Oh, dude, your great-grandmother's first cousin is like... your first cousin thrice removed. Yeah, it's like they're kind of related to you, but like, super distant, you know? So, like, you might not even know they exist until you do one of those intense family tree projects for school or something.
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Well, isn't that a lovely family tree question! Your great-grandmother's first cousin is actually your first cousin thrice removed. Isn't that just a happy little connection to discover in your family history?
Your great uncle's granddaughter is your second cousin.
You and your first cousins share the same Grandmother.
Your aunt's grandson is the son of your First Cousin and is therefore your First Cousin once removed Ref: http://freepages.genealogy.rootsweb.ancestry.com/~gentutor/chart.html
There is no relationship in the English language that is called a "half cousin." Everyone who is your cousin is related to you through one and only one parent, so, unlike a "half brother" there can be no such thing as a "half cousin." If you have a first cousin (child of your aunt or uncle), then your child and that first cousin are first cousins once removed. If you have a second cousin, they your child and that second cousin are second cousins once removed.
Your first cousin's daughter is first cousin once removed to your grandson, and he to her. Your second cousin's daughter is second cousin once removed to your grandson, and he to her.