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En morsom geometri nøtt

Given a triangle [tex]ABC[/tex], find a triangle [tex]A_1B_1C_1[/tex], so that
(1) [tex]A_1\in BC[/tex],[tex]B_1\in CA[/tex],[tex]C_1\in AB[/tex];
(2) the centroids of [tex]△ABC[/tex] and [tex]△A_1B_1C_1[/tex] coincide; and
subject to (1) and (2), [tex]△A_1B_1C_1[/tex] has minimal area.

Hint;

Det er tre reelle tall, [tex]\alpha ,\beta ,\gamma[/tex], i (0,1) som bekrefter at;

[tex]\vec{BA_1}=\alpha \vec{BC}\: \: \vec{CB_1}=\beta \vec{CA}\: \: og\: \: \vec{AC_1}=\gamma \vec{AB}[/tex]