{
  "schemaVersion": "1.0",
  "discourse": {
    "id": "euclid-elements-book-xi",
    "name": "Euclid's Elements, Book XI",
    "subject": "solid_geometry",
    "variant": "classical",
    "description": "Solid geometry: planes, perpendiculars, parallelepipeds, prisms. 28 definitions, 39 propositions. Depends on Books I and VI. Source: David E. Joyce.",
    "structure": {
      "books": 11,
      "definitions": 28,
      "propositions": 39,
      "foundationTypes": [
        "foundation"
      ]
    }
  },
  "metadata": {
    "created": "2026-03-18",
    "lastUpdated": "2026-03-18",
    "version": "1.0.0",
    "license": "CC BY 4.0",
    "authors": [
      "Welz, G."
    ],
    "methodology": "Programming Framework",
    "citation": "Welz, G. (2026). Euclid's Elements Book XI Dependency Graph. Programming Framework.",
    "keywords": [
      "Euclid",
      "Elements",
      "Book XI",
      "solid geometry",
      "plane",
      "parallelepiped",
      "prism"
    ]
  },
  "sources": [
    {
      "id": "joyce",
      "type": "digital",
      "authors": "Joyce, David E.",
      "title": "Euclid's Elements, Book XI",
      "year": "1996",
      "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/bookXI.html",
      "notes": "Clark University"
    }
  ],
  "nodes": [
    {
      "id": "BookI",
      "type": "foundation",
      "label": "Book I — Plane geometry",
      "shortLabel": "Book I",
      "short": "Foundation",
      "book": 1,
      "colorClass": "foundation"
    },
    {
      "id": "BookVI",
      "type": "foundation",
      "label": "Book VI — Similar figures",
      "shortLabel": "Book VI",
      "short": "Foundation",
      "book": 6,
      "colorClass": "foundation"
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      "id": "Prop1",
      "type": "proposition",
      "label": "A part of a straight line cannot be in one plane and part in another elevated",
      "shortLabel": "Prop. XI.1",
      "short": "Line part in plane",
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      "number": 1,
      "colorClass": "proposition"
    },
    {
      "id": "Prop2",
      "type": "proposition",
      "label": "If two straight lines cut one another, they lie in one plane; every triangle in one plane",
      "shortLabel": "Prop. XI.2",
      "short": "Two lines cut: one plane",
      "book": 11,
      "number": 2,
      "colorClass": "proposition"
    },
    {
      "id": "Prop3",
      "type": "proposition",
      "label": "If two planes cut one another, their intersection is a straight line",
      "shortLabel": "Prop. XI.3",
      "short": "Planes cut: line",
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      "number": 3,
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    {
      "id": "Prop4",
      "type": "proposition",
      "label": "If line at right angles to two lines cutting at point, also perpendicular to plane through them",
      "shortLabel": "Prop. XI.4",
      "short": "Line perpendicular to plane",
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      "number": 4,
      "colorClass": "proposition"
    },
    {
      "id": "Prop5",
      "type": "proposition",
      "label": "If line at right angles to three lines meeting at point, the three lie in one plane",
      "shortLabel": "Prop. XI.5",
      "short": "Three lines from point",
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      "number": 5,
      "colorClass": "proposition"
    },
    {
      "id": "Prop6",
      "type": "proposition",
      "label": "If two lines at right angles to same plane, they are parallel",
      "shortLabel": "Prop. XI.6",
      "short": "Perpendicular to same plane: parallel",
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      "number": 6,
      "colorClass": "proposition"
    },
    {
      "id": "Prop7",
      "type": "proposition",
      "label": "If two lines parallel, line joining points on each is in same plane",
      "shortLabel": "Prop. XI.7",
      "short": "Parallel lines: join in plane",
      "book": 11,
      "number": 7,
      "colorClass": "proposition"
    },
    {
      "id": "Prop8",
      "type": "proposition",
      "label": "If two lines parallel, one perpendicular to plane, so is the other",
      "shortLabel": "Prop. XI.8",
      "short": "Parallel: one perpendicular",
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      "number": 8,
      "colorClass": "proposition"
    },
    {
      "id": "Prop9",
      "type": "proposition",
      "label": "Lines parallel to same line but not in same plane are parallel to each other",
      "shortLabel": "Prop. XI.9",
      "short": "Parallel to same: parallel",
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      "number": 9,
      "colorClass": "proposition"
    },
    {
      "id": "Prop10",
      "type": "proposition",
      "label": "Two lines meeting parallel to two meeting not in same plane: contain equal angles",
      "shortLabel": "Prop. XI.10",
      "short": "Skew lines: equal angles",
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      "number": 10,
      "colorClass": "proposition"
    },
    {
      "id": "Prop11",
      "type": "proposition",
      "label": "To draw line perpendicular to given plane from given elevated point",
      "shortLabel": "Prop. XI.11",
      "short": "Perpendicular from point to plane",
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      "number": 11,
      "colorClass": "proposition"
    },
    {
      "id": "Prop12",
      "type": "proposition",
      "label": "To set up line at right angles to plane from given point in it",
      "shortLabel": "Prop. XI.12",
      "short": "Perpendicular from point in plane",
      "book": 11,
      "number": 12,
      "colorClass": "proposition"
    },
    {
      "id": "Prop13",
      "type": "proposition",
      "label": "From same point two lines cannot be perpendicular to same plane on same side",
      "shortLabel": "Prop. XI.13",
      "short": "One perpendicular only",
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      "number": 13,
      "colorClass": "proposition"
    },
    {
      "id": "Prop14",
      "type": "proposition",
      "label": "Planes to which same line is perpendicular are parallel",
      "shortLabel": "Prop. XI.14",
      "short": "Planes perpendicular to line: parallel",
      "book": 11,
      "number": 14,
      "colorClass": "proposition"
    },
    {
      "id": "Prop15",
      "type": "proposition",
      "label": "Two lines meeting parallel to two meeting not in same plane: planes through them parallel",
      "shortLabel": "Prop. XI.15",
      "short": "Skew lines: planes parallel",
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      "number": 15,
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    },
    {
      "id": "Prop16",
      "type": "proposition",
      "label": "If two parallel planes cut by any plane, intersections are parallel",
      "shortLabel": "Prop. XI.16",
      "short": "Parallel planes cut: parallel",
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      "number": 16,
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    {
      "id": "Prop17",
      "type": "proposition",
      "label": "If two lines cut by parallel planes, they are cut in same ratios",
      "shortLabel": "Prop. XI.17",
      "short": "Parallel planes: same ratio",
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    },
    {
      "id": "Prop18",
      "type": "proposition",
      "label": "If line perpendicular to plane, all planes through it perpendicular to that plane",
      "shortLabel": "Prop. XI.18",
      "short": "Line perpendicular: planes through it",
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    {
      "id": "Prop19",
      "type": "proposition",
      "label": "If two planes cutting one another perpendicular to plane, intersection perpendicular",
      "shortLabel": "Prop. XI.19",
      "short": "Planes perpendicular: intersection",
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    {
      "id": "Prop20",
      "type": "proposition",
      "label": "Solid angle by three plane angles: sum of any two greater than third",
      "shortLabel": "Prop. XI.20",
      "short": "Solid angle: plane angles",
      "book": 11,
      "number": 20,
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    },
    {
      "id": "Prop21",
      "type": "proposition",
      "label": "Any solid angle contained by plane angles summing to less than four right angles",
      "shortLabel": "Prop. XI.21",
      "short": "Solid angle: less than four right",
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      "number": 21,
      "colorClass": "proposition"
    },
    {
      "id": "Prop22",
      "type": "proposition",
      "label": "Three plane angles with sum of any two greater than third, equal sides: construct triangle",
      "shortLabel": "Prop. XI.22",
      "short": "Three plane angles: construct triangle",
      "book": 11,
      "number": 22,
      "colorClass": "proposition"
    },
    {
      "id": "Prop23",
      "type": "proposition",
      "label": "To construct solid angle from three plane angles (sum of any two greater than third)",
      "shortLabel": "Prop. XI.23",
      "short": "Construct solid angle",
      "book": 11,
      "number": 23,
      "colorClass": "proposition"
    },
    {
      "id": "Prop24",
      "type": "proposition",
      "label": "If solid contained by parallel planes, opposite planes equal and parallelogrammic",
      "shortLabel": "Prop. XI.24",
      "short": "Solid by parallel planes",
      "book": 11,
      "number": 24,
      "colorClass": "proposition"
    },
    {
      "id": "Prop25",
      "type": "proposition",
      "label": "Parallelepiped cut by plane parallel to opposite: base to base as solid to solid",
      "shortLabel": "Prop. XI.25",
      "short": "Parallelepiped cut: base ratio",
      "book": 11,
      "number": 25,
      "colorClass": "proposition"
    },
    {
      "id": "Prop26",
      "type": "proposition",
      "label": "To construct solid angle equal to given on given line at given point",
      "shortLabel": "Prop. XI.26",
      "short": "Construct equal solid angle",
      "book": 11,
      "number": 26,
      "colorClass": "proposition"
    },
    {
      "id": "Prop27",
      "type": "proposition",
      "label": "To describe parallelepiped similar to given on given straight line",
      "shortLabel": "Prop. XI.27",
      "short": "Similar parallelepiped on line",
      "book": 11,
      "number": 27,
      "colorClass": "proposition"
    },
    {
      "id": "Prop28",
      "type": "proposition",
      "label": "Parallelepiped cut by plane through diagonals of opposite planes: bisected",
      "shortLabel": "Prop. XI.28",
      "short": "Parallelepiped: diagonal plane bisects",
      "book": 11,
      "number": 28,
      "colorClass": "proposition"
    },
    {
      "id": "Prop29",
      "type": "proposition",
      "label": "Parallelepipeds same base, height, ends on same lines: equal",
      "shortLabel": "Prop. XI.29",
      "short": "Same base, height, same lines: equal",
      "book": 11,
      "number": 29,
      "colorClass": "proposition"
    },
    {
      "id": "Prop30",
      "type": "proposition",
      "label": "Parallelepipeds same base, height, ends not on same lines: equal",
      "shortLabel": "Prop. XI.30",
      "short": "Same base, height, different lines: equal",
      "book": 11,
      "number": 30,
      "colorClass": "proposition"
    },
    {
      "id": "Prop31",
      "type": "proposition",
      "label": "Parallelepipeds on equal bases, same height: equal",
      "shortLabel": "Prop. XI.31",
      "short": "Equal bases, same height: equal",
      "book": 11,
      "number": 31,
      "colorClass": "proposition"
    },
    {
      "id": "Prop32",
      "type": "proposition",
      "label": "Parallelepipeds same height: to one another as bases",
      "shortLabel": "Prop. XI.32",
      "short": "Same height: as bases",
      "book": 11,
      "number": 32,
      "colorClass": "proposition"
    },
    {
      "id": "Prop33",
      "type": "proposition",
      "label": "Similar parallelepipeds: to one another in triplicate ratio of corresponding sides",
      "shortLabel": "Prop. XI.33",
      "short": "Similar: triplicate ratio",
      "book": 11,
      "number": 33,
      "colorClass": "proposition"
    },
    {
      "id": "Prop34",
      "type": "proposition",
      "label": "Equal parallelepipeds: bases reciprocally proportional to heights",
      "shortLabel": "Prop. XI.34",
      "short": "Equal: bases reciprocally proportional",
      "book": 11,
      "number": 34,
      "colorClass": "proposition"
    },
    {
      "id": "Prop35",
      "type": "proposition",
      "label": "Equal plane angles, elevated lines with equal angles: perpendiculars, joins",
      "shortLabel": "Prop. XI.35",
      "short": "Equal plane angles: elevated lines",
      "book": 11,
      "number": 35,
      "colorClass": "proposition"
    },
    {
      "id": "Prop36",
      "type": "proposition",
      "label": "Three proportional lines: parallelepiped from three equals that on mean equilateral",
      "shortLabel": "Prop. XI.36",
      "short": "Three proportional: parallelepiped",
      "book": 11,
      "number": 36,
      "colorClass": "proposition"
    },
    {
      "id": "Prop37",
      "type": "proposition",
      "label": "Four proportional: similar parallelepipeds proportional; converse",
      "shortLabel": "Prop. XI.37",
      "short": "Four proportional: parallelepipeds",
      "book": 11,
      "number": 37,
      "colorClass": "proposition"
    },
    {
      "id": "Prop38",
      "type": "proposition",
      "label": "Cube opposite sides bisected, planes through: intersection and diameter bisect each other",
      "shortLabel": "Prop. XI.38",
      "short": "Cube: bisected by planes",
      "book": 11,
      "number": 38,
      "colorClass": "proposition"
    },
    {
      "id": "Prop39",
      "type": "proposition",
      "label": "Two prisms equal height, parallelogram and triangle bases, parallelogram double: equal",
      "shortLabel": "Prop. XI.39",
      "short": "Prisms: parallelogram, triangle",
      "book": 11,
      "number": 39,
      "colorClass": "proposition"
    }
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