# Simson Line Theorem

Simson Line Theorem

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H . Takahashi

## Simson Line Theorem

Simson Line Theorem

Given a triangle ABE and a point P on the circumcircle of the triangle, the three closest points on each line AB, BC and CE are collinear.

## 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 16:18:51.

## Construction

Construction

## Verification

Verification

We prove that points X, Y and Z are collinear if P is on the circumcircle of ABE.

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Goal[SquaredDistance["O","E"]==SquaredDistance["O","P"]CollinearQ[{"X","Y","Z"}]];

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map={(*"A"{0,0},"B"{1,0},"C"{1,1},"D"{0,1},*)"E"{u1,v1},"F"{u2,v1},"G"{u3,v3},"P"{u4,v4}};

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Prove["Simson theorem",Mappingmap]

Proof is successful.

Simson Line Theorem/Origami: Step 11

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EndSession[];