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{"sheet":"Simulation","index":0,"n_rows":102,"n_cols":14,"rows_kept":102,"cut":false,"hidden":false,"merged":[[0,1,0,3],[0,4,0,6],[0,9,1,9]],"flags":["formulas_cached_only"]}
[null,"Three Random Numbers",null,null,"Sort the Numbers",{"$nulls":4},"Is it a triangle? Yes - 1, No - 0"]
[null,"A, B, C - in some order",null,null,"C","B","A","A+2","B+2",null,null,"Hit F9 to generate new values"]
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[91,2.451191871059052,0.4337271413178456,1.1063518688433795,2.451191871059052,1.1063518688433795,0.4337271413178456,2.4337271413178456,3.1063518688433795,1]
[92,3.8592987747753194,3.075896859491226,2.5489230092719986,3.8592987747753194,3.075896859491226,2.5489230092719986,0.5489230092719986,1.075896859491226,0]
[93,0.4925892022881282,0.08569881475060015,1.7031877964369544,1.7031877964369544,0.4925892022881282,0.08569881475060015,2.0856988147506,2.492589202288128,0]
[94,0.19337328383048558,1.2619293006410501,0.6567567832603274,1.2619293006410501,0.6567567832603274,0.19337328383048558,2.1933732838304856,2.6567567832603274,0]
[95,3.845779100982366,0.5449101828238803,3.725606615800956,3.845779100982366,3.725606615800956,0.5449101828238803,2.5449101828238803,1.725606615800956,0]
[96,3.6075013249686805,1.762056707279985,1.7932643088447726,3.6075013249686805,1.7932643088447726,1.762056707279985,3.762056707279985,3.7932643088447726,0]
[97,0.8569030218599707,1.4289121439406713,2.7795873265354922,2.7795873265354922,1.4289121439406713,0.8569030218599707,2.8569030218599707,3.4289121439406713,0]
[98,0.8170180532962954,3.187604979630539,1.0731229654109065,3.187604979630539,1.0731229654109065,0.8170180532962954,2.8170180532962954,3.0731229654109065,0]
[99,2.3843174920473658,2.875467048752874,2.727960735050716,2.875467048752874,2.727960735050716,2.3843174920473658,0.38431749204736576,0.7279607350507158,0]
[100,3.4072915945203457,3.047747367254434,2.076042430718008,3.4072915945203457,3.047747367254434,2.076042430718008,0.07604243071800809,1.0477473672544342,0]

{"sheet":"Notes","index":1,"n_rows":25,"n_cols":3,"rows_kept":25,"cut":false,"hidden":false,"merged":[],"flags":["formulas_cached_only"]}
[]
[null,"Columns B, C, and D choose three values from 0 to 4."]
[null,"They represent three points A, B, C that are randomly chosen."]
[null,null,"If the value is between 0 and 1, the point lies on the bottom side."]
[null,null,"If the value is between 1 and 2, the point lies on the rightside."]
[null,null,"If the value is between 2 and 3, the point lies on the top side."]
[null,null,"If the value is between 3 and 4, the point lies on the left side."]
[]
[null,"Columns E, F, G sort A, B, C with A < B < C."]
[]
[null,"Columns H and I show the reflected point for A and B."]
[null,null,"This determines the region in which C can occur to form a triangle containing the center point P."]
[]
[null,"Column J checks to see if the triangle contains P."]
[null,null,"A 1 is returned if it does; a 0 is returned if it doesn't."]
[null,null,"The formula in column J uses nested IF's to check the various cases. It's a mess, but I'm fairly certain it's correct."]
[]
[null,"Cells L4 to N5 (blue with yellow text) show the compiled results."]
[null,null,"L5 is the number of cases (10,000)."]
[null,null,"M5 is the number of triangles with P inside (column J contains a 1 if this is true, so the value of M5 is the sum of all 1's in column J)."]
[null,null,"N5 is the probability, given by M5/L5."]
[]
[null,"Cells L9-N17 was just a check for me. I used it to test Case 1 from the sheet to verify that it was giving correct results."]
[null,null,"It gives the x- and y-coordinates of A, B, and C, plots them in the chart, and then connects them. This provides a visual check that P (red dot) occurs within the triangle."]
[null,null,"Note that some \"triangles\" will not appear in the chart -- for instance, when all of A, B, C lie along the same side."]