Sondat's Theorem

Author

Eric W. Weisstein
December 29, 2002

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

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

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

Initialization

<<PlaneGeometry`

Figure

aa=Triangle[{{0,0},{.4,1.2},{2,0}}];
x=Line[{{.05,-.1},{1,.5}}];
p=Intersections[x,#]&/@Sides[aa];
lines=PerpendicularLine[Sequence@@#,2]&/@Transpose[{p,Sides[aa]}];
para=RotateLeft[Intersections@@@Partition[lines,2,1,1]];
bb=Triangle[Coordinates/@para];

ha=Orthocenter[aa];
hb=Orthocenter[bb];
Show[Graphics[{
aa,
PointLabels[Transpose[{Point/@Vertices[aa],{"(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,.2],
PointLabels[Transpose[{p,{"(Subscript[X, 1])","(Subscript[X, 2])","(Subscript[X, 3])"}}],LabelOffset->{-.15,0.05}]
},
{DarkGreen,
PointLabels[{{ha,"(Subscript[H, A])"},{hb,"(Subscript[H, B])"}}],
Line[Coordinates/@{ha,hb}],
EqualSideTicksAboutMidpoint[Line[Coordinates/@{ha,hb}],1,.04]
}
},AspectRatio->Automatic,PlotRange->All]
]

PointLineDistance[#,x]&/@{ha,hb}