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  "body": "\n## Summary\n\nSynthetic geometry is geometry done from axioms about figures, without coordinates or formulas.\nIts proofs reason directly with points, lines, circles and the relations between them, as in [[Euclid's Elements|83WHDAV68J]].\nIt is opposite to analytic geometry, which translates figures into equations.\nAxiom systems such as [[Hilbert's axioms|KM1FB8XZ2R]] and the theory of [[projective planes|DANFNN8C4Y]] are modern forms of synthetic geometry.\n\n## Explanations\n\n### L1 human\n\nThere are two ways to prove that the medians of a triangle meet in one point.\nOne can give the vertices coordinates and solve equations; that is analytic geometry.\nOr one can argue with the figure itself, using similar triangles; that is synthetic geometry.\nSynthetic proofs often show why a result is true, while analytic proofs are often more mechanical.\n",
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