# Nine-point circle

Nine-point circle

Tetsuo Ida

## Load Eos

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 16:10:08.

## Nine-point circle

Nine-point circle

The nine-point circle is a circle that can be constructed by the nine concyclic points defined from the triangle. These nine points are :

1

.The midpoint of each side of the triangle.

2

.The foot of each altitude.

3

.The midpoint of the line segment from each vertex of the triangle to the orthocenter.

### Construction

Construction

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EosSession["nine-point circle"];

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NewOrigami[10]

nine-point circle/Origami: Step 1

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NewPoint["E"{6,7}]

nine-point circle/Origami: Step 1

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

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HO["E","A",Mark{{"EA","F1"},{"AB","F2"}}]!

nine-point circle/Origami: Step 3

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HO["A","B",Mark{{"AB","G1"},{"F1F2","O"}}]!

nine-point circle/Origami: Step 5

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HO["B","E",Mark{{"BE","H1"}}]!

nine-point circle/Origami: Step 7

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Point O is circumcenter

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circle1={Thick,Green,GraphicsCircle["O","OA"]};

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ShowOrigami[More{circle1,triangle}]

nine-point circle/Origami: Step 7

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HO["EA","B",Mark{{"EA","J1"}}]!

nine-point circle/Origami: Step 9

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HO["BA","E",Mark{{"BA","K1"},{"J1B","H"}}]!

nine-point circle/Origami: Step 11

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HO["EB","A",Mark{{"EB","L1"}}]!

nine-point circle/Origami: Step 13

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HO["A","H",Mark->{{"AH","M"}}]!

nine-point circle/Origami: Step 15

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HO["B","H",Mark->{{"BH","N"}}]!

nine-point circle/Origami: Step 17

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HO["E","H",Mark->{{"EH","P"}}]!

nine-point circle/Origami: Step 19

### Verification

Verification