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While performing the Hajj rituals, a pilgrim should follow the exact example of the Prophet (peace and blessings be upon him) who said, "Learn your rituals from me." As known, the Tawaf rounds are to be started at the Black Stone.

Focusing more on this issue, the late Saudi scholar Sheikh Ibn `Uthaymeen, (may Allah shower mercy on his soul) issued the following fatwa:

The one who starts his Tawaf from the door of the Ka`bah and completes it on this basis is not regarded as having completed Tawaf because Allah says: (And circumambulate the Ancient House (the Ka`bah)) (Al-Hajj 21:29).

The Prophet (peace and blessings be upon him) started his Tawaf at the Black Stone and said to the people: “Learn from me your rituals (of Hajj).” (Reported by Muslim)

If he starts Tawaf at the door of the Ka`bah, or not in line with the Black Stone, even if he is only off by a little, then this first circuit is invalid because it is not complete. He has to repeat it if he remembers shortly afterwards; otherwise he has to do the whole Tawaf over again, starting in line with the Black Stone.

A line has been placed on the ground in line with the Black Stone, extending to the edge of the area for Tawaf, as a sign of where to start and end Tawaf. Since this line was drawn, people’s mistakes in doing Tawaf have become fewer, but there are still a few ignorant people. Whatever the case, people have to pay attention to this mistake, lest they fall into the serious danger of their Tawaf not being complete.

Excerpted, with slight modifications, from: www.islam-qa.com
Allah Almighty knows best.