Is Ophelia a tool that is used to prove that Hamlet is really mad? Once she is mad, does she deliver thoughtful speeches just like Hamlet?

An important consideration when approaching the question
of madness in characters in drama from the Renaissance is that there were rules of a
sort about using form and structure in language to signal aspects of character, and
these rules applied to the playwright's use of verse and/or
prose.

For example, characters
speaking in a formal and regal way always spoke in
verse--the poetic text written in Shakespeare's plays in
iambic pentameter.  Any character (especially characters in love) who uses descriptive
and poetic language speaks in verse.  This was simply a "rule" of drama in Shakespeare's
day.

For prose, the most
common usage was by the clowns, the "low-born" comic characters, but prose was also a
very common choice to denote the state of mind in a character being "madness."  King
Lear speaks in prose during his "mad" scenes on the heath, Hamlet speaks in prose when
he is onstage with characters he wishes to believe that he is "mad," and Ophelia, in Act
IV, once Polonius is dead, speaks in prose.

Why all the
information about prose vesus verse?  Well, in Hamlet, it is the
best way I know of to settle the question of Hamlet's madness.  The key to
the question of Hamlet's being "really mad" or not is not how he treats Ophelia or
Polonius or Rosencrantz and Guildenstern.  It is his use of prose when he is onstage
with these characters versus his sublime and lucid use of verse when he is onstage alone
with the audience and when he is onstage with Horatio (who is in on his
ruse).

Hamlet is pretending to be mad, just
as he said he would in Act I, scene v:

readability="15">

Here, as before, never, so help you
mercy,

How strange or odd some'er I bear myself
--

As I perchance hereafter shall think
meet

To put an antic disposition on. .
.

And Ophelia is quite mad
indeed when we see her post-Polonius' murder in Act IV.  There is, in fact, nothing at
all like Hamlet's thoughtful, penetrating soliloquies delivered by Ophelia.  She speaks
of her "ladies" and her "coach," and sings disjointed bawdy songs.  All in
prose, indicating her madness.  Hamlet's soliloquies are in
verse, indicating the lucid, high-born beauty of his
speech, and reminding the audience, when he is alone with them, that he
is but "mad in craft."

For more on madness and the use of
prose in Hamlet, please follow the links
below.