Now, as to your question: of course, Aragorn looks Éowyn in the eye -- to do
less would be impolite. But he does not "flirt" with her; in fact, he's
deeply troubled when he realises that she is attracted to him:
"As she stood before Aragorn she paused suddenly and looked upon him, and her
eyes were shining. And he looked down upon her fair face and smiled; but
as he took the cup, his hand met hers, and he knew that she trembled at the
touch. 'Hail Aragorn son of Arathorn!' she said. 'Hail Lady of
Rohan!' he answered, but his face now was troubled and he did not smile."
Unfortunately, he's spoken for; and he also realises that this is going to be a
sticky situation. From now on -- until when they meet after she falls in
love with Faramir -- Aragorn remains strictly polite and formal towards
her. He does not flirt with her at any
time.
__________________________________
All that is gold does not glitter,
Not all those who wander are lost;
The old that is strong does not wither,
Deep roots are not reached by the frost.
From the ashes a fire shall be woken,
A light from the shadows shall spring;
Renewed shall be blade that was broken,
The crownless again shall be king.
