Fontené Theorems

Author

Eric W. Weisstein
March 19, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/FonteneTheorems.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FonteneTheorems.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Fontené's First Theorem

t=Triangle[{{0,0},{1,0},{.3,.7}}];
p=Point[{.53,.3}];
mt=MedialTriangle[t];
pt=PedalTriangle[t,p];
def=Intersections@@@Transpose[{Sides[mt],Sides[pt]}];
ls=LongestLine[#,1]&/@Transpose[{Coordinates/@def,Vertices[pt]}];

Show[Graphics[{
t,
{Red,mt,Circumcircle[mt]},
{Blue,pt,Circumcircle[pt]},
{Purple,ls},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[mt],{"A'","B'","C'"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[pt],{"X","Y","Z"}}],
Append[#,Purple]&/@Transpose[{def,{"D","E","F"}}],
{
{p,"P",Blue},
{Intersections@@Take[ls,2],"L",DarkGreen}
}
],PointSize[.025],LabelOffset->.03]
}],AspectRatio->Automatic,PlotRange->{{-.1,1.1},{-.3,.8}}
]

Redo graphic by hand

g=Import["/Users/eww/Documents/Troves/MathOzTeX/graphics/eps/FontenesFirstTheorem.eps"]

g/.Style[{JoinedCurve[{{{0, 2, 0}}}, {{x_, 
      x_}}, CurveClosed -> {0}]}, s__]:>{s,AbsolutePointSize[8],Point[x]}

Fontené's Second Theorem

t=Triangle[{{0,0},{1,0},{.4,.7}}];
l={{-1,-1},Coordinates[Circumcenter[t]]};
f=Intersections[NinePointCircle[t],PedalCircle[t,Point[l[[1]]-.9 Subtract@@l]]][[2]];

movie=Show/@Table[Graphics[{
t,
{Dashing[{.02}],Sides[t,1]},
{Blue,LongestLine[l,2]},
{Red,PedalTriangle[t,p=Point[l[[1]]-d Subtract@@l]],
PerpSquare[#,.03,2]&/@Transpose[{Sides[t],PedalLines[t,p]}],
Dashing[{.02}],PedalLines[t,p]
},
{Orange,PedalCircle[t,p]},
{Purple,NinePointCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{p,"P",Red},
{Circumcenter[t],"O",Blue},
{f,"F",Purple}
}
],PointSize[.025],LabelOffset->.04]
},AspectRatio->Automatic,PlotRange->{{-.1,1.1},{-.3,.8}}],{d,.6,1.3,.01}]

<<GIF`;

Export[gifs<>"fontene2.gif",movie]//Timing