In the given figure, side \( B C \) of \( \triangle A B C \) is bisected at \( D \) and \( O \) ...

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In the given figure, side \( B C \) of \( \triangle A B C \) is bisected at \( D \) and \( O \) is any point on \( A D \). \( B O \) and \( C O \) produced meet \( A C \) and \( A B \) at \( E \) and \( F \) respectively, and \( A D \) is produced to \( X \) so that \( D \) is the midpoint of \( O X \). Prove that \( A O: A X=A F: A B \) and show that \( E F \| B C \).
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