{
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    "id": "euclid-elements-book-vi",
    "name": "Euclid's Elements, Book VI",
    "subject": "geometry",
    "variant": "classical",
    "description": "Similar figures. 4 definitions, 33 propositions. Depends on Book I and Book V. VI.1 is basis for most. Source: David E. Joyce.",
    "structure": {
      "books": 6,
      "definitions": 4,
      "propositions": 33,
      "foundationTypes": [
        "definition",
        "foundation"
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    }
  },
  "metadata": {
    "created": "2026-03-15",
    "lastUpdated": "2026-03-15",
    "version": "1.0.0",
    "license": "CC BY 4.0",
    "authors": [
      "Welz, G."
    ],
    "methodology": "Programming Framework",
    "citation": "Welz, G. (2026). Euclid's Elements Book VI Dependency Graph. Programming Framework.",
    "keywords": [
      "Euclid",
      "Elements",
      "Book VI",
      "similar",
      "proportion",
      "triangles"
    ]
  },
  "sources": [
    {
      "id": "joyce",
      "type": "digital",
      "authors": "Joyce, David E.",
      "title": "Euclid's Elements, Book VI",
      "year": "1996",
      "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookVI/bookVI.html",
      "notes": "Clark University; VI.1 basis for most"
    }
  ],
  "nodes": [
    {
      "id": "BookI",
      "type": "foundation",
      "label": "Book I — Fundamentals of plane geometry",
      "shortLabel": "Book I",
      "short": "Foundation",
      "book": 1,
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      "id": "BookV",
      "type": "foundation",
      "label": "Book V — Theory of proportions",
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      "id": "Def1",
      "type": "definition",
      "label": "Similar rectilinear figures have equal angles and proportional sides",
      "shortLabel": "Def. VI.1",
      "short": "Similar rectilinear",
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    {
      "id": "Def2",
      "type": "definition",
      "label": "Figures reciprocally proportional when sides are proportional inversely",
      "shortLabel": "Def. VI.2",
      "short": "Reciprocally proportional",
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      "id": "Def3",
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      "label": "Straight line is mean proportional when first to it as it to third",
      "shortLabel": "Def. VI.3",
      "short": "Mean proportional",
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      "number": 3,
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    {
      "id": "Def4",
      "type": "definition",
      "label": "Duplicate ratio is the ratio of the squares on corresponding sides",
      "shortLabel": "Def. VI.4",
      "short": "Duplicate ratio",
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    {
      "id": "Prop1",
      "type": "proposition",
      "label": "Triangles and parallelograms under same height are as their bases",
      "shortLabel": "Prop. VI.1",
      "short": "Triangles under same height",
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    {
      "id": "Prop2",
      "type": "proposition",
      "label": "Line parallel to side cuts sides proportionally; converse",
      "shortLabel": "Prop. VI.2",
      "short": "Parallel cuts sides proportionally",
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      "id": "Prop3",
      "type": "proposition",
      "label": "Angle bisector: segments of base proportionally as remaining sides",
      "shortLabel": "Prop. VI.3",
      "short": "Angle bisector divides base",
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      "id": "Prop4",
      "type": "proposition",
      "label": "Equiangular triangles: sides about equal angles proportional",
      "shortLabel": "Prop. VI.4",
      "short": "Equiangular: sides proportional",
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      "id": "Prop5",
      "type": "proposition",
      "label": "If sides proportional, triangles equiangular",
      "shortLabel": "Prop. VI.5",
      "short": "Sides proportional: equiangular",
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      "id": "Prop6",
      "type": "proposition",
      "label": "One angle equal, sides about it proportional: equiangular",
      "shortLabel": "Prop. VI.6",
      "short": "One angle equal, sides proportional",
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      "id": "Prop7",
      "type": "proposition",
      "label": "One angle equal, sides about others proportional: equiangular",
      "shortLabel": "Prop. VI.7",
      "short": "One angle equal, other sides proportional",
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    {
      "id": "Prop8",
      "type": "proposition",
      "label": "Perpendicular from right angle: triangles similar to whole",
      "shortLabel": "Prop. VI.8",
      "short": "Altitude in right triangle",
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    {
      "id": "Prop9",
      "type": "proposition",
      "label": "To cut off a prescribed part from a given straight line",
      "shortLabel": "Prop. VI.9",
      "short": "Cut off prescribed part",
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      "number": 9,
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    {
      "id": "Prop10",
      "type": "proposition",
      "label": "To cut a given line similarly to a given cut line",
      "shortLabel": "Prop. VI.10",
      "short": "Cut line similarly",
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      "number": 10,
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    {
      "id": "Prop11",
      "type": "proposition",
      "label": "To find a third proportional to two given lines",
      "shortLabel": "Prop. VI.11",
      "short": "Third proportional",
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      "number": 11,
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    {
      "id": "Prop12",
      "type": "proposition",
      "label": "To find a fourth proportional to three given lines",
      "shortLabel": "Prop. VI.12",
      "short": "Fourth proportional",
      "book": 6,
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    {
      "id": "Prop13",
      "type": "proposition",
      "label": "To find a mean proportional to two given lines",
      "shortLabel": "Prop. VI.13",
      "short": "Mean proportional",
      "book": 6,
      "number": 13,
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    {
      "id": "Prop14",
      "type": "proposition",
      "label": "Equal equiangular parallelograms: sides reciprocally proportional",
      "shortLabel": "Prop. VI.14",
      "short": "Parallelograms reciprocally proportional",
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    {
      "id": "Prop15",
      "type": "proposition",
      "label": "Equal triangles, one angle equal: sides reciprocally proportional",
      "shortLabel": "Prop. VI.15",
      "short": "Triangles reciprocally proportional",
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      "number": 15,
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    {
      "id": "Prop16",
      "type": "proposition",
      "label": "Four lines proportional iff rectangle extremes = rectangle means",
      "shortLabel": "Prop. VI.16",
      "short": "Four lines proportional: rectangle",
      "book": 6,
      "number": 16,
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    {
      "id": "Prop17",
      "type": "proposition",
      "label": "Three lines proportional iff rectangle extremes = square on mean",
      "shortLabel": "Prop. VI.17",
      "short": "Three lines proportional: rectangle",
      "book": 6,
      "number": 17,
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    {
      "id": "Prop18",
      "type": "proposition",
      "label": "To describe figure similar to given on given line",
      "shortLabel": "Prop. VI.18",
      "short": "Similar figure on given line",
      "book": 6,
      "number": 18,
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    {
      "id": "Prop19",
      "type": "proposition",
      "label": "Similar triangles in duplicate ratio of corresponding sides",
      "shortLabel": "Prop. VI.19",
      "short": "Similar triangles: duplicate ratio",
      "book": 6,
      "number": 19,
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    {
      "id": "Prop20",
      "type": "proposition",
      "label": "Similar polygons in duplicate ratio of corresponding sides",
      "shortLabel": "Prop. VI.20",
      "short": "Similar polygons: duplicate ratio",
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      "number": 20,
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    {
      "id": "Prop21",
      "type": "proposition",
      "label": "Figures similar to same rectilinear figure are similar",
      "shortLabel": "Prop. VI.21",
      "short": "Similar to same: similar",
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      "number": 21,
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    },
    {
      "id": "Prop22",
      "type": "proposition",
      "label": "Four lines proportional iff similar figures on them proportional",
      "shortLabel": "Prop. VI.22",
      "short": "Four lines proportional: figures",
      "book": 6,
      "number": 22,
      "colorClass": "proposition"
    },
    {
      "id": "Prop23",
      "type": "proposition",
      "label": "Equiangular parallelograms: ratio compounded of sides",
      "shortLabel": "Prop. VI.23",
      "short": "Equiangular parallelograms: compound ratio",
      "book": 6,
      "number": 23,
      "colorClass": "proposition"
    },
    {
      "id": "Prop24",
      "type": "proposition",
      "label": "Parallelograms about diameter similar to whole",
      "shortLabel": "Prop. VI.24",
      "short": "Parallelograms about diameter",
      "book": 6,
      "number": 24,
      "colorClass": "proposition"
    },
    {
      "id": "Prop25",
      "type": "proposition",
      "label": "To construct figure similar to one and equal to another",
      "shortLabel": "Prop. VI.25",
      "short": "Similar and equal to another",
      "book": 6,
      "number": 25,
      "colorClass": "proposition"
    },
    {
      "id": "Prop26",
      "type": "proposition",
      "label": "Parallelogram similar to whole, common angle: about same diameter",
      "shortLabel": "Prop. VI.26",
      "short": "Parallelogram similar to difference",
      "book": 6,
      "number": 26,
      "colorClass": "proposition"
    },
    {
      "id": "Prop27",
      "type": "proposition",
      "label": "Of parallelograms applied to line, greatest is on half",
      "shortLabel": "Prop. VI.27",
      "short": "Greatest parallelogram applied",
      "book": 6,
      "number": 27,
      "colorClass": "proposition"
    },
    {
      "id": "Prop28",
      "type": "proposition",
      "label": "To apply parallelogram equal to figure, deficient by similar",
      "shortLabel": "Prop. VI.28",
      "short": "Apply parallelogram: deficient",
      "book": 6,
      "number": 28,
      "colorClass": "proposition"
    },
    {
      "id": "Prop29",
      "type": "proposition",
      "label": "To apply parallelogram equal to figure, exceeding by similar",
      "shortLabel": "Prop. VI.29",
      "short": "Apply parallelogram: exceeding",
      "book": 6,
      "number": 29,
      "colorClass": "proposition"
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    {
      "id": "Prop30",
      "type": "proposition",
      "label": "To cut a given line in extreme and mean ratio",
      "shortLabel": "Prop. VI.30",
      "short": "Extreme and mean ratio",
      "book": 6,
      "number": 30,
      "colorClass": "proposition"
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    {
      "id": "Prop31",
      "type": "proposition",
      "label": "In right triangle, figure on hypotenuse = sum of similar on sides",
      "shortLabel": "Prop. VI.31",
      "short": "Right triangle: similar figures",
      "book": 6,
      "number": 31,
      "colorClass": "proposition"
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    {
      "id": "Prop32",
      "type": "proposition",
      "label": "Two triangles, two sides proportional, placed with parallel sides",
      "shortLabel": "Prop. VI.32",
      "short": "Two triangles: sides parallel",
      "book": 6,
      "number": 32,
      "colorClass": "proposition"
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    {
      "id": "Prop33",
      "type": "proposition",
      "label": "Angles in equal circles have ratio of circumferences",
      "shortLabel": "Prop. VI.33",
      "short": "Angles in circles: ratio of arcs",
      "book": 6,
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