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Let $P$ be a point inside a triangle $A B C$, and $A_1 B_2, B_1 C_2, C_1 A_2$ be segments through $P$ parallel to $A B, B C_1 C A$ respectively, where points $A_1, A_2$ lie on $B C, B_1, B_2$ on $C A$, and $C_1, C_2$ on $A B$. Prove that
$\operatorname{Area}\left(A_1 A_2 B_1 B_2 C_1 C_2\right) \geq \frac{2}{3} \operatorname{Area}(A B C)$
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