Simson Line

Author

Eric W. Weisstein
January 27, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/SimsonLine.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SimsonLine.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Graphic

New

Circle[Circumcircle]

Factor/@ToSidelengths[SimsonLine[Trilinears[Incenter]]]

Factor/@ToSidelengths[SimsonLine[Trilinears[TarryPoint]]]

((#)/(PolynomialGCD@@#))&@{-((a^3) (a-b) (b^2) (a+b) (a-c) (((a-b-c))^2) (((a+b-c))^2) (c^2) (a+c) (((a-b+c))^2) (((a+b+c))^2))/(4 ((((a^4)+(b^4)-(a^2) (c^2)-(b^2) (c^2)))^2) ((a^2) (b^2)-(b^4)+(a^2) (c^2)-(c^4)) ((((a^4)-(a^2) (b^2)-(b^2) (c^2)+(c^4)))^2)),-((a^2) (a-b) (b^3) (a+b) (((a-b-c))^2) (b-c) (((a+b-c))^2) (c^2) (((a-b+c))^2) (b+c) (((a+b+c))^2))/(4 ((((a^4)+(b^4)-(a^2) (c^2)-(b^2) (c^2)))^2) ((((a^2) (b^2)-(b^4)+(a^2) (c^2)-(c^4)))^2) ((a^4)-(a^2) (b^2)-(b^2) (c^2)+(c^4))),((a^2) (b^2) (a-c) (((a-b-c))^2) (b-c) (((a+b-c))^2) (c^3) (a+c) (((a-b+c))^2) (b+c) (((a+b+c))^2))/(4 ((a^4)+(b^4)-(a^2) (c^2)-(b^2) (c^2)) ((((a^2) (b^2)-(b^4)+(a^2) (c^2)-(c^4)))^2) ((((a^4)-(a^2) (b^2)-(b^2) (c^2)+(c^4)))^2))}

Line[SimsonLine[Trilinears[Incenter]]]

t=Triangle[{{0,0},{1,0},{.2,.7}}];
Show[Graphics[{
t,
SimsonLine[Trilinears[TarryPoint]][t]
}],AspectRatio->Automatic,PlotRange->{{-1,1},{-1,1}}]

Old

tri={{0.,0.},{1.,1.},{1.8,-.2}};
s[i_][tri_]:=SimsonLinePoints[tri,45][[i]];
p=SimsonLinePole[tri,45];
pts={Circumcenter,"O",Orthocenter,"H",
		s[1],"(Subscript[P, 1])",s[2],"(Subscript[P, 2])",s[3],"(Subscript[P, 3])",
	p,"P",
	tri[[1]],"A",tri[[2]],"B",tri[[3]],"C"};

Show[Graphics[{
	Triangle[tri],
	PointLabels[tri,pts,.02,{.04,.06}],
	Circumcircle[tri],
	{Thickness[.01],
				Line[LongestLine[SimsonLinePoints[tri,45] /. Point->Identity]]},
	{Dashing[{.02,.02}],SimsonLineFeet[tri,45],
				NinePointCircle[tri],
			Line[{Orthocenter[tri],p} /. Point->Identity]}
}],AspectRatio->Automatic]

Envelope of Simson Lines

tri={{0.,0.},{1.,.7},{1.6,-.9}};
circ=Circumcircle[tri];
delt={};
movie=Table[Show[Graphics[{Circle@@circ,
delt=AppendTo[delt,SimsonLine[tri,i,5]],
{PointSize[.04],CircumcirclePoint[tri,i]},
Triangle[tri],
{Dashing[{.02,.02}],SimsonLineFeet[tri,i]}}],
AspectRatio->Automatic,PlotRange->{{-1.2,4},{-2,2.5}}],{i,0,2Pi-.001,2Pi/20}];

Export["simsdelt.gif",movie,"GIF"]

delt=Table[SimsonLine[tri,i,5],{i,0,2Pi-.001,2Pi/20}];
Show[Graphics[{Circle@@circ,
delt,
{Thickness[.012],RGBColor[1,0,0],Triangle[tri]},
{RGBColor[1,0,0],SimsonLineFeet[tri,0]},
{PointSize[.04],CircumcirclePoint[tri,0]}
}],
AspectRatio->Automatic,PlotRange->{{-1.2,4},{-2,2.5}}]