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book-03-proposition-02.json
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book-03-proposition-02.json
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{
"title": "Proposition 2",
"prose": "If two points are taken at random on the circumference of a circle then the straight-line joining the points will fall inside the circle.\n\nLet {ABC circle D} be a circle, and let two points {A point} and {B point} have been taken at random on its circumference.\nI say that the straight-line joining {A point} to {B point} will fall inside the circle.\n\nFor (if) not then, if possible, let it fall outside (the circle), like {AEB given} (in the figure).\nAnd let the center of the circle {ABC circle D} have been found [Prop. 3.1], and let it be (at point) {D point}.\nAnd let {DA line} and {DB line} have been joined, and let {DFE line} have been drawn through.\n\nTherefore, since {DA line} is equal to {DB line}, the angle {DAE given} (is) thus also equal to {DBE given} [Prop. 1.5].\nAnd since in triangle {AED given} the one side, {AEB given}, has been produced, angle {DEB given} (is) thus greater than {DAE given} [Prop. 1.16].\nAnd {DAE given} (is) equal to {DBE given} [Prop. 1.5].\nThus, {DEB given} (is) greater than {DBE given}.\nAnd the greater angle is subtended by the greater side [Prop. 1.19].\nThus, {DB line} (is) greater than {DE line}.\nAnd {DB line} (is) equal to {DF line}.\nThus, {DF line} (is) greater than {DE line}, the lesser than the greater.\nThe very thing is impossible.\nThus, the straight-line joining {A point} to {B point} will not fall outside the circle.\nSo, similarly, we can show that neither (will it fall) on the circumference itself.\nThus, (it will fall) inside (the circle).\n\nThus, if two points are taken at random on the circumference of a circle then the straight-line joining the points will fall inside the circle.\n(Which is) the very thing it was required to show.\n",
"prose3": "If two points are taken at random on the circumference of a circle then the straight-line joining the points will fall inside the circle.\n\nLet {circle D ABC} be a circle, and let two points {point A} and {point B} have been taken at random on its circumference.\nI say that the straight-line joining {point A} to {point B} will fall inside the circle.\n\nFor (if) not then, if possible, let it fall outside (the circle), like {given AEB} (in the figure).\nAnd let the center of the circle {circle D ABC} have been found [Prop. 3.1], and let it be (at point) {point D}.\nAnd let {line DA} and {line DB} have been joined, and let {line DFE} have been drawn through.\n\nTherefore, since {line DA} is equal to {line DB}, the angle {given DAE} (is) thus also equal to {given DBE} [Prop. 1.5].\nAnd since in triangle {given AED} the one side, {given AEB}, has been produced, angle {given DEB} (is) thus greater than {given DAE} [Prop. 1.16].\nAnd {given DAE} (is) equal to {given DBE} [Prop. 1.5].\nThus, {given DEB} (is) greater than {given DBE}.\nAnd the greater angle is subtended by the greater side [Prop. 1.19].\nThus, {line DB} (is) greater than {line DE}.\nAnd {line DB} (is) equal to {line DF}.\nThus, {line DF} (is) greater than {line DE}, the lesser than the greater.\nThe very thing is impossible.\nThus, the straight-line joining {point A} to {point B} will not fall outside the circle.\nSo, similarly, we can show that neither (will it fall) on the circumference itself.\nThus, (it will fall) inside (the circle).\n\nThus, if two points are taken at random on the circumference of a circle then the straight-line joining the points will fall inside the circle.\n(Which is) the very thing it was required to show.\n",
"prose2": "Let {circle D ABC} be a circle, and let two points {point A} and {point B} have been taken at random on its circumference.\nI say that the straight-line joining {point A} to {point B} will fall inside the circle.\n",
"points": {
"A": [107.18279480282831, 224.79998503464563],
"B": [274.7999850346456, 354.8172051971717],
"C": [347.9360580479465, 87.47674814364645],
"D": [256, 206],
"E": [153.4061233984813, 338.2632287433936],
"F": [164.0639419520535, 324.52325185635357]
},
"shapes": [
["circle", [256, 206], 300],
["line", [256, 206], [153.4061233984813, 338.2632287433936]],
["line", [256, 206], [107.18279480282831, 224.79998503464563]],
["line", [256, 206], [274.7999850346456, 354.8172051971717]],
[
"arc",
[228.41918258561606, 241.55697555690608],
[274.7999850346456, 354.8172051971717],
[107.18279480282831, 224.79998503464563]
]
],
"given": {
"AEB": [
[
"arc",
[228.41918258561606, 241.55697555690608],
[274.7999850346456, 354.8172051971717],
[107.18279480282831, 224.79998503464563]
]
],
"AED": [
["line", [107.18279480282831, 224.79998503464563], [256, 206]],
[
"arc",
[228.41918258561606, 241.55697555690608],
[153.4061233984813, 338.2632287433936],
[107.18279480282831, 224.79998503464563]
],
["line", [153.4061233984813, 338.2632287433936], [256, 206]]
],
"DAE": [
["line", [107.18279480282831, 224.79998503464563], [256, 206]],
[
"arc",
[228.41918258561606, 241.55697555690608],
[153.4061233984813, 338.2632287433936],
[107.18279480282831, 224.79998503464563]
],
[
"anglecurve",
[256, 206],
[107.18279480282831, 224.79998503464563],
[106.46707846715543, 347.3152556318218]
]
],
"DEB": [
["line", [153.4061233984813, 338.2632287433936], [256, 206]],
[
"arc",
[228.41918258561606, 241.55697555690608],
[274.7999850346456, 354.8172051971717],
[153.4061233984813, 338.2632287433936]
],
[
"anglecurve",
[256, 206],
[153.4061233984813, 338.2632287433936],
[259.2244950080179, 400.0125284096931]
]
],
"DBE": [
["line", [274.7999850346456, 354.8172051971717], [256, 206]],
[
"arc",
[228.41918258561606, 241.55697555690608],
[274.7999850346456, 354.8172051971717],
[153.4061233984813, 338.2632287433936]
],
[
"anglecurve",
[256, 206],
[274.7999850346456, 354.8172051971717],
[156.31174836786363, 385.97874480883246]
]
]
},
"letters": {
"A": [3],
"B": [5.5],
"C": [0],
"D": [1],
"E": [4, 1.2],
"F": [1, 1.2]
},
"id": "3.2"
}