"Το αυγό είναι πίσω από το πιάτο."
Translation:The egg is behind the plate.
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"In back of" means "behind another thing/person" so it fits here. "In the back of" at "the back part/the end".
"We stood in the back of the theater because there were no seats and we were in back of many people."
"The vase was in back of the bottle."
But: "The tea cup was in "the" back of the cupboard." (The part furthest from the opening of the cupboard.) Hope that helps.
Του/της/τους etc are the weak genitive forms of the pronouns and are used for indirect objects. If we use the strong/analytic form though, we have to use preposition+accusative for the indirect object (genitive is used in some cases, but not in cases like this one).
So we have:
Το τραπέζι είναι πίσω από τον άντρα=The table is behind the man
Το τραπέζι είναι πίσω του=The table is behind him
Η γάτα είναι πίσω από τα κορίτσια=The cat is behind the girls
Η γάτα είναι πίσω τους=the cat is behind them
Το βιβλίο είναι μπροστά από τη γυναίκα=The book is in front of the woman
Το βιβλίο είναι μπροστά της=The book is in front of her.