They didn't break up because of sasuke . Technically they didn't even break up they became rivals
Their relationship developed to Rivals . Rivals can be best friends Like gai and kakashi for example .(ino X sakura relationship was paralleled to gai X kakashi and naruto X sasuke in the manga)
Before sakura even like sasuke she was ino shadow
She couldn't stand up for herself .ino is the one who defended her while sakura couldn't do the same .what she had with ino couldn't be called friendship on sakura part .
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She always felt less than ino even before liking sasuke
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Ino liking sasuke Is just the turning point for the two to compete on something .
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She Become a rival to be her equal . Friends supposed to be equal but before they became rivals ino didn't see sakura as her equal and sakura was only the girl who follow her around.
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Sakura didn't want that .she respect and love ino she was her role model ,but she wants to be her equal .
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(This is the same thing naruto did with sasuke to prove they were equal)
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Ino finally gets what sakura meant.
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Now they can be called friends . They are equal their relationship is better than before.