Simson Theorem
Simson Theorem
Header
Header
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<<"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
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EosSession["Simson Theorem"];
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NewOrigami[10];
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NewPoint[{"E"{2,4},"F"{10,4},"G"{5,7}}]
Simson Theorem: Step 1
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triangle={Red,Thickness[0.01],Line[{"E","F","G","E"}]};
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ShowOrigami[More{triangle}]
Simson Theorem: Step 1
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(*HO["GE"]!*)
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HO["E","F",Mark{{"EG","H"},{"EF","I"}}]!
Simson Theorem: Step 3
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(*HO["GF"]!*)
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HO["F","G",Mark{{"HI","O"}}]!
Simson Theorem: Step 5
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circle={Green,Thickness[0.01],GraphicsCircle["O","OE"]};
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ShowOrigami[More{circle,triangle}]
Simson Theorem: Step 5
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NewPoint["P"{7.5,7.0}](*closetothecircumference*)
Simson Theorem: Step 5
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ShowOrigami[More{circle,triangle}]
Simson Theorem: Step 5
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HO["GE","P",Mark{{"EG","X"}}]!
Simson Theorem: Step 7
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HO["EF","P",Mark{{"EF","Y"}}]!
Simson Theorem: Step 9
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HO["GF","P",Mark{{"GF","Z"}}]!
Simson Theorem: Step 11
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simsonLine={Black,Thickness[0.01],Line[{"X","Z","Y"}]};
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ShowOrigami[MarkPoints{"X","Y","Z","E","F","G","O","P"},More{circle,triangle,simsonLine}]
Simson Theorem: Step 11
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Verification
Verification
We prove that points X, Y and Z are collinear if P is on the circumcircle of ABE.
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Goal[SquaredDistance["O","E"]==SquaredDistance["O","P"]CollinearQ[{"X","Y","Z"}]];
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map={(*"A"{0,0},"B"{1,0},"C"{1,1},"D"{0,1},*)"E"{u1,v1},"F"{u2,v1},"G"{u3,v3},"P"{u4,v4}};
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Prove["Simson theorem",Mappingmap]