Half-Altitude Circle

Author

Eric W. Weisstein
December 19, 2004

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.2,.75}}];
Show[Graphics[{
t,
{Green,ht=HalfAltitudeTriangle[t]},
{Red,HalfAltitudeCircle[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'"}}],
{
{Center[HalfAltitudeCircle[t]],"X",Red}
}
],PointSize[.025],LabelOffset->.03]
}],AspectRatio->Automatic,PlotRange->All]

Radius

CircleRadius[HalfAltitudeCircle]//TraditionalForm

CircleRadius[HalfAltitudeCircle]//InputForm

Center

Center[HalfAltitudeCircle]//TraditionalForm

Center[HalfAltitudeCircle][[1]]//InputForm

Circle Function

CircleFunction[HalfAltitudeCircle]//TraditionalForm

CircleFunction[HalfAltitudeCircle]//InputForm

Incident Points

IncidentCenters[KimberlingCenters,HalfAltitudeCircle]//Timing

Derivation

v=ToSidelengths[TrilinearVertexMatrix[HalfAltitudeTriangle]]

circ1=Together//@Circumcircle@@v;

circ=circ1/.Circle[ctr_List,r_]:>Circle[Symmetrize/@ctr,Together//@ExpandAll[Together//@r]]

circ=circ1/.Circle[ctr_List,r_]:>Circle[Symmetrize/@ctr,Together//@ExpandAll[Simplify[Together//@r],TriangleQ[]]]

circ=circ1/.Circle[ctr_List,r_]:>Circle[Symmetrize/@ctr,Simplify[Together//@ExpandAll[Simplify[Together//@r],TriangleQ[]],TriangleQ[]]]

circ2=circ/.Circle[ctr_List,r_]:>Circle[ctr,FullSimplify[r,TriangleQ[]]]

r=Refine[ToAngles[Factor//@Refine[CircleRadius[circ2],TriangleQ[]]],TriangleQ[]]

InputForm[r]

IncidentCenters[KimberlingCenters,Center[circ]]//Timing

Symmetrize[Center[circ2][[1]]]//InputForm

eqn=Symmetrize[Numerator[Together//@ToSidelengths[TrilinearEquation[circ2]]]]

eqn=Symmetrize[eqn/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]]

Factor[ToAngles[Factor[CircleFunction[eqn]]]]

InputForm[%[[1]]]

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Blue,HalfAltitudeTriangle[t]},
{Red,HalfAltitudeCircle[t]}
}],AspectRatio->Automatic]