garywelz/programming_framework
0
1{2 "schemaVersion": "1.0",3 "discourse": {4 "id": "euclid-elements-book-iii",5 "name": "Euclid's Elements, Book III",6 "subject": "geometry",7 "variant": "classical",8 "description": "Theory of circles: 11 definitions, 37 propositions. All depend on Book I. III.35 uses II.5. Source: David E. Joyce.",9 "structure": {10 "books": 3,11 "definitions": 11,12 "propositions": 37,13 "foundationTypes": [14 "definition",15 "foundation"16 ]17 }18 },19 "metadata": {20 "created": "2026-03-15",21 "lastUpdated": "2026-03-15",22 "version": "1.0.0",23 "license": "CC BY 4.0",24 "authors": [25 "Welz, G."26 ],27 "methodology": "Programming Framework",28 "citation": "Welz, G. (2026). Euclid's Elements Book III Dependency Graph. Programming Framework.",29 "keywords": [30 "Euclid",31 "Elements",32 "Book III",33 "circles",34 "chords",35 "tangents"36 ]37 },38 "sources": [39 {40 "id": "joyce",41 "type": "digital",42 "authors": "Joyce, David E.",43 "title": "Euclid's Elements, Book III",44 "year": "1996",45 "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookIII/bookIII.html",46 "notes": "Clark University"47 }48 ],49 "nodes": [50 {51 "id": "BookI",52 "type": "foundation",53 "label": "Book I — Fundamentals of plane geometry",54 "shortLabel": "Book I",55 "short": "Foundation",56 "book": 1,57 "colorClass": "foundation"58 },59 {60 "id": "PropII5",61 "type": "foundation",62 "label": "Prop. II.5 — Rectangle + square = square on half",63 "shortLabel": "Prop. II.5",64 "short": "From Book II",65 "book": 2,66 "colorClass": "foundation"67 },68 {69 "id": "Def1",70 "type": "definition",71 "label": "Equal circles are those with equal radii",72 "shortLabel": "Def. III.1",73 "short": "Equal circles",74 "book": 3,75 "number": 1,76 "colorClass": "definition"77 },78 {79 "id": "Def2",80 "type": "definition",81 "label": "A straight line touches a circle if it meets but does not cut it",82 "shortLabel": "Def. III.2",83 "short": "Tangent",84 "book": 3,85 "number": 2,86 "colorClass": "definition"87 },88 {89 "id": "Def3",90 "type": "definition",91 "label": "Circles touch one another if they meet but do not cut",92 "shortLabel": "Def. III.3",93 "short": "Circles touching",94 "book": 3,95 "number": 3,96 "colorClass": "definition"97 },98 {99 "id": "Def4",100 "type": "definition",101 "label": "Lines equally distant from center when perpendiculars from center equal",102 "shortLabel": "Def. III.4",103 "short": "Equally distant from center",104 "book": 3,105 "number": 4,106 "colorClass": "definition"107 },108 {109 "id": "Def5",110 "type": "definition",111 "label": "Greater distance when greater perpendicular falls",112 "shortLabel": "Def. III.5",113 "short": "Greater distance",114 "book": 3,115 "number": 5,116 "colorClass": "definition"117 },118 {119 "id": "Def6",120 "type": "definition",121 "label": "Segment of circle: figure contained by straight line and circumference",122 "shortLabel": "Def. III.6",123 "short": "Segment of circle",124 "book": 3,125 "number": 6,126 "colorClass": "definition"127 },128 {129 "id": "Def7",130 "type": "definition",131 "label": "Angle of segment: contained by straight line and circumference",132 "shortLabel": "Def. III.7",133 "short": "Angle of segment",134 "book": 3,135 "number": 7,136 "colorClass": "definition"137 },138 {139 "id": "Def8",140 "type": "definition",141 "label": "Angle in segment: contained by straight lines joining circumference",142 "shortLabel": "Def. III.8",143 "short": "Angle in segment",144 "book": 3,145 "number": 8,146 "colorClass": "definition"147 },148 {149 "id": "Def9",150 "type": "definition",151 "label": "Angle stands on circumference when lines cut off that circumference",152 "shortLabel": "Def. III.9",153 "short": "Angle stands on circumference",154 "book": 3,155 "number": 9,156 "colorClass": "definition"157 },158 {159 "id": "Def10",160 "type": "definition",161 "label": "Sector: figure contained by two radii and circumference between them",162 "shortLabel": "Def. III.10",163 "short": "Sector",164 "book": 3,165 "number": 10,166 "colorClass": "definition"167 },168 {169 "id": "Def11",170 "type": "definition",171 "label": "Similar segments are those which admit equal angles",172 "shortLabel": "Def. III.11",173 "short": "Similar segments",174 "book": 3,175 "number": 11,176 "colorClass": "definition"177 },178 {179 "id": "Prop1",180 "type": "proposition",181 "label": "To find the center of a given circle",182 "shortLabel": "Prop. III.1",183 "short": "Find center of circle",184 "book": 3,185 "number": 1,186 "colorClass": "proposition"187 },188 {189 "id": "Prop2",190 "type": "proposition",191 "label": "Straight line joining two points on circumference falls within circle",192 "shortLabel": "Prop. III.2",193 "short": "Chord falls within circle",194 "book": 3,195 "number": 2,196 "colorClass": "proposition"197 },198 {199 "id": "Prop3",200 "type": "proposition",201 "label": "If diameter bisects chord not through center, it cuts at right angles",202 "shortLabel": "Prop. III.3",203 "short": "Diameter bisects chord at right angles",204 "book": 3,205 "number": 3,206 "colorClass": "proposition"207 },208 {209 "id": "Prop4",210 "type": "proposition",211 "label": "Two non-diameters cutting one another do not bisect",212 "shortLabel": "Prop. III.4",213 "short": "Non-diameters do not bisect",214 "book": 3,215 "number": 4,216 "colorClass": "proposition"217 },218 {219 "id": "Prop5",220 "type": "proposition",221 "label": "If two circles cut one another, they do not have same center",222 "shortLabel": "Prop. III.5",223 "short": "Cutting circles do not share center",224 "book": 3,225 "number": 5,226 "colorClass": "proposition"227 },228 {229 "id": "Prop6",230 "type": "proposition",231 "label": "If two circles touch, they do not have same center",232 "shortLabel": "Prop. III.6",233 "short": "Touching circles do not share center",234 "book": 3,235 "number": 6,236 "colorClass": "proposition"237 },238 {239 "id": "Prop7",240 "type": "proposition",241 "label": "From point on diameter: greatest through center, least is remainder",242 "shortLabel": "Prop. III.7",243 "short": "Greatest/shortest from point on diameter",244 "book": 3,245 "number": 7,246 "colorClass": "proposition"247 },248 {249 "id": "Prop8",250 "type": "proposition",251 "label": "From point outside: through center greatest; between point and diameter least",252 "shortLabel": "Prop. III.8",253 "short": "Lines from point outside circle",254 "book": 3,255 "number": 8,256 "colorClass": "proposition"257 },258 {259 "id": "Prop9",260 "type": "proposition",261 "label": "If more than two equal lines fall from point on circle, point is center",262 "shortLabel": "Prop. III.9",263 "short": "Three equal lines imply center",264 "book": 3,265 "number": 9,266 "colorClass": "proposition"267 },268 {269 "id": "Prop10",270 "type": "proposition",271 "label": "A circle does not cut another at more than two points",272 "shortLabel": "Prop. III.10",273 "short": "Circles cut at most two points",274 "book": 3,275 "number": 10,276 "colorClass": "proposition"277 },278 {279 "id": "Prop11",280 "type": "proposition",281 "label": "Line joining centers of internally touching circles passes through contact",282 "shortLabel": "Prop. III.11",283 "short": "Internally touching circles",284 "book": 3,285 "number": 11,286 "colorClass": "proposition"287 },288 {289 "id": "Prop12",290 "type": "proposition",291 "label": "Line joining centers of externally touching circles passes through contact",292 "shortLabel": "Prop. III.12",293 "short": "Externally touching circles",294 "book": 3,295 "number": 12,296 "colorClass": "proposition"297 },298 {299 "id": "Prop13",300 "type": "proposition",301 "label": "Circle does not touch another at more than one point",302 "shortLabel": "Prop. III.13",303 "short": "Circles touch at most one point",304 "book": 3,305 "number": 13,306 "colorClass": "proposition"307 },308 {309 "id": "Prop14",310 "type": "proposition",311 "label": "Equal chords equally distant from center, and conversely",312 "shortLabel": "Prop. III.14",313 "short": "Equal chords equally distant",314 "book": 3,315 "number": 14,316 "colorClass": "proposition"317 },318 {319 "id": "Prop15",320 "type": "proposition",321 "label": "Diameter greatest; nearer to center greater than more remote",322 "shortLabel": "Prop. III.15",323 "short": "Diameter greatest",324 "book": 3,325 "number": 15,326 "colorClass": "proposition"327 },328 {329 "id": "Prop16",330 "type": "proposition",331 "label": "Perpendicular at end of diameter falls outside; horn angle",332 "shortLabel": "Prop. III.16",333 "short": "Tangent at end of diameter",334 "book": 3,335 "number": 16,336 "colorClass": "proposition"337 },338 {339 "id": "Prop17",340 "type": "proposition",341 "label": "From given point to draw straight line touching given circle",342 "shortLabel": "Prop. III.17",343 "short": "Draw tangent from point",344 "book": 3,345 "number": 17,346 "colorClass": "proposition"347 },348 {349 "id": "Prop18",350 "type": "proposition",351 "label": "Radius to point of contact perpendicular to tangent",352 "shortLabel": "Prop. III.18",353 "short": "Radius to tangent perpendicular",354 "book": 3,355 "number": 18,356 "colorClass": "proposition"357 },358 {359 "id": "Prop19",360 "type": "proposition",361 "label": "Perpendicular from contact to tangent passes through center",362 "shortLabel": "Prop. III.19",363 "short": "Perpendicular from contact to center",364 "book": 3,365 "number": 19,366 "colorClass": "proposition"367 },368 {369 "id": "Prop20",370 "type": "proposition",371 "label": "Angle at center double angle at circumference on same base",372 "shortLabel": "Prop. III.20",373 "short": "Angle at center double angle at circumference",374 "book": 3,375 "number": 20,376 "colorClass": "proposition"377 },378 {379 "id": "Prop21",380 "type": "proposition",381 "label": "In a circle angles in same segment equal one another",382 "shortLabel": "Prop. III.21",383 "short": "Angles in same segment equal",384 "book": 3,385 "number": 21,386 "colorClass": "proposition"387 },388 {389 "id": "Prop22",390 "type": "proposition",391 "label": "Sum of opposite angles of cyclic quadrilateral equals two right angles",392 "shortLabel": "Prop. III.22",393 "short": "Opposite angles of cyclic quadrilateral",394 "book": 3,395 "number": 22,396 "colorClass": "proposition"397 },398 {399 "id": "Prop23",400 "type": "proposition",401 "label": "On same line cannot construct two similar unequal segments on same side",402 "shortLabel": "Prop. III.23",403 "short": "Same line, two similar unequal segments",404 "book": 3,405 "number": 23,406 "colorClass": "proposition"407 },408 {409 "id": "Prop24",410 "type": "proposition",411 "label": "Similar segments on equal straight lines equal one another",412 "shortLabel": "Prop. III.24",413 "short": "Similar segments on equal lines equal",414 "book": 3,415 "number": 24,416 "colorClass": "proposition"417 },418 {419 "id": "Prop25",420 "type": "proposition",421 "label": "Given segment of circle, describe complete circle",422 "shortLabel": "Prop. III.25",423 "short": "Complete circle from segment",424 "book": 3,425 "number": 25,426 "colorClass": "proposition"427 },428 {429 "id": "Prop26",430 "type": "proposition",431 "label": "In equal circles equal angles stand on equal circumferences",432 "shortLabel": "Prop. III.26",433 "short": "Equal angles stand on equal arcs",434 "book": 3,435 "number": 26,436 "colorClass": "proposition"437 },438 {439 "id": "Prop27",440 "type": "proposition",441 "label": "In equal circles angles on equal circumferences equal one another",442 "shortLabel": "Prop. III.27",443 "short": "Equal arcs imply equal angles",444 "book": 3,445 "number": 27,446 "colorClass": "proposition"447 },448 {449 "id": "Prop28",450 "type": "proposition",451 "label": "In equal circles equal chords cut off equal circumferences",452 "shortLabel": "Prop. III.28",453 "short": "Equal chords cut off equal arcs",454 "book": 3,455 "number": 28,456 "colorClass": "proposition"457 },458 {459 "id": "Prop29",460 "type": "proposition",461 "label": "In equal circles chords cutting equal circumferences are equal",462 "shortLabel": "Prop. III.29",463 "short": "Equal arcs imply equal chords",464 "book": 3,465 "number": 29,466 "colorClass": "proposition"467 },468 {469 "id": "Prop30",470 "type": "proposition",471 "label": "To bisect a given circumference",472 "shortLabel": "Prop. III.30",473 "short": "Bisect given circumference",474 "book": 3,475 "number": 30,476 "colorClass": "proposition"477 },478 {479 "id": "Prop31",480 "type": "proposition",481 "label": "Angle in semicircle right; in greater segment less; in less greater",482 "shortLabel": "Prop. III.31",483 "short": "Angle in semicircle is right",484 "book": 3,485 "number": 31,486 "colorClass": "proposition"487 },488 {489 "id": "Prop32",490 "type": "proposition",491 "label": "Angle with tangent equals angle in alternate segment",492 "shortLabel": "Prop. III.32",493 "short": "Tangent-chord angle equals alternate segment",494 "book": 3,495 "number": 32,496 "colorClass": "proposition"497 },498 {499 "id": "Prop33",500 "type": "proposition",501 "label": "On given line describe segment admitting angle equal to given",502 "shortLabel": "Prop. 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