# Centroid

Centroid

## Load Eos

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)

## 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.

EosSession["Centroid"];

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NewOrigami[10]

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Centroid: Step 1

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NewPoint["E"{7,8}]

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Centroid: Step 1

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Suppose that we are given triangle ΔABE.

triangle={Red,Thick,Line[{"A","B","E","A"}]};

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ShowOrigami[Moretriangle]

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Centroid: Step 1

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

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Centroid: Step 3

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HO["BB2"]!

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Centroid: Step 5

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

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Centroid: Step 7

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

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Centroid: Step 9

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

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Centroid: Step 11

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HO["E2E"]!

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Centroid: Step 13

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ShowOrigami[Moretriangle]

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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)

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}}]

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Proof is successful.

Centroid: Step 13

Success,0.011866,

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We can further prove that the medians trisects each other.

## 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.