triangle-centers.nb

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This demonstration defines a Manipulate environment for displaying several \
different classical triangle centers as well as some auxiliary lines. The \
vertices of the triangle can be clicked and dragged to change the shape of \
the triangle, and the tick boxes can be used to choose what to show or hide. \
The following triangle centers are included:\
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Cell["Orthocenter: Intersection of the three altitudes.", "Item",
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Circumcenter: Center of the circle circumscribed about the triangle.\
\>", "Item",
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Centroid: Intersection of the three lines connecting each vertex to its \
opposite side\[CloseCurlyQuote]s midpoint.\
\>", "Item",
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Incenter: Center of the circle inscribed inside the triangle.\
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Cell["\<\
The construction lines and inscribed/circumscribed circles can be toggled on \
or off, as can the Euler line, along which the orthocenter, circumcenter, and \
centroid (but not necessarily the incenter) always lie. Note that all four \
centers coincide for an equilateral triangle.\
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    False, $CellContext`circcircleshow$$ = False, $CellContext`circshow$$ = 
    False, $CellContext`eulershow$$ = True, $CellContext`insccircleshow$$ = 
    False, $CellContext`inshow$$ = False, $CellContext`orthshow$$ = False, 
    Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, 
    Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 
    1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{
       Hold[$CellContext`a$$], {-0.2, 0.7}}, Automatic}, {{
       Hold[$CellContext`b$$], {0.7, -0.1}}, Automatic}, {{
       Hold[$CellContext`c$$], {-0.5, -0.3}}, Automatic}, {{
       Hold[$CellContext`eulershow$$], True, "Show Euler Line"}, {
      True -> "On", False -> "Off"}}, {{
       Hold[$CellContext`orthshow$$], False, 
       "Show Orthocenter Construction Lines"}, {
      True -> "On", False -> "Off"}}, {{
       Hold[$CellContext`circshow$$], False, 
       "Show Circumcenter Construction Lines"}, {
      True -> "On", False -> "Off"}}, {{
       Hold[$CellContext`centshow$$], False, 
       "Show Centroid Construction Lines"}, {True -> "On", False -> "Off"}}, {{
       Hold[$CellContext`inshow$$], False, 
       "Show Incenter Construction Lines"}, {True -> "On", False -> "Off"}}, {{
       Hold[$CellContext`circcircleshow$$], False, 
       "Show Circumscribed Circle"}, {True -> "On", False -> "Off"}}, {{
       Hold[$CellContext`insccircleshow$$], False, "Show Inscribed Circle"}, {
      True -> "On", False -> "Off"}}}, Typeset`size$$ = {360., {177., 183.}}, 
    Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = 
    True, $CellContext`eulershow$17774$$ = 
    False, $CellContext`orthshow$17775$$ = 
    False, $CellContext`circshow$17776$$ = 
    False, $CellContext`centshow$17777$$ = 
    False, $CellContext`inshow$17778$$ = 
    False, $CellContext`circcircleshow$17779$$ = 
    False, $CellContext`insccircleshow$17780$$ = False}, 
    DynamicBox[Manipulate`ManipulateBoxes[
     1, StandardForm, 
      "Variables" :> {$CellContext`a$$ = {-0.2, 0.7}, $CellContext`b$$ = {
         0.7, -0.1}, $CellContext`c$$ = {-0.5, -0.3}, $CellContext`centshow$$ = 
        False, $CellContext`circcircleshow$$ = False, $CellContext`circshow$$ = 
        False, $CellContext`eulershow$$ = True, $CellContext`insccircleshow$$ = 
        False, $CellContext`inshow$$ = False, $CellContext`orthshow$$ = 
        False}, "ControllerVariables" :> {
        Hold[$CellContext`eulershow$$, $CellContext`eulershow$17774$$, False], 
        Hold[$CellContext`orthshow$$, $CellContext`orthshow$17775$$, False], 
        Hold[$CellContext`circshow$$, $CellContext`circshow$17776$$, False], 
        Hold[$CellContext`centshow$$, $CellContext`centshow$17777$$, False], 
        Hold[$CellContext`inshow$$, $CellContext`inshow$17778$$, False], 
        Hold[$CellContext`circcircleshow$$, \
$CellContext`circcircleshow$17779$$, False], 
        Hold[$CellContext`insccircleshow$$, \
$CellContext`insccircleshow$17780$$, False]}, 
      "OtherVariables" :> {
       Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 
        Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 
        Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$,
         Typeset`skipInitDone$$}, "Body" :> 
      Module[{$CellContext`mab$, $CellContext`mbc$, $CellContext`mca$, \
$CellContext`orthmalta$, $CellContext`orthmaltb$, $CellContext`orthmaltc$, \
$CellContext`eqab$, $CellContext`eqbc$, $CellContext`eqca$, \
$CellContext`ortheqalta$, $CellContext`ortheqaltb$, $CellContext`ortheqaltc$, \
$CellContext`orthfoota$ = {0, 0}, $CellContext`orthfootb$ = {0, 
          0}, $CellContext`orthfootc$ = {0, 0}, $CellContext`orthocenter$ = {
          0, 0}, $CellContext`midab$, $CellContext`midbc$, \
$CellContext`midca$, $CellContext`circnormasq$, $CellContext`circnormbsq$, \
$CellContext`circnormcsq$, $CellContext`circumcenter$, $CellContext`circsx$, \
$CellContext`circsy$, $CellContext`circad$, $CellContext`circbd$, \
$CellContext`circr$, $CellContext`centroid$, $CellContext`inmbia$, \
$CellContext`inmbib$, $CellContext`inmbic$, $CellContext`ineqbia$, \
$CellContext`ineqbib$, $CellContext`ineqbic$, $CellContext`inintab$ = {0, 
          0}, $CellContext`inintbc$ = {0, 0}, $CellContext`inintca$ = {0, 
          0}, $CellContext`incenter$ = {0, 
          0}, $CellContext`inr$, $CellContext`indsqab$, \
$CellContext`indsqbc$, $CellContext`indsqca$}, $CellContext`mab$ = 
         If[Part[$CellContext`a$$, 1] != 
           Part[$CellContext`b$$, 1], (Part[$CellContext`b$$, 2] - 
            Part[$CellContext`a$$, 2])/(Part[$CellContext`b$$, 1] - 
           Part[$CellContext`a$$, 1]), Infinity]; $CellContext`mbc$ = 
         If[Part[$CellContext`b$$, 1] != 
           Part[$CellContext`c$$, 1], (Part[$CellContext`c$$, 2] - 
            Part[$CellContext`b$$, 2])/(Part[$CellContext`c$$, 1] - 
           Part[$CellContext`b$$, 1]), Infinity]; $CellContext`mca$ = 
         If[Part[$CellContext`a$$, 1] != 
           Part[$CellContext`c$$, 1], (Part[$CellContext`a$$, 2] - 
            Part[$CellContext`c$$, 2])/(Part[$CellContext`a$$, 1] - 
           Part[$CellContext`c$$, 1]), Infinity]; $CellContext`orthmalta$ = 
         If[$CellContext`mbc$ != 0, -(1/$CellContext`mbc$), 
           Infinity]; $CellContext`orthmaltb$ = 
         If[$CellContext`mca$ != 0, -(1/$CellContext`mca$), 
           Infinity]; $CellContext`orthmaltc$ = 
         If[$CellContext`mab$ != 0, -(1/$CellContext`mab$), 
           Infinity]; $CellContext`eqab$ = $CellContext`mab$ (# - 
            Part[$CellContext`a$$, 1]) + 
          Part[$CellContext`a$$, 
            2]& ; $CellContext`eqbc$ = $CellContext`mbc$ (# - 
            Part[$CellContext`b$$, 1]) + 
          Part[$CellContext`b$$, 
            2]& ; $CellContext`eqca$ = $CellContext`mca$ (# - 
            Part[$CellContext`c$$, 1]) + 
          Part[$CellContext`c$$, 
            2]& ; $CellContext`ortheqalta$ = $CellContext`orthmalta$ (# - 
            Part[$CellContext`a$$, 1]) + 
          Part[$CellContext`a$$, 
            2]& ; $CellContext`ortheqaltb$ = $CellContext`orthmaltb$ (# - 
            Part[$CellContext`b$$, 1]) + 
          Part[$CellContext`b$$, 
            2]& ; $CellContext`ortheqaltc$ = $CellContext`orthmaltc$ (# - 
            Part[$CellContext`c$$, 1]) + Part[$CellContext`c$$, 2]& ; 
        Part[$CellContext`orthfoota$, 1] = ReplaceAll[$CellContext`x, 
           Part[
            
            Solve[$CellContext`ortheqalta$[$CellContext`x] == \
$CellContext`eqbc$[$CellContext`x], $CellContext`x], 1]]; 
        Part[$CellContext`orthfoota$, 2] = $CellContext`ortheqalta$[
           Part[$CellContext`orthfoota$, 1]]; 
        Part[$CellContext`orthfootb$, 1] = ReplaceAll[$CellContext`x, 
           Part[
            
            Solve[$CellContext`ortheqaltb$[$CellContext`x] == \
$CellContext`eqca$[$CellContext`x], $CellContext`x], 1]]; 
        Part[$CellContext`orthfootb$, 2] = $CellContext`ortheqaltb$[
           Part[$CellContext`orthfootb$, 1]]; 
        Part[$CellContext`orthfootc$, 1] = ReplaceAll[$CellContext`x, 
           Part[
            
            Solve[$CellContext`ortheqaltc$[$CellContext`x] == \
$CellContext`eqab$[$CellContext`x], $CellContext`x], 1]]; 
        Part[$CellContext`orthfootc$, 2] = $CellContext`ortheqaltc$[
           Part[$CellContext`orthfootc$, 1]]; 
        Part[$CellContext`orthocenter$, 1] = ReplaceAll[$CellContext`x, 
           Part[
            
            Solve[$CellContext`ortheqalta$[$CellContext`x] == \
$CellContext`ortheqaltb$[$CellContext`x], $CellContext`x], 1]]; 
        Part[$CellContext`orthocenter$, 2] = $CellContext`ortheqalta$[
           Part[$CellContext`orthocenter$, 1]]; $CellContext`midab$ = 
         Mean[{$CellContext`a$$, $CellContext`b$$}]; $CellContext`midbc$ = 
         Mean[{$CellContext`b$$, $CellContext`c$$}]; $CellContext`midca$ = 
         Mean[{$CellContext`c$$, $CellContext`a$$}]; \
$CellContext`circnormasq$ = 
         Part[$CellContext`a$$, 1]^2 + 
          Part[$CellContext`a$$, 2]^2; $CellContext`circnormbsq$ = 
         Part[$CellContext`b$$, 1]^2 + 
          Part[$CellContext`b$$, 2]^2; $CellContext`circnormcsq$ = 
         Part[$CellContext`c$$, 1]^2 + 
          Part[$CellContext`c$$, 2]^2; $CellContext`circsx$ = (1/2) 
          Det[{{$CellContext`circnormasq$, 
              Part[$CellContext`a$$, 2], 1}, {$CellContext`circnormbsq$, 
              Part[$CellContext`b$$, 2], 1}, {$CellContext`circnormcsq$, 
              Part[$CellContext`c$$, 2], 1}}]; $CellContext`circsy$ = (1/2) 
          Det[{{
              Part[$CellContext`a$$, 1], $CellContext`circnormasq$, 1}, {
              Part[$CellContext`b$$, 1], $CellContext`circnormbsq$, 1}, {
              Part[$CellContext`c$$, 1], $CellContext`circnormcsq$, 
              1}}]; $CellContext`circad$ = Det[{{
             Part[$CellContext`a$$, 1], 
             Part[$CellContext`a$$, 2], 1}, {
             Part[$CellContext`b$$, 1], 
             Part[$CellContext`b$$, 2], 1}, {
             Part[$CellContext`c$$, 1], 
             Part[$CellContext`c$$, 2], 1}}]; $CellContext`circbd$ = Det[{{
             Part[$CellContext`a$$, 1], 
             Part[$CellContext`a$$, 2], $CellContext`circnormasq$}, {
             Part[$CellContext`b$$, 1], 
             Part[$CellContext`b$$, 2], $CellContext`circnormbsq$}, {
             Part[$CellContext`c$$, 1], 
             
             Part[$CellContext`c$$, 
              2], $CellContext`circnormcsq$}}]; $CellContext`circumcenter$ = \
{$CellContext`circsx$, $CellContext`circsy$}/$CellContext`circad$; \
$CellContext`circr$ = 
         Sqrt[$CellContext`circbd$/$CellContext`circad$ + \
($CellContext`circsx$^2 + $CellContext`circsy$^2)/$CellContext`circad$^2]; \
$CellContext`centroid$ = 
         Mean[{$CellContext`a$$, $CellContext`b$$, $CellContext`c$$}]; \
$CellContext`inmbia$ = 
         Tan[(ArcTan[$CellContext`mab$] + ArcTan[$CellContext`mca$])/
           2]; $CellContext`inmbib$ = 
         Tan[(ArcTan[$CellContext`mab$] + ArcTan[$CellContext`mbc$])/
           2]; $CellContext`inmbic$ = 
         Tan[(ArcTan[$CellContext`mbc$] + ArcTan[$CellContext`mca$])/
           2]; $CellContext`ineqbia$ = $CellContext`inmbia$ (# - 
            Part[$CellContext`a$$, 1]) + 
          Part[$CellContext`a$$, 
            2]& ; $CellContext`ineqbib$ = $CellContext`inmbib$ (# - 
            Part[$CellContext`b$$, 1]) + 
          Part[$CellContext`b$$, 
            2]& ; $CellContext`ineqbic$ = $CellContext`inmbic$ (# - 
            Part[$CellContext`c$$, 1]) + Part[$CellContext`c$$, 2]& ; 
        Part[$CellContext`inintbc$, 1] = ReplaceAll[$CellContext`x, 
           Part[
            
            Solve[$CellContext`eqbc$[$CellContext`x] == \
$CellContext`ineqbia$[$CellContext`x], $CellContext`x], 1]]; 
        Part[$CellContext`inintbc$, 2] = $CellContext`eqbc$[
           Part[$CellContext`inintbc$, 1]]; If[
          Or[
           And[
           Part[$CellContext`inintbc$, 1] < Part[$CellContext`b$$, 1], 
            Part[$CellContext`inintbc$, 1] < Part[$CellContext`c$$, 1]], 
           And[
           Part[$CellContext`inintbc$, 1] > Part[$CellContext`b$$, 1], 
            Part[$CellContext`inintbc$, 1] > Part[$CellContext`c$$, 1]], 
           And[
           Part[$CellContext`inintbc$, 2] < Part[$CellContext`b$$, 2], 
            Part[$CellContext`inintbc$, 2] < Part[$CellContext`c$$, 2]], 
           And[
           Part[$CellContext`inintbc$, 2] > Part[$CellContext`b$$, 2], 
            Part[$CellContext`inintbc$, 2] > 
            Part[$CellContext`c$$, 2]]], $CellContext`inmbia$ = 
           If[$CellContext`inmbia$ != 0, -(1/$CellContext`inmbia$), Infinity]; 
          Part[$CellContext`inintbc$, 1] = ReplaceAll[$CellContext`x, 
             Part[
              
              Solve[$CellContext`eqbc$[$CellContext`x] == \
$CellContext`ineqbia$[$CellContext`x], $CellContext`x], 1]]; 
          Part[$CellContext`inintbc$, 2] = $CellContext`eqbc$[
             Part[$CellContext`inintbc$, 1]]; Null]; 
        Part[$CellContext`inintca$, 1] = ReplaceAll[$CellContext`x, 
           Part[
            
            Solve[$CellContext`eqca$[$CellContext`x] == \
$CellContext`ineqbib$[$CellContext`x], $CellContext`x], 1]]; 
        Part[$CellContext`inintca$, 2] = $CellContext`eqca$[
           Part[$CellContext`inintca$, 1]]; If[
          Or[
           And[
           Part[$CellContext`inintca$, 1] < Part[$CellContext`a$$, 1], 
            Part[$CellContext`inintca$, 1] < Part[$CellContext`c$$, 1]], 
           And[
           Part[$CellContext`inintca$, 1] > Part[$CellContext`a$$, 1], 
            Part[$CellContext`inintca$, 1] > Part[$CellContext`c$$, 1]], 
           And[
           Part[$CellContext`inintca$, 2] < Part[$CellContext`a$$, 2], 
            Part[$CellContext`inintca$, 2] < Part[$CellContext`c$$, 2]], 
           And[
           Part[$CellContext`inintca$, 2] > Part[$CellContext`a$$, 2], 
            Part[$CellContext`inintca$, 2] > 
            Part[$CellContext`c$$, 2]]], $CellContext`inmbib$ = 
           If[$CellContext`inmbib$ != 0, -(1/$CellContext`inmbib$), Infinity]; 
          Part[$CellContext`inintca$, 1] = ReplaceAll[$CellContext`x, 
             Part[
              
              Solve[$CellContext`eqca$[$CellContext`x] == \
$CellContext`ineqbib$[$CellContext`x], $CellContext`x], 1]]; 
          Part[$CellContext`inintca$, 2] = $CellContext`eqca$[
             Part[$CellContext`inintca$, 1]]; Null]; 
        Part[$CellContext`inintab$, 1] = ReplaceAll[$CellContext`x, 
           Part[
            
            Solve[$CellContext`eqab$[$CellContext`x] == \
$CellContext`ineqbic$[$CellContext`x], $CellContext`x], 1]]; 
        Part[$CellContext`inintab$, 2] = $CellContext`eqab$[
           Part[$CellContext`inintab$, 1]]; If[
          Or[
           And[
           Part[$CellContext`inintab$, 1] < Part[$CellContext`a$$, 1], 
            Part[$CellContext`inintab$, 1] < Part[$CellContext`b$$, 1]], 
           And[
           Part[$CellContext`inintab$, 1] > Part[$CellContext`a$$, 1], 
            Part[$CellContext`inintab$, 1] > Part[$CellContext`b$$, 1]], 
           And[
           Part[$CellContext`inintab$, 2] < Part[$CellContext`a$$, 2], 
            Part[$CellContext`inintab$, 2] < Part[$CellContext`b$$, 2]], 
           And[
           Part[$CellContext`inintab$, 2] > Part[$CellContext`a$$, 2], 
            Part[$CellContext`inintab$, 2] > 
            Part[$CellContext`b$$, 2]]], $CellContext`inmbic$ = 
           If[$CellContext`inmbic$ != 0, -(1/$CellContext`inmbic$), Infinity]; 
          Part[$CellContext`inintab$, 1] = ReplaceAll[$CellContext`x, 
             Part[
              
              Solve[$CellContext`eqab$[$CellContext`x] == \
$CellContext`ineqbic$[$CellContext`x], $CellContext`x], 1]]; 
          Part[$CellContext`inintab$, 2] = $CellContext`eqab$[
             Part[$CellContext`inintab$, 1]]; Null]; 
        Part[$CellContext`incenter$, 1] = ReplaceAll[$CellContext`x, 
           Part[
            
            Solve[$CellContext`ineqbia$[$CellContext`x] == \
$CellContext`ineqbib$[$CellContext`x], $CellContext`x], 1]]; 
        Part[$CellContext`incenter$, 2] = $CellContext`ineqbia$[
           
           Part[$CellContext`incenter$, 1]]; $CellContext`indsqab$ = (
            Part[$CellContext`inintab$, 1] - 
            Part[$CellContext`incenter$, 1])^2 + (
            Part[$CellContext`inintab$, 2] - 
            Part[$CellContext`incenter$, 2])^2; $CellContext`indsqbc$ = (
            Part[$CellContext`inintbc$, 1] - 
            Part[$CellContext`incenter$, 1])^2 + (
            Part[$CellContext`inintbc$, 2] - 
            Part[$CellContext`incenter$, 2])^2; $CellContext`indsqca$ = (
            Part[$CellContext`inintca$, 1] - 
            Part[$CellContext`incenter$, 1])^2 + (
            Part[$CellContext`inintca$, 2] - 
            Part[$CellContext`incenter$, 2])^2; $CellContext`inr$ = Sqrt[
           
           Min[$CellContext`indsqab$, $CellContext`indsqbc$, \
$CellContext`indsqca$]]; Graphics[
          Join[{Black, 
            
            Line[{$CellContext`a$$, $CellContext`b$$, $CellContext`c$$, \
$CellContext`a$$}]}, 
           If[
            
            And[$CellContext`eulershow$$ == 
             True, $CellContext`orthocenter$ != $CellContext`circumcenter$], {
             
             InfiniteLine[{$CellContext`orthocenter$, \
$CellContext`circumcenter$}]}, {}], {
            PointSize[Large], Blue, 
            Point[$CellContext`orthocenter$]}, 
           If[$CellContext`orthshow$$ == True, {
             InfiniteLine[{$CellContext`a$$, $CellContext`orthfoota$}], 
             Point[$CellContext`orthfoota$], 
             InfiniteLine[{$CellContext`b$$, $CellContext`orthfootb$}], 
             Point[$CellContext`orthfootb$], 
             InfiniteLine[{$CellContext`c$$, $CellContext`orthfootc$}], 
             Point[$CellContext`orthfootc$]}, {}], {Green, 
            Point[$CellContext`circumcenter$]}, 
           If[$CellContext`circshow$$ == True, {
             InfiniteLine[{$CellContext`midbc$, $CellContext`circumcenter$}], 
             Point[$CellContext`midbc$], 
             InfiniteLine[{$CellContext`midca$, $CellContext`circumcenter$}], 
             Point[$CellContext`midca$], 
             InfiniteLine[{$CellContext`midab$, $CellContext`circumcenter$}], 
             Point[$CellContext`midab$]}, {}], 
           If[$CellContext`circcircleshow$$ == True, {
             Circle[$CellContext`circumcenter$, $CellContext`circr$]}, {}], {
           Red, 
            Point[$CellContext`centroid$]}, 
           If[$CellContext`centshow$$ == True, {
             InfiniteLine[{$CellContext`a$$, $CellContext`midbc$}], 
             Point[$CellContext`midbc$], 
             InfiniteLine[{$CellContext`b$$, $CellContext`midca$}], 
             Point[$CellContext`midca$], 
             InfiniteLine[{$CellContext`c$$, $CellContext`midab$}], 
             Point[$CellContext`midab$]}, {}], {Orange, 
            Point[$CellContext`incenter$]}, 
           If[$CellContext`inshow$$ == True, {
             InfiniteLine[{$CellContext`a$$, $CellContext`inintbc$}], 
             Point[$CellContext`inintbc$], 
             InfiniteLine[{$CellContext`b$$, $CellContext`inintca$}], 
             Point[$CellContext`inintca$], 
             InfiniteLine[{$CellContext`c$$, $CellContext`inintab$}], 
             Point[$CellContext`inintab$]}, {}], 
           If[$CellContext`insccircleshow$$ == True, {
             Circle[$CellContext`incenter$, $CellContext`inr$]}, {}]], 
          PlotRange -> 1]], 
      "Specifications" :> {{{$CellContext`a$$, {-0.2, 0.7}}, Automatic, 
         ControlType -> Locator}, {{$CellContext`b$$, {0.7, -0.1}}, Automatic,
          ControlType -> Locator}, {{$CellContext`c$$, {-0.5, -0.3}}, 
         Automatic, ControlType -> 
         Locator}, {{$CellContext`eulershow$$, True, "Show Euler Line"}, {
         True -> "On", False -> "Off"}}, {{$CellContext`orthshow$$, False, 
          "Show Orthocenter Construction Lines"}, {
         True -> "On", False -> "Off"}}, {{$CellContext`circshow$$, False, 
          "Show Circumcenter Construction Lines"}, {
         True -> "On", False -> "Off"}}, {{$CellContext`centshow$$, False, 
          "Show Centroid Construction Lines"}, {
         True -> "On", False -> "Off"}}, {{$CellContext`inshow$$, False, 
          "Show Incenter Construction Lines"}, {
         True -> "On", False -> "Off"}}, {{$CellContext`circcircleshow$$, 
          False, "Show Circumscribed Circle"}, {
         True -> "On", False -> "Off"}}, {{$CellContext`insccircleshow$$, 
          False, "Show Inscribed Circle"}, {True -> "On", False -> "Off"}}}, 
      "Options" :> {ControlType -> Checkbox}, "DefaultOptions" :> {}],
     ImageSizeCache->{417., {276., 282.}},
     SingleEvaluation->True],
    Deinitialization:>None,
    DynamicModuleValues:>{},
    SynchronousInitialization->True,
    UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$},
    UnsavedVariables:>{Typeset`initDone$$},
    UntrackedVariables:>{Typeset`size$$}], "Manipulate",
   Deployed->True,
   StripOnInput->False],
  Manipulate`InterpretManipulate[1]]], "Output",
 CellChangeTimes->{3.7770757181270256`*^9}]
}, {2}]]
}, Open  ]]
}, Open  ]]
}, Open  ]]
},
WindowSize->{759, 833},
WindowMargins->{{99, Automatic}, {Automatic, 70}},
FrontEndVersion->"10.4 for Microsoft Windows (64-bit) (April 11, 2016)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

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