{
  "schemaVersion": "1.0",
  "discourse": {
    "id": "euclid-elements-book-iii",
    "name": "Euclid's Elements, Book III",
    "subject": "geometry",
    "variant": "classical",
    "description": "Theory of circles: 11 definitions, 37 propositions. All depend on Book I. III.35 uses II.5. Source: David E. Joyce.",
    "structure": {
      "books": 3,
      "definitions": 11,
      "propositions": 37,
      "foundationTypes": [
        "definition",
        "foundation"
      ]
    }
  },
  "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 III Dependency Graph. Programming Framework.",
    "keywords": [
      "Euclid",
      "Elements",
      "Book III",
      "circles",
      "chords",
      "tangents"
    ]
  },
  "sources": [
    {
      "id": "joyce",
      "type": "digital",
      "authors": "Joyce, David E.",
      "title": "Euclid's Elements, Book III",
      "year": "1996",
      "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookIII/bookIII.html",
      "notes": "Clark University"
    }
  ],
  "nodes": [
    {
      "id": "BookI",
      "type": "foundation",
      "label": "Book I — Fundamentals of plane geometry",
      "shortLabel": "Book I",
      "short": "Foundation",
      "book": 1,
      "colorClass": "foundation"
    },
    {
      "id": "PropII5",
      "type": "foundation",
      "label": "Prop. II.5 — Rectangle + square = square on half",
      "shortLabel": "Prop. II.5",
      "short": "From Book II",
      "book": 2,
      "colorClass": "foundation"
    },
    {
      "id": "Def1",
      "type": "definition",
      "label": "Equal circles are those with equal radii",
      "shortLabel": "Def. III.1",
      "short": "Equal circles",
      "book": 3,
      "number": 1,
      "colorClass": "definition"
    },
    {
      "id": "Def2",
      "type": "definition",
      "label": "A straight line touches a circle if it meets but does not cut it",
      "shortLabel": "Def. III.2",
      "short": "Tangent",
      "book": 3,
      "number": 2,
      "colorClass": "definition"
    },
    {
      "id": "Def3",
      "type": "definition",
      "label": "Circles touch one another if they meet but do not cut",
      "shortLabel": "Def. III.3",
      "short": "Circles touching",
      "book": 3,
      "number": 3,
      "colorClass": "definition"
    },
    {
      "id": "Def4",
      "type": "definition",
      "label": "Lines equally distant from center when perpendiculars from center equal",
      "shortLabel": "Def. III.4",
      "short": "Equally distant from center",
      "book": 3,
      "number": 4,
      "colorClass": "definition"
    },
    {
      "id": "Def5",
      "type": "definition",
      "label": "Greater distance when greater perpendicular falls",
      "shortLabel": "Def. III.5",
      "short": "Greater distance",
      "book": 3,
      "number": 5,
      "colorClass": "definition"
    },
    {
      "id": "Def6",
      "type": "definition",
      "label": "Segment of circle: figure contained by straight line and circumference",
      "shortLabel": "Def. III.6",
      "short": "Segment of circle",
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      "number": 6,
      "colorClass": "definition"
    },
    {
      "id": "Def7",
      "type": "definition",
      "label": "Angle of segment: contained by straight line and circumference",
      "shortLabel": "Def. III.7",
      "short": "Angle of segment",
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      "number": 7,
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    },
    {
      "id": "Def8",
      "type": "definition",
      "label": "Angle in segment: contained by straight lines joining circumference",
      "shortLabel": "Def. III.8",
      "short": "Angle in segment",
      "book": 3,
      "number": 8,
      "colorClass": "definition"
    },
    {
      "id": "Def9",
      "type": "definition",
      "label": "Angle stands on circumference when lines cut off that circumference",
      "shortLabel": "Def. III.9",
      "short": "Angle stands on circumference",
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      "number": 9,
      "colorClass": "definition"
    },
    {
      "id": "Def10",
      "type": "definition",
      "label": "Sector: figure contained by two radii and circumference between them",
      "shortLabel": "Def. III.10",
      "short": "Sector",
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      "number": 10,
      "colorClass": "definition"
    },
    {
      "id": "Def11",
      "type": "definition",
      "label": "Similar segments are those which admit equal angles",
      "shortLabel": "Def. III.11",
      "short": "Similar segments",
      "book": 3,
      "number": 11,
      "colorClass": "definition"
    },
    {
      "id": "Prop1",
      "type": "proposition",
      "label": "To find the center of a given circle",
      "shortLabel": "Prop. III.1",
      "short": "Find center of circle",
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      "number": 1,
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    },
    {
      "id": "Prop2",
      "type": "proposition",
      "label": "Straight line joining two points on circumference falls within circle",
      "shortLabel": "Prop. III.2",
      "short": "Chord falls within circle",
      "book": 3,
      "number": 2,
      "colorClass": "proposition"
    },
    {
      "id": "Prop3",
      "type": "proposition",
      "label": "If diameter bisects chord not through center, it cuts at right angles",
      "shortLabel": "Prop. III.3",
      "short": "Diameter bisects chord at right angles",
      "book": 3,
      "number": 3,
      "colorClass": "proposition"
    },
    {
      "id": "Prop4",
      "type": "proposition",
      "label": "Two non-diameters cutting one another do not bisect",
      "shortLabel": "Prop. III.4",
      "short": "Non-diameters do not bisect",
      "book": 3,
      "number": 4,
      "colorClass": "proposition"
    },
    {
      "id": "Prop5",
      "type": "proposition",
      "label": "If two circles cut one another, they do not have same center",
      "shortLabel": "Prop. III.5",
      "short": "Cutting circles do not share center",
      "book": 3,
      "number": 5,
      "colorClass": "proposition"
    },
    {
      "id": "Prop6",
      "type": "proposition",
      "label": "If two circles touch, they do not have same center",
      "shortLabel": "Prop. III.6",
      "short": "Touching circles do not share center",
      "book": 3,
      "number": 6,
      "colorClass": "proposition"
    },
    {
      "id": "Prop7",
      "type": "proposition",
      "label": "From point on diameter: greatest through center, least is remainder",
      "shortLabel": "Prop. III.7",
      "short": "Greatest/shortest from point on diameter",
      "book": 3,
      "number": 7,
      "colorClass": "proposition"
    },
    {
      "id": "Prop8",
      "type": "proposition",
      "label": "From point outside: through center greatest; between point and diameter least",
      "shortLabel": "Prop. III.8",
      "short": "Lines from point outside circle",
      "book": 3,
      "number": 8,
      "colorClass": "proposition"
    },
    {
      "id": "Prop9",
      "type": "proposition",
      "label": "If more than two equal lines fall from point on circle, point is center",
      "shortLabel": "Prop. III.9",
      "short": "Three equal lines imply center",
      "book": 3,
      "number": 9,
      "colorClass": "proposition"
    },
    {
      "id": "Prop10",
      "type": "proposition",
      "label": "A circle does not cut another at more than two points",
      "shortLabel": "Prop. III.10",
      "short": "Circles cut at most two points",
      "book": 3,
      "number": 10,
      "colorClass": "proposition"
    },
    {
      "id": "Prop11",
      "type": "proposition",
      "label": "Line joining centers of internally touching circles passes through contact",
      "shortLabel": "Prop. III.11",
      "short": "Internally touching circles",
      "book": 3,
      "number": 11,
      "colorClass": "proposition"
    },
    {
      "id": "Prop12",
      "type": "proposition",
      "label": "Line joining centers of externally touching circles passes through contact",
      "shortLabel": "Prop. III.12",
      "short": "Externally touching circles",
      "book": 3,
      "number": 12,
      "colorClass": "proposition"
    },
    {
      "id": "Prop13",
      "type": "proposition",
      "label": "Circle does not touch another at more than one point",
      "shortLabel": "Prop. III.13",
      "short": "Circles touch at most one point",
      "book": 3,
      "number": 13,
      "colorClass": "proposition"
    },
    {
      "id": "Prop14",
      "type": "proposition",
      "label": "Equal chords equally distant from center, and conversely",
      "shortLabel": "Prop. III.14",
      "short": "Equal chords equally distant",
      "book": 3,
      "number": 14,
      "colorClass": "proposition"
    },
    {
      "id": "Prop15",
      "type": "proposition",
      "label": "Diameter greatest; nearer to center greater than more remote",
      "shortLabel": "Prop. III.15",
      "short": "Diameter greatest",
      "book": 3,
      "number": 15,
      "colorClass": "proposition"
    },
    {
      "id": "Prop16",
      "type": "proposition",
      "label": "Perpendicular at end of diameter falls outside; horn angle",
      "shortLabel": "Prop. III.16",
      "short": "Tangent at end of diameter",
      "book": 3,
      "number": 16,
      "colorClass": "proposition"
    },
    {
      "id": "Prop17",
      "type": "proposition",
      "label": "From given point to draw straight line touching given circle",
      "shortLabel": "Prop. III.17",
      "short": "Draw tangent from point",
      "book": 3,
      "number": 17,
      "colorClass": "proposition"
    },
    {
      "id": "Prop18",
      "type": "proposition",
      "label": "Radius to point of contact perpendicular to tangent",
      "shortLabel": "Prop. III.18",
      "short": "Radius to tangent perpendicular",
      "book": 3,
      "number": 18,
      "colorClass": "proposition"
    },
    {
      "id": "Prop19",
      "type": "proposition",
      "label": "Perpendicular from contact to tangent passes through center",
      "shortLabel": "Prop. III.19",
      "short": "Perpendicular from contact to center",
      "book": 3,
      "number": 19,
      "colorClass": "proposition"
    },
    {
      "id": "Prop20",
      "type": "proposition",
      "label": "Angle at center double angle at circumference on same base",
      "shortLabel": "Prop. III.20",
      "short": "Angle at center double angle at circumference",
      "book": 3,
      "number": 20,
      "colorClass": "proposition"
    },
    {
      "id": "Prop21",
      "type": "proposition",
      "label": "In a circle angles in same segment equal one another",
      "shortLabel": "Prop. III.21",
      "short": "Angles in same segment equal",
      "book": 3,
      "number": 21,
      "colorClass": "proposition"
    },
    {
      "id": "Prop22",
      "type": "proposition",
      "label": "Sum of opposite angles of cyclic quadrilateral equals two right angles",
      "shortLabel": "Prop. III.22",
      "short": "Opposite angles of cyclic quadrilateral",
      "book": 3,
      "number": 22,
      "colorClass": "proposition"
    },
    {
      "id": "Prop23",
      "type": "proposition",
      "label": "On same line cannot construct two similar unequal segments on same side",
      "shortLabel": "Prop. III.23",
      "short": "Same line, two similar unequal segments",
      "book": 3,
      "number": 23,
      "colorClass": "proposition"
    },
    {
      "id": "Prop24",
      "type": "proposition",
      "label": "Similar segments on equal straight lines equal one another",
      "shortLabel": "Prop. III.24",
      "short": "Similar segments on equal lines equal",
      "book": 3,
      "number": 24,
      "colorClass": "proposition"
    },
    {
      "id": "Prop25",
      "type": "proposition",
      "label": "Given segment of circle, describe complete circle",
      "shortLabel": "Prop. III.25",
      "short": "Complete circle from segment",
      "book": 3,
      "number": 25,
      "colorClass": "proposition"
    },
    {
      "id": "Prop26",
      "type": "proposition",
      "label": "In equal circles equal angles stand on equal circumferences",
      "shortLabel": "Prop. III.26",
      "short": "Equal angles stand on equal arcs",
      "book": 3,
      "number": 26,
      "colorClass": "proposition"
    },
    {
      "id": "Prop27",
      "type": "proposition",
      "label": "In equal circles angles on equal circumferences equal one another",
      "shortLabel": "Prop. III.27",
      "short": "Equal arcs imply equal angles",
      "book": 3,
      "number": 27,
      "colorClass": "proposition"
    },
    {
      "id": "Prop28",
      "type": "proposition",
      "label": "In equal circles equal chords cut off equal circumferences",
      "shortLabel": "Prop. III.28",
      "short": "Equal chords cut off equal arcs",
      "book": 3,
      "number": 28,
      "colorClass": "proposition"
    },
    {
      "id": "Prop29",
      "type": "proposition",
      "label": "In equal circles chords cutting equal circumferences are equal",
      "shortLabel": "Prop. III.29",
      "short": "Equal arcs imply equal chords",
      "book": 3,
      "number": 29,
      "colorClass": "proposition"
    },
    {
      "id": "Prop30",
      "type": "proposition",
      "label": "To bisect a given circumference",
      "shortLabel": "Prop. III.30",
      "short": "Bisect given circumference",
      "book": 3,
      "number": 30,
      "colorClass": "proposition"
    },
    {
      "id": "Prop31",
      "type": "proposition",
      "label": "Angle in semicircle right; in greater segment less; in less greater",
      "shortLabel": "Prop. III.31",
      "short": "Angle in semicircle is right",
      "book": 3,
      "number": 31,
      "colorClass": "proposition"
    },
    {
      "id": "Prop32",
      "type": "proposition",
      "label": "Angle with tangent equals angle in alternate segment",
      "shortLabel": "Prop. III.32",
      "short": "Tangent-chord angle equals alternate segment",
      "book": 3,
      "number": 32,
      "colorClass": "proposition"
    },
    {
      "id": "Prop33",
      "type": "proposition",
      "label": "On given line describe segment admitting angle equal to given",
      "shortLabel": "Prop. III.33",
      "short": "Segment admitting given angle",
      "book": 3,
      "number": 33,
      "colorClass": "proposition"
    },
    {
      "id": "Prop34",
      "type": "proposition",
      "label": "From given circle cut off segment admitting given angle",
      "shortLabel": "Prop. III.34",
      "short": "Cut off segment admitting angle",
      "book": 3,
      "number": 34,
      "colorClass": "proposition"
    },
    {
      "id": "Prop35",
      "type": "proposition",
      "label": "If chords cut one another, rectangle by segments of one equals other",
      "shortLabel": "Prop. III.35",
      "short": "Rectangle from chord segments equal",
      "book": 3,
      "number": 35,
      "colorClass": "proposition"
    },
    {
      "id": "Prop36",
      "type": "proposition",
      "label": "From point outside: tangent squared = secant × external part",
      "shortLabel": "Prop. III.36",
      "short": "Tangent squared = secant × external",
      "book": 3,
      "number": 36,
      "colorClass": "proposition"
    },
    {
      "id": "Prop37",
      "type": "proposition",
      "label": "If rectangle equals square on line, that line touches circle",
      "shortLabel": "Prop. III.37",
      "short": "Converse: tangent if rectangle = square",
      "book": 3,
      "number": 37,
      "colorClass": "proposition"
    }
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