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Euler Line

Load Eos

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<<EosLoader.m
Eos3.7.2 (June 24,2023) running under Mathematica 13.2.1 for Mac OS X ARM (64-bit) (January 27, 2023) on Fri 23 Jun 2023 15:59:21.

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.
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EosSession["Euler line"];
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NewOrigami[10]
Euler line/Origami: Step 1
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NewPoint["E"{7,8}]
Euler line/Origami: Step 1
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HO["A","E",Mark{{"AD","F1"},{"AB","F2"}}]!
Euler line/Origami: Step 3
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HO["B","E",Mark{{"F1F2","O"}}]!
Euler line/Origami: 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}]
Euler line/Origami: Step 5
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O is the circumcenter.
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HO["BE"]!
Euler line/Origami: Step 7
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HO["BE","A",Mark{{"BE","J"}}]!
Euler line/Origami: Step 9
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HO["AE"]!
Euler line/Origami: Step 11
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HO["AE","B",Mark{{"AJ","H"}}]!
Mark
::dup
:Already marked point(s) H exists on the loation.
​
Euler line/Origami: Step 13
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HO["A","E",Mark{{"AE","K"}}]!
Euler line/Origami: Step 15
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HO["B","E",Mark{{"BE","L"}}]!
Euler line/Origami: Step 17
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HO["AL"]!
Euler line/Origami: Step 19
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