In the books they are not related at all as far as what was written. What was originally thought, was that Elena was the reincarnation of Katherine, but this was before anyone knew she was in the tomb underneath fells church
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In the Vampire Diaries, Katherine had a baby before she turned into a vampire. Which was in the 1400s. Which that baby had a baby and so on. Then how many babies afterwards, there came Isobel. Isobel gave birth to Elena, but Isobel gave Elena up and the Gilbert's' took her.
Katherine is Elena's ancestor and Elena is Katherine's descended
It hasn't been revealed why Elena and Katherine look so alike but it's possible Isobel had some connection to Katherine if she's also Elena's birth mother.
It hasn't been revealed yet how they're related
- Yes, Katherine is jealous of the relationship Elena and Stefan have.
Katerina Petrova had a child before she was turned but it was an unmarried child so the family disowned it. Elena was adopted and Isobel, Elena's biological mother, is a descendant of that child.