# Simson Line Theorem

Simson Line Theorem

In[]:=

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

In[]:=

<<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 14:38:57.

## Construction

Construction

## Verification

Verification

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

In[]:=

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

In[]:=

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}};

In[]:=

Prove["Simson theorem",Mappingmap]

Proof is successful.

Simson Line Theorem: Step 11

Out[]=

In[]:=

EndSession[];