Euler Line

Load Eos

<<"EosHeader.m"
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g3 Version 1.2.9
Eos3.31 (November 5,2020) running under Mathematica 12.1.1 for Mac OS X x86 (64-bit) (June 22, 2020)

Euler line

Three centers are collinear

For any triangle ΔABE, the circumcenter, the centroid and the orthocenter are collinear.
The line determined by the three points is called Euler line.
EosSession["Euler line"];
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NewOrigami[10]
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Euler line: Step 1
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NewPoint["E"{7,8}]
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Euler line: Step 1
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HO["A","E",Mark{{"AD","F1"},{"AB","F2"}}]!
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Euler line: Step 3
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HO["B","E",Mark{{"F1F2","O"}}]!
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Euler line: Step 5
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circle={Green,Thickness[0.01],GraphicsCircle["O","OE"]};​​triangle={Red,Thickness[0.01],Line[{"A","B","E","A"}]};
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ShowOrigami[More{triangle,circle}]
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Euler line: Step 5
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O is the circumcenter.
HO["BE"]!
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Euler line: Step 7
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HO["BE","A",Mark{{"BE","J"}}]!
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Euler line: Step 9
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HO["AE"]!
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Euler line: Step 11
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HO["AE","B",Mark{{"AJ","H"}}]!
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Euler line: Step 13
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HO["A","E",Mark{{"AE","K"}}]!
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Euler line: Step 15
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HO["B","E",Mark{{"BE","L"}}]!
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Euler line: Step 17
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HO["AL"]!
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Euler line: Step 19