{
  "aliases": [
    "ruler-and-compass-construction",
    "compass-and-straightedge-construction"
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  "body": "\n## Summary\n\nA straightedge and compass construction draws points, lines and circles in the plane from given points with two ideal tools.\nThe straightedge draws the line through two constructed points, and it has no marks.\nThe compass draws the circle with a constructed centre through a constructed point.\nNew points are the intersections of the lines and circles drawn so far.\nThe [[constructible numbers|0FMN2MN7TM]] describe the points that these steps can reach.\n\n## Remarks\n\nEach step solves linear or quadratic equations.\nSo the coordinates of the new points lie in an extension of degree $1$ or $2$ of the field of the old coordinates.\nThis turns questions about constructions into questions about [[degrees of field extensions|0GS74FGFEK]].\nIt shows that [[doubling the cube|04SD3F4FCR]], [[trisecting the angle|00J3DCCEK5]] and [[squaring the circle|0BSXRER81B]] are impossible with these tools.\n\n## Explanations\n\n### L1 human\n\nThe ancient Greeks asked which figures can be drawn with an unmarked ruler and a compass alone.\nField theory gives the answer: a length can be drawn exactly when it comes from rational numbers by field operations and square roots.\n",
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