Centroid
Centroid
Load Eos
Load Eos
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<<EosLoader`
Eos3.7.1.1 (March 21,2023) running under Mathematica 13.2.0 for Mac OS X ARM (64-bit) (November 18, 2022) on Thu 30 Mar 2023 11:38:42.
Centroid (1)
Centroid (1)
For any triangle ΔABE, the lines that pass through the midpoints of the three edges AB, BE and EA, and vertices E, A, and B, respectively, intersect at the same point G.
Point G is called the centroid (or the center of gravity) of ΔABE. The line that passes through the edge and the midpoint of the opposite side is called a median.
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EosSession["Centroid"];
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NewOrigami[10]
Centroid: Step 1
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NewPoint["E"{7,8}]
Centroid: Step 1
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Suppose that we are given triangle ΔABE.
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triangle={Red,Thick,Line[{"A","B","E","A"}]};
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ShowOrigami[Moretriangle]
Centroid: Step 1
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HO["A","E",Mark{{"AE","B2"}}]!
Centroid: Step 3
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HO["BB2"]!
Centroid: Step 5
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HO["B","E",Mark{{"BE","A2"}}]!
Centroid: Step 7
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HO["AA2",Mark{{"BB2","G"}}]!
Centroid: Step 9
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HO["A","B",Mark{{"AB","E2"}}]!
Centroid: Step 11
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HO["E2E"]!
Centroid: Step 13
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ShowOrigami[Moretriangle]
Centroid: Step 13
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We prove the following
The three medians of any triangle all pass through one point. (see Coxeter P.10)
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Prove["centroid",Goal(¬ColinearQ["A","B","E"]IncidentQ["G","E2E"]),Mapping{"A"{0,0},"B"{1,0},"C"{1,1},"D"{0,1},"E"{u,v}}]
Proof is successful.
Centroid: Step 13
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We can further prove that the medians trisects each other.
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Prove["medians trisect each other",Goal(¬ColinearQ["A","B","E"]SquaredDistance["G","E"]4SquaredDistance["G","E2"]),Mapping{"A"{0,0},"B"{1,0},"C"{1,1},"D"{0,1},"E"{u,v}}]
Proof is successful.
Centroid: Step 13
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EndSession[];
Centroid (2)
Centroid (2)
Since the midpoint of the segment on the origami is foldable, Orikoto provides a function that gives the midpoint (P+Q)/2 of the segment PQ. Using the function Orikoto`Midpoint, we can construct the centroid as follows.