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  "body": "\n## Definition\n\nLet $n \\ge 3$ be an integer.\nA *polygon* with $n$ vertices in the [[Euclidean plane|0XW7V9ZZ05]] is a cyclic sequence of points $P_1, \\ldots, P_n$, with indices taken modulo $n$, such that:\n\n1. $P_i \\ne P_{i+1}$ for every $i$.\n2. $P_{i-1}$, $P_i$, $P_{i+1}$ are not [[collinear|0AFNNTRTJE]] for every $i$.\n\nThe points $P_i$ are its *vertices* and the segments $P_i P_{i+1}$ are its *edges*.\nThe polygon is *simple* if two edges meet only when they are consecutive, and then only in their common vertex.\n\n## Remarks\n\nBy the [[Jordan curve theorem|687RGCHBTP]], a simple polygon bounds one bounded region of the plane, its interior.\nMany authors use the word polygon for this closed region rather than for the boundary.\nA polygon with $3$ vertices is a [[triangle|0CTNFGYMM2]], and one with $4$ vertices is a quadrilateral.\nA simple polygon is convex if its interior is a [[convex set|02AS45T3H9]].\n\nExample: $(0, 0), (1, 0), (1, 1), (0, 1)$ is a simple polygon, the unit square.\nConsecutive vertices differ, and no three consecutive ones lie on a line.\n\nNon-example: $(0, 0), (1, 0), (2, 0), (1, 1)$ fails clause 2.\nThe three consecutive vertices $(0, 0), (1, 0), (2, 0)$ are collinear, so the figure is a triangle with an extra point on one edge.\n",
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