This is what I know from what I've read: Shortly after Nathaniel Gist and Daniel Boone went together to explore and settle Kentucky they got into a disagreement and went their separate ways. Nathaniel Gist was later captured by the Cherokee in southeastern Kentucky and spent almost six years living among them. While he lived with the Cherokee in Tennessee he "married" Chief Doublehead's sister and fathered a child, George Gist (Indian name Sequoyah). (Sequoyah later developed the Cherokee alphabet and printed the first Cherokee-language Bible.) This is where the Gists line crosses into Cherokee blood.
Oddly, I am supposed to be doubly related to Sequoyah. On my father's side of the family Nathaniel Gist and Sequoyah are 2nd and 3rd cousins. But on my mother's side of the family Sequoyah is my first cousin. My mom's father is descended from a long line of Cherokee Indian Chiefs, including Chief Doublehead, who was Sequoyah's uncle. It's strange how my mom's tree crosses over into my dad's tree almost 200 years ago.