Euler Points

Author

Eric W. Weisstein
February 27, 2003

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

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

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

Initialization

<<PlaneGeometry`;

Figure

OrthicTriangle[t]

NinePointCircle[t]

MapIndexed[
EqualSideTicksAboutMidpoint[Line[{#1,Coordinates[Orthocenter[t]]}],#2[[1]],.025,.015]&,Vertices[t]
]

MapIndexed[
EqualSideTicksAboutMidpoint[Line[{#1,h}],#2[[1]],.025,.015]&,Vertices[t]]

t=Triangle[{{0.,0.},{2.,-.5},{1.,1.}}];
Show[Graphics[{
t,
EqualSideTicksAboutMidpoints[t,.03,.01],
{
Purple,
NinePointCircle[t]
},
{
Red,
EulerTriangle[t]
},
{
DarkGreen,
Altitudes[t],
PerpSquare[#,.06,4]&/@Transpose[{Sides[t],Altitudes[t]}],
MapIndexed[
EqualSideTicksAboutMidpoint[Line[{#1,Coordinates[Orthocenter[t]]}],#2[[1]],.025,.015]&,Vertices[t]
]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,DarkGreen]&/@Transpose[{VertexPoints[OrthicTriangle[t]],{"(Subscript[H, A])","(Subscript[H, B])","(Subscript[H, C])"}}],
Append[#,Red]&/@Transpose[{EulerPoints[t],{"(Subscript[E, A])","(Subscript[E, B])","(Subscript[E, C])"}}],
Append[#,Blue]&/@Transpose[{Midpoints[t],{"(Subscript[M, A])","(Subscript[M, B])","(Subscript[M, C])"}}],
{
{NinePointCenter[t],"N",Purple},
{Orthocenter[t],"H",DarkGreen}
}
],{PointSize[.025],LabelOffset->.07}
]
}],  AspectRatio->Automatic,PlotRange->All]