Half-Altitude Triangle

Author

Eric W. Weisstein
November 11, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/Half-AltitudeTriangle.nb.

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

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

Figure

t=Triangle[{{0,0},{1,0},{.2,.75}}];
Show[Graphics[{
t,
{Red,ht=HalfAltitudeTriangle[t]},
{Blue,Altitudes[t],
MapIndexed[EqualSideTicksAboutMidpoint[#1,Sequence@@#2,.015,.015]&,Altitudes[t]],
PerpSquare[#,.03,3]&/@Transpose[{Sides[t],Altitudes[t]}]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[ht],{"A'","B'","C'"}}]
],PointSize[.025],LabelOffset->.03]
}],AspectRatio->Automatic,PlotRange->All]

Trilinears

<<Typesetting`

Trilinears[HalfAltitudeTriangle]//MatrixForm//TraditionalForm

Side Lengths

Simplify[Together//@DistanceTrilinears[SideLengths[HalfAltitudeTriangle]] ,TriangleQ[]]

Area

FullSimplify[Together//@DistanceTrilinears[Area[HalfAltitudeTriangle]Δ/Area[]],TriangleQ[]]

LawOfCosines[%]

Centers

(pts={#,KimberlingLookup[#[HalfAltitudeTriangle]]}&/@Take[KimberlingCenters,All]);//Timing

l=Map[First,DeleteCases[pts,_?(Last[#]==={}&)],{2}]

{#[[1]],Name[#[[1]]],X[#[[2,1]]],Name[X[#[[2,1]]]]}&/@Cases[pts,_?(Last[#]=!={}&)]//TableForm