book-01-proposition-47.json

{
  "title": "Proposition 47",
  "prose": "Let {polygon ABC} be a right-angled triangle having the right-angle {angle BAC}.\nI say that the square on {line BC} is equal to the (sum of the) squares on {line BA} and {line AC}.\n\nFor let the square {polygon BDEC} have been described on {line BC}, and (the squares) {polygon GB} and {polygon HC} on {line AB} and {line AC} (respectively) [Prop. 1.46].\nAnd let {line AL} have been drawn through point {point A}, parallel to either of {line BD} or {line CE} [Prop. 1.31].\nAnd let {line AD} and {line FC} have been joined.\nAnd since angles {angle BAC} and {angle BAG} are each right-angles, then two straight-lines {line AC} and {line AG}, not lying on the same side, make the (sum of the) adjacent angles equal to two right-angles at the same point {point A} on some straight-line {line BA}.\nThus, {line CA} is straight-on to {line AG} [Prop. 1.14].\nSo, for the same (reasons), {line BA} is also straight-on to {line AH}.\nAnd since angle {angle DBC} is equal to {angle FBA}, for (they are) both right-angles, let {angle ABC} have been added to both.\nThus, the whole (angle) {angle DBA} is equal to the whole (angle) {angle FBC}.\nAnd since {line DB} is equal to {line BC}, and {line FB} to {line BA}, the two (straight-lines) {line DB}, {line BA} are equal to the two (straight-lines) {line CB}, {line BF}, respectively.\nAnd angle {angle DBA} (is) equal to angle {angle FBC}.\nThus, the base {line AD} [is] equal to the base {line FC}, and the triangle {polygon ABD} is equal to the triangle {polygon FBC} [Prop. 1.4].\nAnd parallelogram {polygon BL} [is] double (the area) of triangle {polygon ABD}.\nFor they have the same base, {line BD}, and are between the same parallels, {line BD} and {line AL} [Prop. 1.41].\nAnd square {polygon GB} is double (the area) of triangle {polygon FBC}.\nFor again they have the same base, {line FB}, and are between the same parallels, {line FB} and {line GC} [Prop. 1.41].\nThus, the parallelogram {polygon BL} is also equal to the square {polygon GB}.\nSo, similarly, {line AE} and {line BK} being joined, the parallelogram {polygon CL} can be shown (to be) equal to the square {polygon HC}.\nThus, the whole square {polygon BDEC} is equal to the (sum of the) two squares {polygon GB} and {polygon HC}.\nAnd the square {polygon BDEC} is described on {line BC}, and the (squares) {polygon GB} and {polygon HC} on {line BA} and {line AC} (respectively).\nThus, the square on the side {line BC} is equal to the (sum of the) squares on the sides {line BA} and {line AC}.\n\nThus, in a right-angled triangle, the square on the side subtending the right-angle is equal to the (sum of the) squares on the sides surrounding the right-[angle].\n(Which is) the very thing it was required to show.\n",
  "points": {
    "A": [180, 193.39745962155615],
    "B": [130, 280],
    "C": [330, 280],
    "D": [130, 480],
    "E": [330, 480],
    "F": [43.39745962155615, 230],
    "G": [93.39745962155615, 143.39745962155615],
    "H": [266.60254037844385, 43.39745962155615],
    "K": [416.60254037844385, 130],
    "L": [180, 480],
    "M": [180, 280]
  },
  "shapes": [
    [
      "polygon",
      [
        [180, 193.39745962155615],
        [93.39745962155615, 143.39745962155615],
        [43.39745962155615, 230],
        [130, 280]
      ]
    ],
    [
      "polygon",
      [
        [130, 280],
        [130, 480],
        [330, 480],
        [330, 280]
      ]
    ],
    [
      "polygon",
      [
        [180, 193.39745962155615],
        [330, 280],
        [416.60254037844385, 130],
        [266.60254037844385, 43.39745962155615]
      ]
    ],
    ["line", [180, 193.39745962155615], [180, 480]],
    ["line", [180, 193.39745962155615], [130, 480]],
    ["line", [180, 193.39745962155615], [330, 480]],
    ["line", [130, 280], [416.60254037844385, 130]],
    ["line", [330, 280], [43.39745962155615, 230]]
  ],
  "letters": {
    "A": [1.3, 1.2],
    "B": [3.5],
    "C": [-1.5],
    "D": [5],
    "E": [5],
    "F": [3],
    "G": [2],
    "H": [0],
    "K": [0],
    "L": [5]
  },
  "polygonl": {
    "GB": "GABF",
    "HC": "HKCA",
    "BL": "BDLM",
    "CL": "CELM"
  },
  "id": "1.47"
}