Equilateral Cevian Triangle Point

Author

Eric W. Weisstein
November 28, 2004

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

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

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

Derivation

CevianTriangle[{α,β,γ}]

eqns=Factor/@Simplify[(1/(a b c)) SideLengths[Evaluate[CevianTriangle[{α,β,γ}]]]^2,TriangleQ[]]

Numerically

t=Triangle[{{0,0},{1,0},{.3,.5}}]

(soln=NSolve[Equal@@eqns&&2Area[]==(a α+b β+c γ)/.Thread[{a,b,c}->SideLengths[Evaluate[t]]],{α,β,γ}]);

soln1=Select[{α,β,γ}/.soln,Equal@@(Sign/@#)&]

eqns/.Thread[{α,β,γ}->First[soln1]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

p=Point[TrilinearToCartesian[t,-{α,β,γ}/.Thread[{α,β,γ}->soln1[[1]]]]]

SideLengths[Evaluate[ct=CevianTriangle[t,p]]]

Show[Graphics[{
t,
{Blue,ct,EqualSideTicks[ct,1,.015]},
{Red,Cevians[t,p]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[ct],{"A'","B'","C'"}}],
{{p,"(Subscript[X, 370])",{Red,LabelOffset->.03}}}
],LabelOffset->.02]
}],AspectRatio->Automatic,PlotRange->All]

Solve

(soln=Solve[Equal@@eqns&&2Area[]==(a α+b β+c γ),{α},{β,γ}])//Timing

Gröbner Basis

eqns2=Numerator[Factor/@(Together/@(Subtract@@@Partition[eqns,2,1]))]

(soln=Factor/@GroebnerBasis[Append[eqns2,4Area[]^2-(a α+b β+c γ)^2],{α},{β,γ},MonomialOrder->EliminationOrder])//Timing

...ran overnight...

PID USER     PRI  NI  SIZE  RSS SHARE STAT %CPU %MEM   TIME CPU COMMAND
 9623 eww       14   0 1284M 1.3G  5268 R    99.0 63.6 346:34   0 MathKernel

Center

CenterName[X[370]]

Side Lengths

SideLengths[CevianTriangle[X[370]]]