Cevian

Author

Eric W. Weisstein
February 15, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Trilinears

CevianTrilinears[alpha]

t=Triangle[{{0,0},{1,0},{.4,.7}}];
alpha={.2,.3,.4};
Show[Graphics[{
t,
{Red,Cevians[t,alpha]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[CevianTriangle[t,alpha]],{"A'","B'","C'"}}],
{{Point[TrilinearToCartesian[t,alpha]],"P",Blue}}
],{PointSize[.025],LabelOffset->.04}
]}],AspectRatio->Automatic,PlotRange->All]

Distances

l=FullSimplify[Distance@@@Transpose[{IdentityMatrix[3],CevianTrilinears[{α,β,γ}]}],TriangleQ[{a,b,c}]&&α>0&&β>0&&γ>0]

l/.Thread[{a,b,c}->SideLengths[t]]/.Thread[{α,β,γ}->alpha]

Sqrt[#.#]&/@Subtract@@@Coordinates/@Cevians[t,alpha]

Ratios

vp=Distance@@@Transpose[{IdentityMatrix[3],Table[{α,β,γ},{3}]}];

ps=Distance@@@Transpose[{CevianTrilinears[{α,β,γ}],Table[{α,β,γ},{3}]}];

ratios=Divide@@@Transpose[{vp,ps}];

Sum

sum=FullSimplify[Plus@@ratios,TriangleQ[{a,b,c}]&&α>0&&β>0&&γ>0]

Experimental`Resolve[ForAll[{a,b,c,α,β,γ},TriangleQ[{a,b,c}]&&And@@(0<#<1&/@{α,β,γ}),a ((α/(b β))+(α/(c γ)))+((b β)/(c γ))+((c γ)/(b β))+((b β+c γ)/(a α))>=6
]]//Timing

Experimental`Resolve[Exists[{a,b,c,α,β,γ},TriangleQ[{a,b,c}]&&And@@(0<#<1&/@{α,β,γ}),a ((α/(b β))+(α/(c γ)))+((b β)/(c γ))+((c γ)/(b β))+((b β+c γ)/(a α))==6
]]//Timing

Timing[eq=Reduce[a ((α/(b β))+(α/(c γ)))+((b β)/(c γ))+((c γ)/(b β))+((b β+c γ)/(a α))==6&&
TriangleQ[{a,b,c}]&&And@@(0<#<1&/@{α,β,γ})/.a->1,
ReplaceValues->True]]

Product

product=FullSimplify[Times@@ratios,TriangleQ[{a,b,c}]&&α>0&&β>0&&γ>0]

Experimental`Resolve[ForAll[
{a,b,c,α,β,γ},
TriangleQ[{a,b,c}]&&And@@(0<#<1&/@{α,β,γ}),
(((a α+b β) (a α+c γ) (b β+c γ))/(a b c α β γ))>=8
]]//Timing

Experimental`Resolve[Exists[
{a,b,c,α,β,γ},
TriangleQ[{a,b,c}]&&And@@(0<#<1&/@{α,β,γ}),
(((a α+b β) (a α+c γ) (b β+c γ))/(a b c α β γ))==8
]]//Timing