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  "body": "\n## Statement\n\nLet $G$ be a finite simple [[planar graph|0NC9R64TXP]].\nThen $G$ has an [[embedding|CG91WNMG2T]] in the plane in which every edge is a straight line segment.\n\n## Remarks\n\nThe idea of the proof is induction on the number of vertices.\nAdd edges until $G$ is maximal planar, so that every face, including the outer one, is a triangle.\nFor at least four vertices, Euler's formula gives a vertex $v$ of degree at most $5$ that is not on the outer triangle.\nRemove $v$, triangulate the face it leaves, and draw the smaller graph with straight edges by induction.\nDelete the added diagonals; the face left by $v$ is a polygon with at most $5$ sides.\nSuch a polygon always contains a point that sees all its corners, so $v$ can be placed there and joined by straight segments.\n\nThe theorem is named after István Fáry; Klaus Wagner had proved it earlier, and Sherman Stein proved it independently.\nThe result needs $G$ to be simple: a multigraph with two parallel edges cannot be drawn with both edges straight.\n\n## Explanations\n\n### L1 human\n\nIf a network can be drawn on paper without crossing lines, using curvy lines if needed, then it can also be drawn without crossings using only straight lines.\nSo curves give no extra power for planar drawings.\n\n### L2 human\n\nFáry's theorem lets one treat planar graphs as plane [[geometric graphs|HFJCAPWWD4]].\nStronger versions exist: every planar graph with $n$ vertices has a straight-line drawing with vertices on an $O(n) \\times O(n)$ integer grid.\nBy the Koebe–Andreev–Thurston theorem, every planar graph is the contact graph of a family of discs with disjoint interiors.\nTutte's spring embedding gives a straight-line drawing of every $3$-connected planar graph with convex faces.\n",
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