garywelz/programming_framework
0
1{2 "schemaVersion": "1.0",3 "discourse": {4 "id": "euclid-elements-book-vi",5 "name": "Euclid's Elements, Book VI",6 "subject": "geometry",7 "variant": "classical",8 "description": "Similar figures. 4 definitions, 33 propositions. Depends on Book I and Book V. VI.1 is basis for most. Source: David E. Joyce.",9 "structure": {10 "books": 6,11 "definitions": 4,12 "propositions": 33,13 "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 VI Dependency Graph. Programming Framework.",29 "keywords": [30 "Euclid",31 "Elements",32 "Book VI",33 "similar",34 "proportion",35 "triangles"36 ]37 },38 "sources": [39 {40 "id": "joyce",41 "type": "digital",42 "authors": "Joyce, David E.",43 "title": "Euclid's Elements, Book VI",44 "year": "1996",45 "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookVI/bookVI.html",46 "notes": "Clark University; VI.1 basis for most"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": "BookV",61 "type": "foundation",62 "label": "Book V — Theory of proportions",63 "shortLabel": "Book V",64 "short": "Foundation",65 "book": 5,66 "colorClass": "foundation"67 },68 {69 "id": "Def1",70 "type": "definition",71 "label": "Similar rectilinear figures have equal angles and proportional sides",72 "shortLabel": "Def. VI.1",73 "short": "Similar rectilinear",74 "book": 6,75 "number": 1,76 "colorClass": "definition"77 },78 {79 "id": "Def2",80 "type": "definition",81 "label": "Figures reciprocally proportional when sides are proportional inversely",82 "shortLabel": "Def. VI.2",83 "short": "Reciprocally proportional",84 "book": 6,85 "number": 2,86 "colorClass": "definition"87 },88 {89 "id": "Def3",90 "type": "definition",91 "label": "Straight line is mean proportional when first to it as it to third",92 "shortLabel": "Def. VI.3",93 "short": "Mean proportional",94 "book": 6,95 "number": 3,96 "colorClass": "definition"97 },98 {99 "id": "Def4",100 "type": "definition",101 "label": "Duplicate ratio is the ratio of the squares on corresponding sides",102 "shortLabel": "Def. VI.4",103 "short": "Duplicate ratio",104 "book": 6,105 "number": 4,106 "colorClass": "definition"107 },108 {109 "id": "Prop1",110 "type": "proposition",111 "label": "Triangles and parallelograms under same height are as their bases",112 "shortLabel": "Prop. VI.1",113 "short": "Triangles under same height",114 "book": 6,115 "number": 1,116 "colorClass": "proposition"117 },118 {119 "id": "Prop2",120 "type": "proposition",121 "label": "Line parallel to side cuts sides proportionally; converse",122 "shortLabel": "Prop. VI.2",123 "short": "Parallel cuts sides proportionally",124 "book": 6,125 "number": 2,126 "colorClass": "proposition"127 },128 {129 "id": "Prop3",130 "type": "proposition",131 "label": "Angle bisector: segments of base proportionally as remaining sides",132 "shortLabel": "Prop. VI.3",133 "short": "Angle bisector divides base",134 "book": 6,135 "number": 3,136 "colorClass": "proposition"137 },138 {139 "id": "Prop4",140 "type": "proposition",141 "label": "Equiangular triangles: sides about equal angles proportional",142 "shortLabel": "Prop. VI.4",143 "short": "Equiangular: sides proportional",144 "book": 6,145 "number": 4,146 "colorClass": "proposition"147 },148 {149 "id": "Prop5",150 "type": "proposition",151 "label": "If sides proportional, triangles equiangular",152 "shortLabel": "Prop. VI.5",153 "short": "Sides proportional: equiangular",154 "book": 6,155 "number": 5,156 "colorClass": "proposition"157 },158 {159 "id": "Prop6",160 "type": "proposition",161 "label": "One angle equal, sides about it proportional: equiangular",162 "shortLabel": "Prop. 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VI.9",193 "short": "Cut off prescribed part",194 "book": 6,195 "number": 9,196 "colorClass": "proposition"197 },198 {199 "id": "Prop10",200 "type": "proposition",201 "label": "To cut a given line similarly to a given cut line",202 "shortLabel": "Prop. VI.10",203 "short": "Cut line similarly",204 "book": 6,205 "number": 10,206 "colorClass": "proposition"207 },208 {209 "id": "Prop11",210 "type": "proposition",211 "label": "To find a third proportional to two given lines",212 "shortLabel": "Prop. VI.11",213 "short": "Third proportional",214 "book": 6,215 "number": 11,216 "colorClass": "proposition"217 },218 {219 "id": "Prop12",220 "type": "proposition",221 "label": "To find a fourth proportional to three given lines",222 "shortLabel": "Prop. VI.12",223 "short": "Fourth proportional",224 "book": 6,225 "number": 12,226 "colorClass": "proposition"227 },228 {229 "id": "Prop13",230 "type": "proposition",231 "label": "To find a mean proportional to two given lines",232 "shortLabel": "Prop. VI.13",233 "short": "Mean proportional",234 "book": 6,235 "number": 13,236 "colorClass": "proposition"237 },238 {239 "id": "Prop14",240 "type": "proposition",241 "label": "Equal equiangular parallelograms: sides reciprocally proportional",242 "shortLabel": "Prop. VI.14",243 "short": "Parallelograms reciprocally proportional",244 "book": 6,245 "number": 14,246 "colorClass": "proposition"247 },248 {249 "id": "Prop15",250 "type": "proposition",251 "label": "Equal triangles, one angle equal: sides reciprocally proportional",252 "shortLabel": "Prop. VI.15",253 "short": "Triangles reciprocally proportional",254 "book": 6,255 "number": 15,256 "colorClass": "proposition"257 },258 {259 "id": "Prop16",260 "type": "proposition",261 "label": "Four lines proportional iff rectangle extremes = rectangle means",262 "shortLabel": "Prop. VI.16",263 "short": "Four lines proportional: rectangle",264 "book": 6,265 "number": 16,266 "colorClass": "proposition"267 },268 {269 "id": "Prop17",270 "type": "proposition",271 "label": "Three lines proportional iff rectangle extremes = square on mean",272 "shortLabel": "Prop. VI.17",273 "short": "Three lines proportional: rectangle",274 "book": 6,275 "number": 17,276 "colorClass": "proposition"277 },278 {279 "id": "Prop18",280 "type": "proposition",281 "label": "To describe figure similar to given on given line",282 "shortLabel": "Prop. VI.18",283 "short": "Similar figure on given line",284 "book": 6,285 "number": 18,286 "colorClass": "proposition"287 },288 {289 "id": "Prop19",290 "type": "proposition",291 "label": "Similar triangles in duplicate ratio of corresponding sides",292 "shortLabel": "Prop. VI.19",293 "short": "Similar triangles: duplicate ratio",294 "book": 6,295 "number": 19,296 "colorClass": "proposition"297 },298 {299 "id": "Prop20",300 "type": "proposition",301 "label": "Similar polygons in duplicate ratio of corresponding sides",302 "shortLabel": "Prop. VI.20",303 "short": "Similar polygons: duplicate ratio",304 "book": 6,305 "number": 20,306 "colorClass": "proposition"307 },308 {309 "id": "Prop21",310 "type": "proposition",311 "label": "Figures similar to same rectilinear figure are similar",312 "shortLabel": "Prop. VI.21",313 "short": "Similar to same: similar",314 "book": 6,315 "number": 21,316 "colorClass": "proposition"317 },318 {319 "id": "Prop22",320 "type": "proposition",321 "label": "Four lines proportional iff similar figures on them proportional",322 "shortLabel": "Prop. VI.22",323 "short": "Four lines proportional: figures",324 "book": 6,325 "number": 22,326 "colorClass": "proposition"327 },328 {329 "id": "Prop23",330 "type": "proposition",331 "label": "Equiangular parallelograms: ratio compounded of sides",332 "shortLabel": "Prop. VI.23",333 "short": "Equiangular parallelograms: compound ratio",334 "book": 6,335 "number": 23,336 "colorClass": "proposition"337 },338 {339 "id": "Prop24",340 "type": "proposition",341 "label": "Parallelograms about diameter similar to whole",342 "shortLabel": "Prop. VI.24",343 "short": "Parallelograms about diameter",344 "book": 6,345 "number": 24,346 "colorClass": "proposition"347 },348 {349 "id": "Prop25",350 "type": "proposition",351 "label": "To construct figure similar to one and equal to another",352 "shortLabel": "Prop. VI.25",353 "short": "Similar and equal to another",354 "book": 6,355 "number": 25,356 "colorClass": "proposition"357 },358 {359 "id": "Prop26",360 "type": "proposition",361 "label": "Parallelogram similar to whole, common angle: about same diameter",362 "shortLabel": "Prop. VI.26",363 "short": "Parallelogram similar to difference",364 "book": 6,365 "number": 26,366 "colorClass": "proposition"367 },368 {369 "id": "Prop27",370 "type": "proposition",371 "label": "Of parallelograms applied to line, greatest is on half",372 "shortLabel": "Prop. VI.27",373 "short": "Greatest parallelogram applied",374 "book": 6,375 "number": 27,376 "colorClass": "proposition"377 },378 {379 "id": "Prop28",380 "type": "proposition",381 "label": "To apply parallelogram equal to figure, deficient by similar",382 "shortLabel": "Prop. VI.28",383 "short": "Apply parallelogram: deficient",384 "book": 6,385 "number": 28,386 "colorClass": "proposition"387 },388 {389 "id": "Prop29",390 "type": "proposition",391 "label": "To apply parallelogram equal to figure, exceeding by similar",392 "shortLabel": "Prop. VI.29",393 "short": "Apply parallelogram: exceeding",394 "book": 6,395 "number": 29,396 "colorClass": "proposition"397 },398 {399 "id": "Prop30",400 "type": "proposition",401 "label": "To cut a given line in extreme and mean ratio",402 "shortLabel": "Prop. VI.30",403 "short": "Extreme and mean ratio",404 "book": 6,405 "number": 30,406 "colorClass": "proposition"407 },408 {409 "id": "Prop31",410 "type": "proposition",411 "label": "In right triangle, figure on hypotenuse = sum of similar on sides",412 "shortLabel": "Prop. 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