\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
\(TRIA\) is a tetrahedron. \(E\) is a point on segment \([TA]\), distinct from \(T\) and \(A\). The line through \(E\) parallel to \((AI)\) intersects \((TI)\) at \(F\), and the line through \(E\) parallel to \((RA)\) intersects \((TR)\) at \(G\).
  1. Sketch a figure.
    1. Justify that vectors \(\Vect{TA}\) and \(\Vect{TE}\) are collinear. We then write \(\Vect{TE} = \alpha \Vect{TA}\) where \(\alpha\) is a real number.
    2. Express \(\Vect{TF}\) in terms of \(\Vect{TI}\) and \(\alpha\). Then, express \(\Vect{TG}\) in terms of \(\Vect{TR}\) and \(\alpha\).
    3. Deduce that \(\Vect{GF}\) and \(\Vect{RI}\) are collinear, and conclude about the relative position of lines \((GF)\) and \((RI)\).

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