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Coordinate or analytic geometry is appropriate to use when algebra is used to solve a problem or construct a proof. That usually deals with min/max of polynomials, construction of lines and tangents to objects like circles. Parallel lines and their relation to independence, dependence, or inconsistent solutions, etc. Non-coordinate geometry or plane geometry is useful for proofs of objects like triangles and their components like angles and sides, and congruency and similar triangles.
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