Simson Theorem
Simson Theorem
Header
Header
In[]:=
<<"EosHeader.m"
Simson Theorem
Simson Theorem
Given a triangle ABE and a point P on the circumcircle of the triangle, the three closest points on each line AB, BC and CE are collinear.
Construction
Construction
In[]:=
EosSession["Simson Theorem"];
In[]:=
NewOrigami[10];
In[]:=
NewPoint[{"E"{2,4},"F"{10,4},"G"{5,7}}]
Simson Theorem: Step 1
Out[]=
In[]:=
triangle={Red,Thickness[0.01],Line[{"E","F","G","E"}]};
In[]:=
ShowOrigami[More{triangle}]
Simson Theorem: Step 1
Out[]=
In[]:=
(*HO["GE"]!*)
In[]:=
In[]:=
HO["E","F",Mark{{"EG","H"},{"EF","I"}}]!
Simson Theorem: Step 3
Out[]=
In[]:=
(*HO["GF"]!*)
In[]:=
HO["F","G",Mark{{"HI","O"}}]!
Simson Theorem: Step 5
Out[]=
In[]:=
circle={Green,Thickness[0.01],GraphicsCircle["O","OE"]};
In[]:=
ShowOrigami[More{circle,triangle}]
Simson Theorem: Step 5
Out[]=
In[]:=
NewPoint["P"{7.5,7.0}](*closetothecircumference*)
Simson Theorem: Step 5
Out[]=
In[]:=
ShowOrigami[More{circle,triangle}]
Simson Theorem: Step 5
Out[]=
In[]:=
HO["GE","P",Mark{{"EG","X"}}]!
Simson Theorem: Step 7
Out[]=
In[]:=
HO["EF","P",Mark{{"EF","Y"}}]!
Simson Theorem: Step 9
Out[]=
In[]:=
HO["GF","P",Mark{{"GF","Z"}}]!
Simson Theorem: Step 11
Out[]=
In[]:=
simsonLine={Black,Thickness[0.01],Line[{"X","Z","Y"}]};
In[]:=
ShowOrigami[MarkPoints{"X","Y","Z","E","F","G","O","P"},More{circle,triangle,simsonLine}]
Simson Theorem: Step 11
Out[]=
Verification
Verification
We prove that points X, Y and Z are collinear if P is on the circumcircle of ABE.