Paralogic Triangles

Author

Eric W. Weisstein
December 29, 2002

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

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

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

Initialization

<<PlaneGeometry`

Figure

t=Triangle[{{0,0},{.4,1.2},{2,0}}];
l=Line[{{.05,-.1},{1,.5}}];
p=Intersections[l,#]&/@Sides[t];
lines=PerpendicularLine[Sequence@@#,2]&/@Transpose[{p,Sides[t]}];
para=Intersections@@@Partition[lines,2,1,1];
Show[Graphics[{
t,
PointLabels[Transpose[{Point/@Vertices[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],LabelOffset->{-.15,0.05}],{
Red,
lines,
Thickness[.005],
Triangle[Coordinates/@para],
PointLabels[Transpose[{para,{"(Subscript[B, 3])","(Subscript[B, 1])","(Subscript[B, 2])"}}]]
},
{Blue,
LongestLine[Coordinates/@p,.2],
PointLabels[Transpose[{p,{"(Subscript[X, 1])","(Subscript[X, 2])","(Subscript[X, 3])"}}],LabelOffset->{-.15,0.05}]
}
},AspectRatio->Automatic,PlotRange->All]
]

t=Triangle[{{0,0},{.4,1.2},{2,0}}];
l=Line[{{.05,-.1},{1,.5}}];
p=Intersections[l,#]&/@Sides[t];
lines=PerpendicularLine[Sequence@@#,2]&/@Transpose[{p,Sides[t]}];
para=RotateLeft[Intersections@@@Partition[lines,2,1,1]];
parat=Triangle[Coordinates/@para];
int=Intersections[Circumcircle[parat],Circumcircle[t]];
Show[Graphics[{
t,
PointLabels[Transpose[{Point/@Vertices[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],LabelOffset->{-.15,0.05}],{
Red,
lines,
Thickness[.005],
parat,
PointLabels[Transpose[{para,{"(Subscript[B, 1])","(Subscript[B, 2])","(Subscript[B, 3])"}}]]
},
{Blue,
LongestLine[Coordinates/@p,1],
PointLabels[Transpose[{p,{"(Subscript[X, 1])","(Subscript[X, 2])","(Subscript[X, 3])"}}],LabelOffset->{-.15,0.05}]
},
Circumcircle[t],
{Red,Circumcircle[parat]},
{Green,LongestLine[{Coordinates[int[[2]]],Sequence@@#}]&/@Transpose[Vertices/@{t,parat}],
PointSize[.02],int[[2]]},
PointSize[.02],Violet,int[[1]]
},AspectRatio->Automatic,PlotRange->All]
]