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In Lepage and Brassard's Polygraph, the three characters are intricately woven together.
Francois is introduced as he discusses the history of the Berlin Wall. Interestingly, as we read we find that Francois is stuck in the past: he is haunted by memories of the polygraph test given to him when he was a suspect in his friend's murder. His response in the flashback shows that he is desperate (it would seem) to convey his innocence, even though we later learn that the police were certain of his innocence, but never told him.
(a complete emotional breakdown)
...I didn't kill her!! It wasn't me! It wasn't me! You want to drive me mad, that's it!! You are driving me mad...
Francois is a tenderhearted, yet tortured soul.
David, on the other hand, seems almost soulless. His intellect recognizes that the handling of the polygraph test can damage the subject's psyche.
...such strategies...should be used only with great care and compassion [...because] the psychological response we trigger is so violent as to effect a lasting disorder in the mind of a totally innocent suspect...
But he shows no genuine concern for Francois or anyone else. He cries tears while remember Anna, who he left behind in Berlin—but his tears are not real: they are a trick of the actor's trade. He can understand that Francois might be mentally fragile, but never takes steps to enlighten him or encourage him. David hears Francois' lamenting cries from the waiter's apartment next door to Lucie but does nothing. David is unfazed by the horror that Lucie suffers when she witnesses the young man's suicide in front of the Metro train. He shows no compassion. He is like an automaton. Lucie has to yell at him at the end because he shows no emotion when he learns that Francois and Lucie have slept together.
Lucie has a foot in both of these worlds. She handles the scenes she acts out in the play with a certain detachment until she learns that the dead woman (upon whom her role is based) was a friend of Francois. At this point, she cries without needing the fake tears. She dates David. She struggles with keeping David's information about the polygraph from Francois. Lucie sleeps with Francois in an effort to comfort and reassure him. When David shows no reaction, she demands that he show his feelings:
David...React!...Feel something!...If you want to cry, cry!
In terms of how the plot is developed, each character has a specific function. When David finds himself in the company of a man he administered the polygraph to six years before, we are given insight into why Francois seems so tormented and masochistic. He can't even recall if he is innocent or not: David's presence is necessary to explain the core of Francois' character, his suffering.
Francois lives next door to Lucie. Lucie knows Francois' boyfriend, and Francois has an extra key to Lucie's apartment in case she loses hers. They have a genuine friendship. Lucie witnesses a suicide; this experience brings David into her social circle, allowing David to see Francois again, years after they first met. (Francois does not recognize David.)
In interacting with Francois, we not only learn of the rape and murder of Francois' friend, Marie, but we also recognize how emotionally detached David is from his career, Anna, and even Lucie.
Each character's interaction with the others galvanizes the plot forward and explains the conflicts central to the play.
In Polygraph, the characters are only minimally related to each other: they are related through circumstances, not through personal involvement with each other's lives. The relationship revolves around Francois who is suspected of murder. The other character performs the autopsy while the other character is an actress auditioning for the film being made about the murder. The film leads to Francois' contemplations about how fiction can override truth and adds to his self-doubt that has arisen after he is not told whether he passed or failed the lie detector test.
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