*.
>
>Sincerely,
>Vladimir  Dubrovsky


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