Katniss didn't want Peete to fall asleep, cause she was afraid that if he would fall asleep he would never wake up.
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Katniss showed loyalty to Peeta when she wouldn't leave him in the cave. He was injured and would be a burden to her, but she was loyal to him and helped him recover.
Peeta fell in love with Katniss when they were young, and Katniss didn't return the feelings. Peeta was in love with Katniss before the story started, and Katniss began to like him back during the "cave scene" near the middle of the first novel.
They dry up the nearby water supply.
Peeta tells Katniss that she is not good at lying. He is proven wrong when he is later tricked by Katniss into consuming the sleeping syrup when she tells him it's a mix of sugar berries. He is also duped by Katniss because he thinks her feelings for him are real. He realizes after the Games, that she was only pretending.
Clove punishes him for helping Katniss escape by stabbing him in the leg then abandoning him. Peeta can't find anyone to help him so he camouflages himself by burying himself in mud. Then Caesar that two tributes from the same district can both win, so Katniss looks for Peeta and finds him in the mud with a very high fever. Then she tries to heal him. She goes to some kind of feast that has what she needs and finds the medicine that can cure him.