Trisected Perimeter Point

Author

Eric W. Weisstein
January 12, 2005

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

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

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

Center Function

Yff Form

CenterFunction[X[369]]

Moses Form

CenterFunction[X[369]]

Figure

TrisectedPerimeter[t_Triangle]:=Line[Insert[#1,#2,2]]&@@@Transpose[{Partition[RotateLeft[Vertices[CevianTriangle[t,Trilinears[X[369]]]]],2,1,1],
Vertices[t]}];

t=Triangle[{{0,0},{1,0},{.3,.5}}];
colors={Red,Green,Blue};
Show[Graphics[{
{Dashing[{.02}],
Transpose[{colors,Cevians[t,Trilinears[X[369]]]}]
},
Transpose[{colors,TrisectedPerimeter[t]}],
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Transpose[{VertexPoints[CevianTriangle[t,Trilinears[X[369]]]],{"A'","B'","C'"},colors}],
{{X[369][t],"(Subscript[X, 369])",Purple}}
],LabelOffset->.03
]
}],AspectRatio->Automatic,PlotRange->All]

Derivation

t1=Factor[FullSimplify[Plus@@(Distance[#,{1,0,0}]&/@{{α,0,γ},{α,β,0}}),TriangleQ[]&&And@@(0<#<1&/@{α,β,γ})]]

t2=Factor[FullSimplify[Plus@@(Distance[#,{0,1,0}]&/@{{0,β,γ},{α,β,0}}),TriangleQ[]&&And@@(0<#<1&/@{α,β,γ})]]

t3=Factor[FullSimplify[Plus@@(Distance[#,{0,0,1}]&/@{{0,β,γ},{α,0,γ}}),TriangleQ[]&&And@@(0<#<1&/@{α,β,γ})]]

Solve

(soln=Solve[Equal@@{t1,t2,t3,2Semiperimeter[]/3},{α},{β,γ}])//Timing

(soln=Solve[Equal@@{t1,t2,t3,2Semiperimeter[]/3}&&2Area[]==(a α+b β+c γ),{α,β,γ}])//Timing

...ran overnight...

Groebner Basis

eqns=Numerator[Together[#-2/3Semiperimeter[]]&/@{t1,t2,t3}]

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

myroot=Root[(a^18)-12 (a^16) (b^2)-4 (a^15) (b^3)+60 (a^14) (b^4)+36 (a^13) (b^5)-160 (a^12) (b^6)-132 (a^11) (b^7)+246 (a^10) (b^8)+260 (a^9) (b^9)-216 (a^8) (b^10)-300 (a^7) (b^11)+92 (a^6) (b^12)+204 (a^5) (b^13)-76 (a^3) (b^15)-15 (a^2) (b^16)+12 a (b^17)+4 (b^18)-12 (a^16) b c-12 (a^15) (b^2) c+108 (a^14) (b^3) c+132 (a^13) (b^4) c-372 (a^12) (b^5) c-540 (a^11) (b^6) c+636 (a^10) (b^7) c+1140 (a^9) (b^8) c-540 (a^8) (b^9) c-1380 (a^7) (b^10) c+132 (a^6) (b^11) c+972 (a^5) (b^12) c+132 (a^4) (b^13) c-372 (a^3) (b^14) c-108 (a^2) (b^15) c+60 a (b^16) c+24 (b^17) c-12 (a^16) (c^2)-12 (a^15) b (c^2)+144 (a^14) (b^2) (c^2)+216 (a^13) (b^3) (c^2)-528 (a^12) (b^4) (c^2)-1044 (a^11) (b^5) (c^2)+768 (a^10) (b^6) (c^2)+2304 (a^9) (b^7) (c^2)-216 (a^8) (b^8) (c^2)-2676 (a^7) (b^9) (c^2)-624 (a^6) (b^10) (c^2)+1656 (a^5) (b^11) (c^2)+720 (a^4) (b^12) (c^2)-492 (a^3) (b^13) (c^2)-288 (a^2) (b^14) (c^2)+48 a (b^15) (c^2)+36 (b^16) (c^2)-4 (a^15) (c^3)+108 (a^14) b (c^3)+216 (a^13) (b^2) (c^3)-568 (a^12) (b^3) (c^3)-1356 (a^11) (b^4) (c^3)+708 (a^10) (b^5) (c^3)+3104 (a^9) (b^6) (c^3)+624 (a^8) (b^7) (c^3)-3036 (a^7) (b^8) (c^3)-1996 (a^6) (b^9) (c^3)+888 (a^5) (b^10) (c^3)+1416 (a^4) (b^11) (c^3)+428 (a^3) (b^12) (c^3)-228 (a^2) (b^13) (c^3)-240 a (b^14) (c^3)-64 (b^15) (c^3)+60 (a^14) (c^4)+132 (a^13) b (c^4)-528 (a^12) (b^2) (c^4)-1356 (a^11) (b^3) (c^4)+660 (a^10) (b^4) (c^4)+3432 (a^9) (b^5) (c^4)+1200 (a^8) (b^6) (c^4)-2616 (a^7) (b^7) (c^4)-2556 (a^6) (b^8) (c^4)-780 (a^5) (b^9) (c^4)+912 (a^4) (b^10) (c^4)+1668 (a^3) (b^11) (c^4)+492 (a^2) (b^12) (c^4)-480 a (b^13) (c^4)-240 (b^14) (c^4)+36 (a^13) (c^5)-372 (a^12) b (c^5)-1044 (a^11) (b^2) (c^5)+708 (a^10) (b^3) (c^5)+3432 (a^9) (b^4) (c^5)+1368 (a^8) (b^5) (c^5)-2280 (a^7) (b^6) (c^5)-2232 (a^6) (b^7) (c^5)-1548 (a^5) (b^8) (c^5)-708 (a^4) (b^9) (c^5)+1212 (a^3) (b^10) (c^5)+1332 (a^2) (b^11) (c^5)+192 a (b^12) (c^5)-96 (b^13) (c^5)-160 (a^12) (c^6)-540 (a^11) b (c^6)+768 (a^10) (b^2) (c^6)+3104 (a^9) (b^3) (c^6)+1200 (a^8) (b^4) (c^6)-2280 (a^7) (b^5) (c^6)-2016 (a^6) (b^6) (c^6)-1392 (a^5) (b^7) (c^6)-1632 (a^4) (b^8) (c^6)-604 (a^3) (b^9) (c^6)+864 (a^2) (b^10) (c^6)+1200 a (b^11) (c^6)+464 (b^12) (c^6)-132 (a^11) (c^7)+636 (a^10) b (c^7)+2304 (a^9) (b^2) (c^7)+624 (a^8) (b^3) (c^7)-2616 (a^7) (b^4) (c^7)-2232 (a^6) (b^5) (c^7)-1392 (a^5) (b^6) (c^7)-1680 (a^4) (b^7) (c^7)-1764 (a^3) (b^8) (c^7)-996 (a^2) (b^9) (c^7)+528 a (b^10) (c^7)+576 (b^11) (c^7)+246 (a^10) (c^8)+1140 (a^9) b (c^8)-216 (a^8) (b^2) (c^8)-3036 (a^7) (b^3) (c^8)-2556 (a^6) (b^4) (c^8)-1548 (a^5) (b^5) (c^8)-1632 (a^4) (b^6) (c^8)-1764 (a^3) (b^7) (c^8)-2106 (a^2) (b^8) (c^8)-1320 a (b^9) (c^8)-264 (b^10) (c^8)+260 (a^9) (c^9)-540 (a^8) b (c^9)-2676 (a^7) (b^2) (c^9)-1996 (a^6) (b^3) (c^9)-780 (a^5) (b^4) (c^9)-708 (a^4) (b^5) (c^9)-604 (a^3) (b^6) (c^9)-996 (a^2) (b^7) (c^9)-1320 a (b^8) (c^9)-880 (b^9) (c^9)-216 (a^8) (c^10)-1380 (a^7) b (c^10)-624 (a^6) (b^2) (c^10)+888 (a^5) (b^3) (c^10)+912 (a^4) (b^4) (c^10)+1212 (a^3) (b^5) (c^10)+864 (a^2) (b^6) (c^10)+528 a (b^7) (c^10)-264 (b^8) (c^10)-300 (a^7) (c^11)+132 (a^6) b (c^11)+1656 (a^5) (b^2) (c^11)+1416 (a^4) (b^3) (c^11)+1668 (a^3) (b^4) (c^11)+1332 (a^2) (b^5) (c^11)+1200 a (b^6) (c^11)+576 (b^7) (c^11)+92 (a^6) (c^12)+972 (a^5) b (c^12)+720 (a^4) (b^2) (c^12)+428 (a^3) (b^3) (c^12)+492 (a^2) (b^4) (c^12)+192 a (b^5) (c^12)+464 (b^6) (c^12)+204 (a^5) (c^13)+132 (a^4) b (c^13)-492 (a^3) (b^2) (c^13)-228 (a^2) (b^3) (c^13)-480 a (b^4) (c^13)-96 (b^5) (c^13)-372 (a^3) b (c^14)-288 (a^2) (b^2) (c^14)-240 a (b^3) (c^14)-240 (b^4) (c^14)-76 (a^3) (c^15)-108 (a^2) b (c^15)+48 a (b^2) (c^15)-64 (b^3) (c^15)-15 (a^2) (c^16)+60 a b (c^16)+36 (b^2) (c^16)+12 a (c^17)+24 b (c^17)+4 (c^18)+36 (a^16) ((#1)^2)-48 (a^15) b ((#1)^2)+264 (a^14) (b^2) ((#1)^2)+864 (a^13) (b^3) ((#1)^2)-1572 (a^12) (b^4) ((#1)^2)-3216 (a^11) (b^5) ((#1)^2)+3024 (a^10) (b^6) ((#1)^2)+5184 (a^9) (b^7) ((#1)^2)-2916 (a^8) (b^8) ((#1)^2)-4176 (a^7) (b^9) ((#1)^2)+1608 (a^6) (b^10) ((#1)^2)+1632 (a^5) (b^11) ((#1)^2)-540 (a^4) (b^12) ((#1)^2)-240 (a^3) (b^13) ((#1)^2)+96 (a^2) (b^14) ((#1)^2)-48 (a^15) c ((#1)^2)-48 (a^14) b c ((#1)^2)+2208 (a^13) (b^2) c ((#1)^2)+2160 (a^12) (b^3) c ((#1)^2)-7824 (a^11) (b^4) c ((#1)^2)-7584 (a^10) (b^5) c ((#1)^2)+10176 (a^9) (b^6) c ((#1)^2)+9696 (a^8) (b^7) c ((#1)^2)-4944 (a^7) (b^8) c ((#1)^2)-4464 (a^6) (b^9) c ((#1)^2)-96 (a^5) (b^10) c ((#1)^2)-336 (a^4) (b^11) c ((#1)^2)+528 (a^3) (b^12) c ((#1)^2)+576 (a^2) (b^13) c ((#1)^2)+264 (a^14) (c^2) ((#1)^2)+2208 (a^13) b (c^2) ((#1)^2)+1128 (a^12) (b^2) (c^2) ((#1)^2)-8160 (a^11) (b^3) (c^2) ((#1)^2)-10704 (a^10) (b^4) (c^2) ((#1)^2)+4416 (a^9) (b^5) (c^2) ((#1)^2)+18576 (a^8) (b^6) (c^2) ((#1)^2)+10560 (a^7) (b^7) (c^2) ((#1)^2)-9816 (a^6) (b^8) (c^2) ((#1)^2)-12768 (a^5) (b^9) (c^2) ((#1)^2)-504 (a^4) (b^10) (c^2) ((#1)^2)+3744 (a^3) (b^11) (c^2) ((#1)^2)+1056 (a^2) (b^12) (c^2) ((#1)^2)+864 (a^13) (c^3) ((#1)^2)+2160 (a^12) b (c^3) ((#1)^2)-8160 (a^11) (b^2) (c^3) ((#1)^2)-14016 (a^10) (b^3) (c^3) ((#1)^2)-1344 (a^9) (b^4) (c^3) ((#1)^2)+9504 (a^8) (b^5) (c^3) ((#1)^2)+24384 (a^7) (b^6) (c^3) ((#1)^2)+14016 (a^6) (b^7) (c^3) ((#1)^2)-16416 (a^5) (b^8) (c^3) ((#1)^2)-11280 (a^4) (b^9) (c^3) ((#1)^2)+672 (a^3) (b^10) (c^3) ((#1)^2)-384 (a^2) (b^11) (c^3) ((#1)^2)-1572 (a^12) (c^4) ((#1)^2)-7824 (a^11) b (c^4) ((#1)^2)-10704 (a^10) (b^2) (c^4) ((#1)^2)-1344 (a^9) (b^3) (c^4) ((#1)^2)+7080 (a^8) (b^4) (c^4) ((#1)^2)+23328 (a^7) (b^5) (c^4) ((#1)^2)+32784 (a^6) (b^6) (c^4) ((#1)^2)+4032 (a^5) (b^7) (c^4) ((#1)^2)-17700 (a^4) (b^8) (c^4) ((#1)^2)-12048 (a^3) (b^9) (c^4) ((#1)^2)-3744 (a^2) (b^10) (c^4) ((#1)^2)-3216 (a^11) (c^5) ((#1)^2)-7584 (a^10) b (c^5) ((#1)^2)+4416 (a^9) (b^2) (c^5) ((#1)^2)+9504 (a^8) (b^3) (c^5) ((#1)^2)+23328 (a^7) (b^4) (c^5) ((#1)^2)+30048 (a^6) (b^5) (c^5) ((#1)^2)+23616 (a^5) (b^6) (c^5) ((#1)^2)+11616 (a^4) (b^7) (c^5) ((#1)^2)-8208 (a^3) (b^8) (c^5) ((#1)^2)-3648 (a^2) (b^9) (c^5) ((#1)^2)+3024 (a^10) (c^6) ((#1)^2)+10176 (a^9) b (c^6) ((#1)^2)+18576 (a^8) (b^2) (c^6) ((#1)^2)+24384 (a^7) (b^3) (c^6) ((#1)^2)+32784 (a^6) (b^4) (c^6) ((#1)^2)+23616 (a^5) (b^5) (c^6) ((#1)^2)+37488 (a^4) (b^6) (c^6) ((#1)^2)+15552 (a^3) (b^7) (c^6) ((#1)^2)+2592 (a^2) (b^8) (c^6) ((#1)^2)+5184 (a^9) (c^7) ((#1)^2)+9696 (a^8) b (c^7) ((#1)^2)+10560 (a^7) (b^2) (c^7) ((#1)^2)+14016 (a^6) (b^3) (c^7) ((#1)^2)+4032 (a^5) (b^4) (c^7) ((#1)^2)+11616 (a^4) (b^5) (c^7) ((#1)^2)+15552 (a^3) (b^6) (c^7) ((#1)^2)+6912 (a^2) (b^7) (c^7) ((#1)^2)-2916 (a^8) (c^8) ((#1)^2)-4944 (a^7) b (c^8) ((#1)^2)-9816 (a^6) (b^2) (c^8) ((#1)^2)-16416 (a^5) (b^3) (c^8) ((#1)^2)-17700 (a^4) (b^4) (c^8) ((#1)^2)-8208 (a^3) (b^5) (c^8) ((#1)^2)+2592 (a^2) (b^6) (c^8) ((#1)^2)-4176 (a^7) (c^9) ((#1)^2)-4464 (a^6) b (c^9) ((#1)^2)-12768 (a^5) (b^2) (c^9) ((#1)^2)-11280 (a^4) (b^3) (c^9) ((#1)^2)-12048 (a^3) (b^4) (c^9) ((#1)^2)-3648 (a^2) (b^5) (c^9) ((#1)^2)+1608 (a^6) (c^10) ((#1)^2)-96 (a^5) b (c^10) ((#1)^2)-504 (a^4) (b^2) (c^10) ((#1)^2)+672 (a^3) (b^3) (c^10) ((#1)^2)-3744 (a^2) (b^4) (c^10) ((#1)^2)+1632 (a^5) (c^11) ((#1)^2)-336 (a^4) b (c^11) ((#1)^2)+3744 (a^3) (b^2) (c^11) ((#1)^2)-384 (a^2) (b^3) (c^11) ((#1)^2)-540 (a^4) (c^12) ((#1)^2)+528 (a^3) b (c^12) ((#1)^2)+1056 (a^2) (b^2) (c^12) ((#1)^2)-240 (a^3) (c^13) ((#1)^2)+576 (a^2) b (c^13) ((#1)^2)+96 (a^2) (c^14) ((#1)^2)+384 (a^14) ((#1)^4)-384 (a^13) b ((#1)^4)-5568 (a^12) (b^2) ((#1)^4)-6912 (a^11) (b^3) ((#1)^4)+6144 (a^10) (b^4) ((#1)^4)+15360 (a^9) (b^5) ((#1)^4)+3456 (a^8) (b^6) ((#1)^4)-8448 (a^7) (b^7) ((#1)^4)-4992 (a^6) (b^8) ((#1)^4)+384 (a^5) (b^9) ((#1)^4)+576 (a^4) (b^10) ((#1)^4)-384 (a^13) c ((#1)^4)+35328 (a^12) b c ((#1)^4)-1536 (a^11) (b^2) c ((#1)^4)-117120 (a^10) (b^3) c ((#1)^4)-768 (a^9) (b^4) c ((#1)^4)+131712 (a^8) (b^5) c ((#1)^4)+7680 (a^7) (b^6) c ((#1)^4)-53376 (a^6) (b^7) c ((#1)^4)-4992 (a^5) (b^8) c ((#1)^4)+3456 (a^4) (b^9) c ((#1)^4)-5568 (a^12) (c^2) ((#1)^4)-1536 (a^11) b (c^2) ((#1)^4)+119808 (a^10) (b^2) (c^2) ((#1)^4)+3840 (a^9) (b^3) (c^2) ((#1)^4)-181632 (a^8) (b^4) (c^2) ((#1)^4)+46080 (a^7) (b^5) (c^2) ((#1)^4)+90624 (a^6) (b^6) (c^2) ((#1)^4)-17664 (a^5) (b^7) (c^2) ((#1)^4)+7488 (a^4) (b^8) (c^2) ((#1)^4)-6912 (a^11) (c^3) ((#1)^4)-117120 (a^10) b (c^3) ((#1)^4)+3840 (a^9) (b^2) (c^3) ((#1)^4)+126720 (a^8) (b^3) (c^3) ((#1)^4)+40704 (a^7) (b^4) (c^3) ((#1)^4)+16512 (a^6) (b^5) (c^3) ((#1)^4)-6912 (a^5) (b^6) (c^3) ((#1)^4)+4608 (a^4) (b^7) (c^3) ((#1)^4)+6144 (a^10) (c^4) ((#1)^4)-768 (a^9) b (c^4) ((#1)^4)-181632 (a^8) (b^2) (c^4) ((#1)^4)+40704 (a^7) (b^3) (c^4) ((#1)^4)-244992 (a^6) (b^4) (c^4) ((#1)^4)+29184 (a^5) (b^5) (c^4) ((#1)^4)-8064 (a^4) (b^6) (c^4) ((#1)^4)+15360 (a^9) (c^5) ((#1)^4)+131712 (a^8) b (c^5) ((#1)^4)+46080 (a^7) (b^2) (c^5) ((#1)^4)+16512 (a^6) (b^3) (c^5) ((#1)^4)+29184 (a^5) (b^4) (c^5) ((#1)^4)-16128 (a^4) (b^5) (c^5) ((#1)^4)+3456 (a^8) (c^6) ((#1)^4)+7680 (a^7) b (c^6) ((#1)^4)+90624 (a^6) (b^2) (c^6) ((#1)^4)-6912 (a^5) (b^3) (c^6) ((#1)^4)-8064 (a^4) (b^4) (c^6) ((#1)^4)-8448 (a^7) (c^7) ((#1)^4)-53376 (a^6) b (c^7) ((#1)^4)-17664 (a^5) (b^2) (c^7) ((#1)^4)+4608 (a^4) (b^3) (c^7) ((#1)^4)-4992 (a^6) (c^8) ((#1)^4)-4992 (a^5) b (c^8) ((#1)^4)+7488 (a^4) (b^2) (c^8) ((#1)^4)+384 (a^5) (c^9) ((#1)^4)+3456 (a^4) b (c^9) ((#1)^4)+576 (a^4) (c^10) ((#1)^4)+1024 (a^12) ((#1)^6)+6144 (a^11) b ((#1)^6)+15360 (a^10) (b^2) ((#1)^6)+20480 (a^9) (b^3) ((#1)^6)+15360 (a^8) (b^4) ((#1)^6)+6144 (a^7) (b^5) ((#1)^6)+1024 (a^6) (b^6) ((#1)^6)+6144 (a^11) c ((#1)^6)-24576 (a^10) b c ((#1)^6)-104448 (a^9) (b^2) c ((#1)^6)-104448 (a^8) (b^3) c ((#1)^6)-24576 (a^7) (b^4) c ((#1)^6)+6144 (a^6) (b^5) c ((#1)^6)+15360 (a^10) (c^2) ((#1)^6)-104448 (a^9) b (c^2) ((#1)^6)+506880 (a^8) (b^2) (c^2) ((#1)^6)-104448 (a^7) (b^3) (c^2) ((#1)^6)+15360 (a^6) (b^4) (c^2) ((#1)^6)+20480 (a^9) (c^3) ((#1)^6)-104448 (a^8) b (c^3) ((#1)^6)-104448 (a^7) (b^2) (c^3) ((#1)^6)+20480 (a^6) (b^3) (c^3) ((#1)^6)+15360 (a^8) (c^4) ((#1)^6)-24576 (a^7) b (c^4) ((#1)^6)+15360 (a^6) (b^2) (c^4) ((#1)^6)+6144 (a^7) (c^5) ((#1)^6)+6144 (a^6) b (c^5) ((#1)^6)+1024 (a^6) (c^6) ((#1)^6)&,2];

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

X[369][t]

TrilinearToCartesian[t,CyclicTrilinears[myroot]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]]

Factor/@CoefficientList[myroot[[1]][x],x]//ColumnForm//TraditionalForm

Check Perimeter Trisection

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

X[369][t]

Cartesian[CyclicTrilinears[Module[{U,V,Q,Z},
U = 216 (a - b) (b - c) (c - a) ;V = a^2 + b^2 + c^2 - 10 (a b + b c + c a);
Q = (U + Sqrt[U^2 - 108 V^3])^(1/3);
Z = (a + b + c)/2 + Q/(6 2^(1/3)) + V/(Q 4^(1/3));
b c (Z^2 - (a + 2 c) Z - a^2 + b^2 + 2 c^2 + 2 a b + 2 b c + 3 c a)
]][t]]

TrisectedPerimeter[t]/.Line[x_List]:>Plus@@Norm/@(Subtract@@@Partition[x,2,1])

Yff Form

t1=Plus@@(Distance[#,{1,0,0}]&/@{{α,0,γ},{α,β,0}})/.Thread[{α,β,γ}->Trilinears[X[369]]];

t2=Plus@@(Distance[#,{0,1,0}]&/@{{0,β,γ},{α,β,0}})/.Thread[{α,β,γ}->Trilinears[X[369]]];

t3=Plus@@(Distance[#,{0,0,1}]&/@{{0,β,γ},{α,0,γ}})/.Thread[{α,β,γ}->Trilinears[X[369]]];

{t1,t2,t3}/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

FullSimplify[t1,a>0&&b>0&&c>0&&a+b>c&&b+c>a&&a+c>b]//Timing

root=Root[-(a^2) b-5 a (b^2)-5 (a^2) c-9 a b c-(b^2) c-a (c^2)-5 b (c^2)+(a^2) #1+8 a b #1+(b^2) #1+8 a c #1+8 b c #1+(c^2) #1-3 a ((#1)^2)-3 b ((#1)^2)-3 c ((#1)^2)+2 ((#1)^3)&,1];

eqn=Factor[(c √(-((a-b-c) (a+b-c) (a-b+c) (a+b+c) (((2 (a^2)+3 a b-(b^2)+2 (a+b) c+(c^2)-(2 a+b) Root[-(a^2) b-5 a (b^2)-5 (a^2) c-9 a b c-(b^2) c-a (c^2)-5 b (c^2)+(a^2) #1+8 a b #1+(b^2) #1+8 a c #1+8 b c #1+(c^2) #1-3 a ((#1)^2)-3 b ((#1)^2)-3 c ((#1)^2)+2 ((#1)^3)&,1]+((Root[-(a^2) b-5 a (b^2)-5 (a^2) c-9 a b c-(b^2) c-a (c^2)-5 b (c^2)+(a^2) #1+8 a b #1+(b^2) #1+8 a c #1+8 b c #1+(c^2) #1-3 a ((#1)^2)-3 b ((#1)^2)-3 c ((#1)^2)+2 ((#1)^3)&,1])^2)))^2))/((((a^2)+5 a (b+c)+c (4 b+3 c)-(3 a+b+2 c) Root[-(a^2) b-5 a (b^2)-5 (a^2) c-9 a b c-(b^2) c-a (c^2)-5 b (c^2)+(a^2) #1+8 a b #1+(b^2) #1+8 a c #1+8 b c #1+(c^2) #1-3 a ((#1)^2)-3 b ((#1)^2)-3 c ((#1)^2)+2 ((#1)^3)&,1]+2 ((Root[-(a^2) b-5 a (b^2)-5 (a^2) c-9 a b c-(b^2) c-a (c^2)-5 b (c^2)+(a^2) #1+8 a b #1+(b^2) #1+8 a c #1+8 b c #1+(c^2) #1-3 a ((#1)^2)-3 b ((#1)^2)-3 c ((#1)^2)+2 ((#1)^3)&,1])^2)))^2))+b √(-((a-b-c) (a+b-c) (a-b+c) (a+b+c) ((((a^2)+2 (b^2)+3 b c-(c^2)+2 a (b+c)-(2 b+c) Root[-(a^2) b-5 a (b^2)-5 (a^2) c-9 a b c-(b^2) c-a (c^2)-5 b (c^2)+(a^2) #1+8 a b #1+(b^2) #1+8 a c #1+8 b c #1+(c^2) #1-3 a ((#1)^2)-3 b ((#1)^2)-3 c ((#1)^2)+2 ((#1)^3)&,1]+((Root[-(a^2) b-5 a (b^2)-5 (a^2) c-9 a b c-(b^2) c-a (c^2)-5 b (c^2)+(a^2) #1+8 a b #1+(b^2) #1+8 a c #1+8 b c #1+(c^2) #1-3 a ((#1)^2)-3 b ((#1)^2)-3 c ((#1)^2)+2 ((#1)^3)&,1])^2)))^2))/(((b (4 a+3 b)+5 (a+b) c+(c^2)-(a+2 b+3 c) Root[-(a^2) b-5 a (b^2)-5 (a^2) c-9 a b c-(b^2) c-a (c^2)-5 b (c^2)+(a^2) #1+8 a b #1+(b^2) #1+8 a c #1+8 b c #1+(c^2) #1-3 a ((#1)^2)-3 b ((#1)^2)-3 c ((#1)^2)+2 ((#1)^3)&,1]+2 ((Root[-(a^2) b-5 a (b^2)-5 (a^2) c-9 a b c-(b^2) c-a (c^2)-5 b (c^2)+(a^2) #1+8 a b #1+(b^2) #1+8 a c #1+8 b c #1+(c^2) #1-3 a ((#1)^2)-3 b ((#1)^2)-3 c ((#1)^2)+2 ((#1)^3)&,1])^2)))^2)))/.root->r]

c √(-((a-b-c) (a+b-c) (a-b+c) (a+b+c)Factor[ (((2 (a^2)+3 a b-(b^2)+2 (a+b) c+(c^2)-(2 a+b) r+(r^2)))^2)])/Factor[((((a^2)+5 a (b+c)+c (4 b+3 c)-(3 a+b+2 c) r+2 (r^2)))^2)])+b √(-((a-b-c) (a+b-c) (a-b+c) (a+b+c) Factor[((((a^2)+2 (b^2)+3 b c-(c^2)+2 a (b+c)-(2 b+c) r+(r^2)))^2)])/Factor[(((b (4 a+3 b)+5 (a+b) c+(c^2)-(a+2 b+3 c) r+2 (r^2)))^2)])

FullSimplify[eqn/.r->root,TriangleQ[]]//Timing

Suggestion: simplify should be able to handle this

FullSimplify[Factor[(c (( (2 (a^2)+3 a b-(b^2)+2 (a+b) c+(c^2)-(2 a+b) r+(r^2)))/(((a^2)+5 a (b+c)+c (4 b+3 c)-(3 a+b+2 c) r+2 (r^2))))+b(( ((a^2)+2 (b^2)+3 b c-(c^2)+2 a (b+c)-(2 b+c) r+(r^2)))/((b (4 a+3 b)+5 (a+b) c+(c^2)-(a+2 b+3 c) r+2 (r^2)))))/.r->root]]

FullSimplify[t1-t2,a>0&&b>0&&c>0&&a+b>c&&b+c>a&&a+c>b]//Timing

...ran overnight...

Root Form

t1=Plus@@(Distance[#,{1,0,0}]&/@{{α,0,γ},{α,β,0}})/.Thread[{α,β,γ}->CyclicTrilinears[myroot]];

t2=Plus@@(Distance[#,{0,1,0}]&/@{{0,β,γ},{α,β,0}})/.Thread[{α,β,γ}->CyclicTrilinears[myroot]];

t3=Plus@@(Distance[#,{0,0,1}]&/@{{0,β,γ},{α,0,γ}})/.Thread[{α,β,γ}->CyclicTrilinears[myroot]];

{t1,t2,t3}/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

FullSimplify[t1,a>0&&b>0&&c>0&&a+b>c&&b+c>a&&a+c>b]//Timing

RootReduce[%]