What do “3 different points” have to do with linear dependence in determining a unique circle?
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In learning about the general formula for a circle: $$(x-h)^2 + (y-k)^2 = r^2$$ my book states that $3$ points are sufficient to guarantee the solution (or absence of solution) for an unique circle. It follows this up by explaining that $3$ different points creates $3$ different equations...namely: 1) $(x_1-h)^2 + (y_1-k)^2 = r^2;$ where $;(x_1,y_1);$ is the first point provided 2) $(x_2-h)^2 + (y_2-k)^2 = r^2;$ where $;(x_2,y_2);$ is the second point provided 3) $(x_3-h)^2 + (y_3-k)^2 = r^2;$ where $;(x_3,y_3);$ is the third point provided I'm okay with this as I'm familiar with the whole " $3$ equations for $3$ unknowns". However, the book's explanation concludes with the statement: "The equations can't be linearly dependent since the points that lead to them