# Centroid

Centroid

## Load Eos

Load Eos

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<<EosLoader`

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 16:02:56.

## 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/Origami: Step 1

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NewPoint["E"{7,8}]

Centroid/Origami: 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/Origami: Step 1

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HO["A","E",Mark{{"AE","B2"}}]!

Centroid/Origami: Step 3

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HO["BB2"]!

Centroid/Origami: Step 5

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HO["B","E",Mark{{"BE","A2"}}]!

Centroid/Origami: Step 7

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HO["AA2",Mark{{"BB2","G"}}]!

Centroid/Origami: Step 9

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HO["A","B",Mark{{"AB","E2"}}]!

Centroid/Origami: Step 11

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HO["E2E"]!

Centroid/Origami: Step 13

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ShowOrigami[Moretriangle]

Centroid/Origami: 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/Origami: 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/Origami: 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.