What relation is Abby to the Proctors?
In "The Crucible," written by Arthur Miller, Abigail Williams is not related to the Proctors by blood.
However, Abigail, is quite involved with the Proctors. First of all, she was at one point a servant working for the Proctors.
The major way in which she is "related" to the Proctors is that, while she was working as their servant, she was having an affair with John Proctor.
It is because of that she tries to get Elizabeth Proctor convicted of witchcraft, in addition to attempting to kill her by drinking blood.
Abby is not at all related to the proctors. She is their helper around the house for John's wife Elizabeth. Elizabeth is sick and she has been neglecting her husband and other duties. John hires Abigail to help and then falls for her. They have an affair he obviously "reaches for her" because later in the play she confronts him about it.
This affair leads to the very act of "dancing" in the forest with Tituba. Abigail will stop at nothing to have John Proctor as her own. It is she who throws the dead chicken in the pot. It is she who drinks blood and it is she who wants Goody Proctor dead so that she can have her husband. She is also the first to lie when confronted about their actions.
Therefore, though not related to the proctors, she is intricately intertwined with them and the catalyst for many of their conflicts and the conflicts in the story.