Yiu Circle

Author

Eric W. Weisstein
January 2, 2005

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

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

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

Figure

t=Triangle[Reverse/@{{0,0},{16/100,5/10},{1,0}}]//N[#,50]&;
Show[Graphics[{
t,
{Green,YiuTriangle[t]},
{Red,YiuCircle[t]},
{Blue,YiuCircles[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{Center/@YiuCircles[t],{"(Subscript[O, A])","(Subscript[O, B])","(Subscript[O, C])"}}]]]
}],AspectRatio->Automatic,PlotRange->All]

(TrilinearEquation[YiuCircle]/.Thread[{α,β,γ}->#])&/@Vertices[Triangle[YiuTriangle]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]/.Thread[{A,B,C}->Angles[t]]//Chop

Derivation

v=Factor[Simplify[ToSidelengths[TrilinearVertexMatrix[YiuTriangle]/.Cos[x_]:>TrigExpand[Cos[x]]],TriangleQ[]]]

Factor[ToSidelengths[TrilinearEquation@@v]]

eqn=(-(a^9) (b^7) c (α^2)+4 (a^7) (b^9) c (α^2)-6 (a^5) (b^11) c (α^2)+4 (a^3) (b^13) c (α^2)-a (b^15) c (α^2)+(a^11) (b^3) (c^3) (α^2)-2 (a^9) (b^5) (c^3) (α^2)+7 (a^5) (b^9) (c^3) (α^2)-11 (a^3) (b^11) (c^3) (α^2)+5 a (b^13) (c^3) (α^2)-2 (a^9) (b^3) (c^5) (α^2)+(a^7) (b^5) (c^5) (α^2)-(a^5) (b^7) (c^5) (α^2)+8 (a^3) (b^9) (c^5) (α^2)-9 a (b^11) (c^5) (α^2)-(a^9) b (c^7) (α^2)-(a^5) (b^5) (c^7) (α^2)-2 (a^3) (b^7) (c^7) (α^2)+5 a (b^9) (c^7) (α^2)+4 (a^7) b (c^9) (α^2)+7 (a^5) (b^3) (c^9) (α^2)+8 (a^3) (b^5) (c^9) (α^2)+5 a (b^7) (c^9) (α^2)-6 (a^5) b (c^11) (α^2)-11 (a^3) (b^3) (c^11) (α^2)-9 a (b^5) (c^11) (α^2)+4 (a^3) b (c^13) (α^2)+5 a (b^3) (c^13) (α^2)-a b (c^15) (α^2)-(a^14) (b^2) c α β+4 (a^12) (b^4) c α β-7 (a^10) (b^6) c α β+8 (a^8) (b^8) c α β-7 (a^6) (b^10) c α β+4 (a^4) (b^12) c α β-(a^2) (b^14) c α β+5 (a^12) (b^2) (c^3) α β-10 (a^10) (b^4) (c^3) α β+5 (a^8) (b^6) (c^3) α β+5 (a^6) (b^8) (c^3) α β-10 (a^4) (b^10) (c^3) α β+5 (a^2) (b^12) (c^3) α β+(a^12) (c^5) α β-10 (a^10) (b^2) (c^5) α β+6 (a^8) (b^4) (c^5) α β+6 (a^4) (b^8) (c^5) α β-10 (a^2) (b^10) (c^5) α β+(b^12) (c^5) α β-6 (a^10) (c^7) α β+4 (a^8) (b^2) (c^7) α β-3 (a^6) (b^4) (c^7) α β-3 (a^4) (b^6) (c^7) α β+4 (a^2) (b^8) (c^7) α β-6 (b^10) (c^7) α β+15 (a^8) (c^9) α β+19 (a^6) (b^2) (c^9) α β+23 (a^4) (b^4) (c^9) α β+19 (a^2) (b^6) (c^9) α β+15 (b^8) (c^9) α β-20 (a^6) (c^11) α β-35 (a^4) (b^2) (c^11) α β-35 (a^2) (b^4) (c^11) α β-20 (b^6) (c^11) α β+15 (a^4) (c^13) α β+24 (a^2) (b^2) (c^13) α β+15 (b^4) (c^13) α β-6 (a^2) (c^15) α β-6 (b^2) (c^15) α β+(c^17) α β-(a^15) b c (β^2)+4 (a^13) (b^3) c (β^2)-6 (a^11) (b^5) c (β^2)+4 (a^9) (b^7) c (β^2)-(a^7) (b^9) c (β^2)+5 (a^13) b (c^3) (β^2)-11 (a^11) (b^3) (c^3) (β^2)+7 (a^9) (b^5) (c^3) (β^2)-2 (a^5) (b^9) (c^3) (β^2)+(a^3) (b^11) (c^3) (β^2)-9 (a^11) b (c^5) (β^2)+8 (a^9) (b^3) (c^5) (β^2)-(a^7) (b^5) (c^5) (β^2)+(a^5) (b^7) (c^5) (β^2)-2 (a^3) (b^9) (c^5) (β^2)+5 (a^9) b (c^7) (β^2)-2 (a^7) (b^3) (c^7) (β^2)-(a^5) (b^5) (c^7) (β^2)-a (b^9) (c^7) (β^2)+5 (a^7) b (c^9) (β^2)+8 (a^5) (b^3) (c^9) (β^2)+7 (a^3) (b^5) (c^9) (β^2)+4 a (b^7) (c^9) (β^2)-9 (a^5) b (c^11) (β^2)-11 (a^3) (b^3) (c^11) (β^2)-6 a (b^5) (c^11) (β^2)+5 (a^3) b (c^13) (β^2)+4 a (b^3) (c^13) (β^2)-a b (c^15) (β^2)+(a^12) (b^5) α γ-6 (a^10) (b^7) α γ+15 (a^8) (b^9) α γ-20 (a^6) (b^11) α γ+15 (a^4) (b^13) α γ-6 (a^2) (b^15) α γ+(b^17) α γ-(a^14) b (c^2) α γ+5 (a^12) (b^3) (c^2) α γ-10 (a^10) (b^5) (c^2) α γ+4 (a^8) (b^7) (c^2) α γ+19 (a^6) (b^9) (c^2) α γ-35 (a^4) (b^11) (c^2) α γ+24 (a^2) (b^13) (c^2) α γ-6 (b^15) (c^2) α γ+4 (a^12) b (c^4) α γ-10 (a^10) (b^3) (c^4) α γ+6 (a^8) (b^5) (c^4) α γ-3 (a^6) (b^7) (c^4) α γ+23 (a^4) (b^9) (c^4) α γ-35 (a^2) (b^11) (c^4) α γ+15 (b^13) (c^4) α γ-7 (a^10) b (c^6) α γ+5 (a^8) (b^3) (c^6) α γ-3 (a^4) (b^7) (c^6) α γ+19 (a^2) (b^9) (c^6) α γ-20 (b^11) (c^6) α γ+8 (a^8) b (c^8) α γ+5 (a^6) (b^3) (c^8) α γ+6 (a^4) (b^5) (c^8) α γ+4 (a^2) (b^7) (c^8) α γ+15 (b^9) (c^8) α γ-7 (a^6) b (c^10) α γ-10 (a^4) (b^3) (c^10) α γ-10 (a^2) (b^5) (c^10) α γ-6 (b^7) (c^10) α γ+4 (a^4) b (c^12) α γ+5 (a^2) (b^3) (c^12) α γ+(b^5) (c^12) α γ-(a^2) b (c^14) α γ+(a^17) β γ-6 (a^15) (b^2) β γ+15 (a^13) (b^4) β γ-20 (a^11) (b^6) β γ+15 (a^9) (b^8) β γ-6 (a^7) (b^10) β γ+(a^5) (b^12) β γ-6 (a^15) (c^2) β γ+24 (a^13) (b^2) (c^2) β γ-35 (a^11) (b^4) (c^2) β γ+19 (a^9) (b^6) (c^2) β γ+4 (a^7) (b^8) (c^2) β γ-10 (a^5) (b^10) (c^2) β γ+5 (a^3) (b^12) (c^2) β γ-a (b^14) (c^2) β γ+15 (a^13) (c^4) β γ-35 (a^11) (b^2) (c^4) β γ+23 (a^9) (b^4) (c^4) β γ-3 (a^7) (b^6) (c^4) β γ+6 (a^5) (b^8) (c^4) β γ-10 (a^3) (b^10) (c^4) β γ+4 a (b^12) (c^4) β γ-20 (a^11) (c^6) β γ+19 (a^9) (b^2) (c^6) β γ-3 (a^7) (b^4) (c^6) β γ+5 (a^3) (b^8) (c^6) β γ-7 a (b^10) (c^6) β γ+15 (a^9) (c^8) β γ+4 (a^7) (b^2) (c^8) β γ+6 (a^5) (b^4) (c^8) β γ+5 (a^3) (b^6) (c^8) β γ+8 a (b^8) (c^8) β γ-6 (a^7) (c^10) β γ-10 (a^5) (b^2) (c^10) β γ-10 (a^3) (b^4) (c^10) β γ-7 a (b^6) (c^10) β γ+(a^5) (c^12) β γ+5 (a^3) (b^2) (c^12) β γ+4 a (b^4) (c^12) β γ-a (b^2) (c^14) β γ-(a^15) b c (γ^2)+5 (a^13) (b^3) c (γ^2)-9 (a^11) (b^5) c (γ^2)+5 (a^9) (b^7) c (γ^2)+5 (a^7) (b^9) c (γ^2)-9 (a^5) (b^11) c (γ^2)+5 (a^3) (b^13) c (γ^2)-a (b^15) c (γ^2)+4 (a^13) b (c^3) (γ^2)-11 (a^11) (b^3) (c^3) (γ^2)+8 (a^9) (b^5) (c^3) (γ^2)-2 (a^7) (b^7) (c^3) (γ^2)+8 (a^5) (b^9) (c^3) (γ^2)-11 (a^3) (b^11) (c^3) (γ^2)+4 a (b^13) (c^3) (γ^2)-6 (a^11) b (c^5) (γ^2)+7 (a^9) (b^3) (c^5) (γ^2)-(a^7) (b^5) (c^5) (γ^2)-(a^5) (b^7) (c^5) (γ^2)+7 (a^3) (b^9) (c^5) (γ^2)-6 a (b^11) (c^5) (γ^2)+4 (a^9) b (c^7) (γ^2)+(a^5) (b^5) (c^7) (γ^2)+4 a (b^9) (c^7) (γ^2)-(a^7) b (c^9) (γ^2)-2 (a^5) (b^3) (c^9) (γ^2)-2 (a^3) (b^5) (c^9) (γ^2)-a (b^7) (c^9) (γ^2)+(a^3) (b^3) (c^11) (γ^2));

Center

Symmetrize[Center[eqn][[1]]]

InputForm[%]

Circle Function

Factor[ToAngles[Factor[CircleFunction[eqn][[1]]]]]

InputForm[%]

Radius

r=ToSidelengths[CircleRadius[eqn]];

Simplify[r,TriangleQ[]]//Timing

Aborted after running overnight

Simplify[Together//@r,TriangleQ[]]

r2=Factor[Together//@(r^2)]

poly1=((a^34)-12 (a^32) (b^2)+66 (a^30) (b^4)-220 (a^28) (b^6)+495 (a^26) (b^8)-792 (a^24) (b^10)+924 (a^22) (b^12)-792 (a^20) (b^14)+495 (a^18) (b^16)-220 (a^16) (b^18)+66 (a^14) (b^20)-12 (a^12) (b^22)+(a^10) (b^24)-2 (a^31) (b^2) c+20 (a^29) (b^4) c-88 (a^27) (b^6) c+220 (a^25) (b^8) c-330 (a^23) (b^10) c+264 (a^21) (b^12) c-264 (a^17) (b^16) c+330 (a^15) (b^18) c-220 (a^13) (b^20) c+88 (a^11) (b^22) c-20 (a^9) (b^24) c+2 (a^7) (b^26) c-12 (a^32) (c^2)+116 (a^30) (b^2) (c^2)-501 (a^28) (b^4) (c^2)+1270 (a^26) (b^6) (c^2)-2084 (a^24) (b^8) (c^2)+2294 (a^22) (b^10) (c^2)-1677 (a^20) (b^12) (c^2)+732 (a^18) (b^14) (c^2)-60 (a^16) (b^16) (c^2)-192 (a^14) (b^18) (c^2)+193 (a^12) (b^20) (c^2)-114 (a^10) (b^22) (c^2)+44 (a^8) (b^24) (c^2)-10 (a^6) (b^26) (c^2)+(a^4) (b^28) (c^2)+18 (a^29) (b^2) (c^3)-140 (a^27) (b^4) (c^3)+462 (a^25) (b^6) (c^3)-824 (a^23) (b^8) (c^3)+818 (a^21) (b^10) (c^3)-378 (a^19) (b^12) (c^3)-12 (a^17) (b^14) (c^3)+24 (a^15) (b^16) (c^3)+126 (a^13) (b^18) (c^3)-168 (a^11) (b^20) (c^3)+110 (a^9) (b^22) (c^3)-48 (a^7) (b^24) (c^3)+14 (a^5) (b^26) (c^3)-2 (a^3) (b^28) (c^3)+66 (a^30) (c^4)-502 (a^28) (b^2) (c^4)+1654 (a^26) (b^4) (c^4)-3072 (a^24) (b^6) (c^4)+3493 (a^22) (b^8) (c^4)-2448 (a^20) (b^10) (c^4)+985 (a^18) (b^12) (c^4)-200 (a^16) (b^14) (c^4)+124 (a^14) (b^16) (c^4)-254 (a^12) (b^18) (c^4)+272 (a^10) (b^20) (c^4)-168 (a^8) (b^22) (c^4)+61 (a^6) (b^24) (c^4)-12 (a^4) (b^26) (c^4)+(a^2) (b^28) (c^4)+2 (a^29) (c^5)-86 (a^27) (b^2) (c^5)+464 (a^25) (b^4) (c^5)-1074 (a^23) (b^6) (c^5)+1292 (a^21) (b^8) (c^5)-838 (a^19) (b^10) (c^5)+336 (a^17) (b^12) (c^5)-212 (a^15) (b^14) (c^5)+170 (a^13) (b^16) (c^5)-2 (a^11) (b^18) (c^5)-144 (a^9) (b^20) (c^5)+150 (a^7) (b^22) (c^5)-72 (a^5) (b^24) (c^5)+14 (a^3) (b^26) (c^5)-220 (a^28) (c^6)+1276 (a^26) (b^2) (c^6)-3083 (a^24) (b^4) (c^6)+3976 (a^22) (b^6) (c^6)-2874 (a^20) (b^8) (c^6)+1044 (a^18) (b^10) (c^6)-19 (a^16) (b^12) (c^6)-208 (a^14) (b^14) (c^6)+249 (a^12) (b^16) (c^6)-306 (a^10) (b^18) (c^6)+286 (a^8) (b^20) (c^6)-172 (a^6) (b^22) (c^6)+61 (a^4) (b^24) (c^6)-10 (a^2) (b^26) (c^6)-24 (a^27) (c^7)+294 (a^25) (b^2) (c^7)-984 (a^23) (b^4) (c^7)+1470 (a^21) (b^6) (c^7)-1110 (a^19) (b^8) (c^7)+446 (a^17) (b^10) (c^7)-94 (a^15) (b^12) (c^7)-68 (a^13) (b^14) (c^7)+118 (a^11) (b^16) (c^7)+10 (a^9) (b^18) (c^7)-162 (a^7) (b^20) (c^7)+150 (a^5) (b^22) (c^7)-48 (a^3) (b^24) (c^7)+2 a (b^26) (c^7)+495 (a^26) (c^8)-2092 (a^24) (b^2) (c^8)+3493 (a^22) (b^4) (c^8)-2856 (a^20) (b^6) (c^8)+1069 (a^18) (b^8) (c^8)-28 (a^16) (b^10) (c^8)-87 (a^14) (b^12) (c^8)-64 (a^12) (b^14) (c^8)+204 (a^10) (b^16) (c^8)-296 (a^8) (b^18) (c^8)+286 (a^6) (b^20) (c^8)-168 (a^4) (b^22) (c^8)+44 (a^2) (b^24) (c^8)+132 (a^25) (c^9)-742 (a^23) (b^2) (c^9)+1432 (a^21) (b^4) (c^9)-1250 (a^19) (b^6) (c^9)+528 (a^17) (b^8) (c^9)-104 (a^15) (b^10) (c^9)+12 (a^13) (b^12) (c^9)-56 (a^11) (b^14) (c^9)+92 (a^9) (b^16) (c^9)+10 (a^7) (b^18) (c^9)-144 (a^5) (b^20) (c^9)+110 (a^3) (b^22) (c^9)-20 a (b^24) (c^9)-791 (a^24) (c^10)+2260 (a^22) (b^2) (c^10)-2356 (a^20) (b^4) (c^10)+986 (a^18) (b^6) (c^10)-28 (a^16) (b^8) (c^10)-82 (a^14) (b^10) (c^10)+18 (a^12) (b^12) (c^10)-64 (a^10) (b^14) (c^10)+204 (a^8) (b^16) (c^10)-306 (a^6) (b^18) (c^10)+272 (a^4) (b^20) (c^10)-114 (a^2) (b^22) (c^10)+(b^24) (c^10)-440 (a^23) (c^11)+1302 (a^21) (b^2) (c^11)-1332 (a^19) (b^4) (c^11)+588 (a^17) (b^6) (c^11)-114 (a^15) (b^8) (c^11)+8 (a^13) (b^10) (c^11)+8 (a^11) (b^12) (c^11)-56 (a^9) (b^14) (c^11)+118 (a^7) (b^16) (c^11)-2 (a^5) (b^18) (c^11)-168 (a^3) (b^20) (c^11)+88 a (b^22) (c^11)+912 (a^22) (c^12)-1514 (a^20) (b^2) (c^12)+803 (a^18) (b^4) (c^12)+4 (a^16) (b^6) (c^12)-71 (a^14) (b^8) (c^12)+24 (a^12) (b^10) (c^12)+18 (a^10) (b^12) (c^12)-64 (a^8) (b^14) (c^12)+249 (a^6) (b^16) (c^12)-254 (a^4) (b^18) (c^12)+193 (a^2) (b^20) (c^12)-12 (b^22) (c^12)+990 (a^21) (c^13)-1426 (a^19) (b^2) (c^13)+644 (a^17) (b^4) (c^13)-108 (a^15) (b^6) (c^13)+16 (a^13) (b^8) (c^13)+8 (a^11) (b^10) (c^13)+12 (a^9) (b^12) (c^13)-68 (a^7) (b^14) (c^13)+170 (a^5) (b^16) (c^13)+126 (a^3) (b^18) (c^13)-220 a (b^20) (c^13)-726 (a^20) (c^14)+444 (a^18) (b^2) (c^14)-76 (a^16) (b^4) (c^14)-176 (a^14) (b^6) (c^14)-71 (a^12) (b^8) (c^14)-82 (a^10) (b^10) (c^14)-87 (a^8) (b^12) (c^14)-208 (a^6) (b^14) (c^14)+124 (a^4) (b^16) (c^14)-192 (a^2) (b^18) (c^14)+66 (b^20) (c^14)-1584 (a^19) (c^15)+642 (a^17) (b^2) (c^15)-208 (a^15) (b^4) (c^15)-108 (a^13) (b^6) (c^15)-114 (a^11) (b^8) (c^15)-104 (a^9) (b^10) (c^15)-94 (a^7) (b^12) (c^15)-212 (a^5) (b^14) (c^15)+24 (a^3) (b^16) (c^15)+330 a (b^18) (c^15)+275 (a^18) (c^16)+48 (a^16) (b^2) (c^16)-76 (a^14) (b^4) (c^16)+4 (a^12) (b^6) (c^16)-28 (a^10) (b^8) (c^16)-28 (a^8) (b^10) (c^16)-19 (a^6) (b^12) (c^16)-200 (a^4) (b^14) (c^16)-60 (a^2) (b^16) (c^16)-220 (b^18) (c^16)+1848 (a^17) (c^17)+642 (a^15) (b^2) (c^17)+644 (a^13) (b^4) (c^17)+588 (a^11) (b^6) (c^17)+528 (a^9) (b^8) (c^17)+446 (a^7) (b^10) (c^17)+336 (a^5) (b^12) (c^17)-12 (a^3) (b^14) (c^17)-264 a (b^16) (c^17)+275 (a^16) (c^18)+444 (a^14) (b^2) (c^18)+803 (a^12) (b^4) (c^18)+986 (a^10) (b^6) (c^18)+1069 (a^8) (b^8) (c^18)+1044 (a^6) (b^10) (c^18)+985 (a^4) (b^12) (c^18)+732 (a^2) (b^14) (c^18)+495 (b^16) (c^18)-1584 (a^15) (c^19)-1426 (a^13) (b^2) (c^19)-1332 (a^11) (b^4) (c^19)-1250 (a^9) (b^6) (c^19)-1110 (a^7) (b^8) (c^19)-838 (a^5) (b^10) (c^19)-378 (a^3) (b^12) (c^19)-726 (a^14) (c^20)-1514 (a^12) (b^2) (c^20)-2356 (a^10) (b^4) (c^20)-2856 (a^8) (b^6) (c^20)-2874 (a^6) (b^8) (c^20)-2448 (a^4) (b^10) (c^20)-1677 (a^2) (b^12) (c^20)-792 (b^14) (c^20)+990 (a^13) (c^21)+1302 (a^11) (b^2) (c^21)+1432 (a^9) (b^4) (c^21)+1470 (a^7) (b^6) (c^21)+1292 (a^5) (b^8) (c^21)+818 (a^3) (b^10) (c^21)+264 a (b^12) (c^21)+912 (a^12) (c^22)+2260 (a^10) (b^2) (c^22)+3493 (a^8) (b^4) (c^22)+3976 (a^6) (b^6) (c^22)+3493 (a^4) (b^8) (c^22)+2294 (a^2) (b^10) (c^22)+924 (b^12) (c^22)-440 (a^11) (c^23)-742 (a^9) (b^2) (c^23)-984 (a^7) (b^4) (c^23)-1074 (a^5) (b^6) (c^23)-824 (a^3) (b^8) (c^23)-330 a (b^10) (c^23)-791 (a^10) (c^24)-2092 (a^8) (b^2) (c^24)-3083 (a^6) (b^4) (c^24)-3072 (a^4) (b^6) (c^24)-2084 (a^2) (b^8) (c^24)-792 (b^10) (c^24)+132 (a^9) (c^25)+294 (a^7) (b^2) (c^25)+464 (a^5) (b^4) (c^25)+462 (a^3) (b^6) (c^25)+220 a (b^8) (c^25)+495 (a^8) (c^26)+1276 (a^6) (b^2) (c^26)+1654 (a^4) (b^4) (c^26)+1270 (a^2) (b^6) (c^26)+495 (b^8) (c^26)-24 (a^7) (c^27)-86 (a^5) (b^2) (c^27)-140 (a^3) (b^4) (c^27)-88 a (b^6) (c^27)-220 (a^6) (c^28)-502 (a^4) (b^2) (c^28)-501 (a^2) (b^4) (c^28)-220 (b^6) (c^28)+2 (a^5) (c^29)+18 (a^3) (b^2) (c^29)+20 a (b^4) (c^29)+66 (a^4) (c^30)+116 (a^2) (b^2) (c^30)+66 (b^4) (c^30)-2 a (b^2) (c^31)-12 (a^2) (c^32)-12 (b^2) (c^32)+(c^34));

poly2=((a^28)-9 (a^26) (b^2)+36 (a^24) (b^4)-84 (a^22) (b^6)+126 (a^20) (b^8)-126 (a^18) (b^10)+84 (a^16) (b^12)-36 (a^14) (b^14)+9 (a^12) (b^16)-(a^10) (b^18)-7 (a^26) (c^2)+52 (a^24) (b^2) (c^2)-165 (a^22) (b^4) (c^2)+287 (a^20) (b^6) (c^2)-286 (a^18) (b^8) (c^2)+140 (a^16) (b^10) (c^2)+14 (a^14) (b^12) (c^2)-70 (a^12) (b^14) (c^2)+53 (a^10) (b^16) (c^2)-24 (a^8) (b^18) (c^2)+7 (a^6) (b^20) (c^2)-(a^4) (b^22) (c^2)+20 (a^24) (c^4)-122 (a^22) (b^2) (c^4)+301 (a^20) (b^4) (c^4)-378 (a^18) (b^6) (c^4)+243 (a^16) (b^8) (c^4)-65 (a^14) (b^10) (c^4)+13 (a^12) (b^12) (c^4)-41 (a^10) (b^14) (c^4)+53 (a^8) (b^16) (c^4)-33 (a^6) (b^18) (c^4)+10 (a^4) (b^20) (c^4)-(a^2) (b^22) (c^4)-2 (a^17) (b^6) (c^5)+8 (a^15) (b^8) (c^5)-14 (a^13) (b^10) (c^5)+16 (a^11) (b^12) (c^5)-14 (a^9) (b^14) (c^5)+8 (a^7) (b^16) (c^5)-2 (a^5) (b^18) (c^5)-32 (a^22) (c^6)+154 (a^20) (b^2) (c^6)-281 (a^18) (b^4) (c^6)+245 (a^16) (b^6) (c^6)-109 (a^14) (b^8) (c^6)+28 (a^12) (b^10) (c^6)+9 (a^10) (b^12) (c^6)-46 (a^8) (b^14) (c^6)+58 (a^6) (b^16) (c^6)-33 (a^4) (b^18) (c^6)+7 (a^2) (b^20) (c^6)+8 (a^15) (b^6) (c^7)-16 (a^13) (b^8) (c^7)+8 (a^11) (b^10) (c^7)+8 (a^9) (b^12) (c^7)-16 (a^7) (b^14) (c^7)+8 (a^5) (b^16) (c^7)+46 (a^20) (c^8)-130 (a^18) (b^2) (c^8)+136 (a^16) (b^4) (c^8)-77 (a^14) (b^6) (c^8)+31 (a^12) (b^8) (c^8)-13 (a^10) (b^10) (c^8)+24 (a^8) (b^12) (c^8)-46 (a^6) (b^14) (c^8)+53 (a^4) (b^16) (c^8)-24 (a^2) (b^18) (c^8)-14 (a^13) (b^6) (c^9)+8 (a^11) (b^8) (c^9)-6 (a^9) (b^10) (c^9)+8 (a^7) (b^12) (c^9)-14 (a^5) (b^14) (c^9)-97 (a^18) (c^10)+95 (a^16) (b^2) (c^10)-2 (a^14) (b^4) (c^10)+7 (a^12) (b^6) (c^10)-10 (a^10) (b^8) (c^10)-13 (a^8) (b^10) (c^10)+9 (a^6) (b^12) (c^10)-41 (a^4) (b^14) (c^10)+53 (a^2) (b^16) (c^10)-(b^18) (c^10)+16 (a^11) (b^6) (c^11)+8 (a^9) (b^8) (c^11)+8 (a^7) (b^10) (c^11)+16 (a^5) (b^12) (c^11)+189 (a^16) (c^12)-40 (a^14) (b^2) (c^12)-50 (a^12) (b^4) (c^12)+7 (a^10) (b^6) (c^12)+31 (a^8) (b^8) (c^12)+28 (a^6) (b^10) (c^12)+13 (a^4) (b^12) (c^12)-70 (a^2) (b^14) (c^12)+9 (b^16) (c^12)-14 (a^9) (b^6) (c^13)-16 (a^7) (b^8) (c^13)-14 (a^5) (b^10) (c^13)-240 (a^14) (c^14)-40 (a^12) (b^2) (c^14)-2 (a^10) (b^4) (c^14)-77 (a^8) (b^6) (c^14)-109 (a^6) (b^8) (c^14)-65 (a^4) (b^10) (c^14)+14 (a^2) (b^12) (c^14)-36 (b^14) (c^14)+8 (a^7) (b^6) (c^15)+8 (a^5) (b^8) (c^15)+189 (a^12) (c^16)+95 (a^10) (b^2) (c^16)+136 (a^8) (b^4) (c^16)+245 (a^6) (b^6) (c^16)+243 (a^4) (b^8) (c^16)+140 (a^2) (b^10) (c^16)+84 (b^12) (c^16)-2 (a^5) (b^6) (c^17)-97 (a^10) (c^18)-130 (a^8) (b^2) (c^18)-281 (a^6) (b^4) (c^18)-378 (a^4) (b^6) (c^18)-286 (a^2) (b^8) (c^18)-126 (b^10) (c^18)+46 (a^8) (c^20)+154 (a^6) (b^2) (c^20)+301 (a^4) (b^4) (c^20)+287 (a^2) (b^6) (c^20)+126 (b^8) (c^20)-32 (a^6) (c^22)-122 (a^4) (b^2) (c^22)-165 (a^2) (b^4) (c^22)-84 (b^6) (c^22)+20 (a^4) (c^24)+52 (a^2) (b^2) (c^24)+36 (b^4) (c^24)-7 (a^2) (c^26)-9 (b^2) (c^26)+(c^28));

n=2 (a^6) (b^6) (c^6) ((a^12)-4 (a^10) (b^2)+7 (a^8) (b^4)-8 (a^6) (b^6)+7 (a^4) (b^8)-4 (a^2) (b^10)+(b^12)-4 (a^10) (c^2)+8 (a^8) (b^2) (c^2)-4 (a^6) (b^4) (c^2)-4 (a^4) (b^6) (c^2)+8 (a^2) (b^8) (c^2)-4 (b^10) (c^2)+7 (a^8) (c^4)-4 (a^6) (b^2) (c^4)+3 (a^4) (b^4) (c^4)-4 (a^2) (b^6) (c^4)+7 (b^8) (c^4)-8 (a^6) (c^6)-4 (a^4) (b^2) (c^6)-4 (a^2) (b^4) (c^6)-8 (b^6) (c^6)+7 (a^4) (c^8)+8 (a^2) (b^2) (c^8)+7 (b^4) (c^8)-4 (a^2) (c^10)-4 (b^2) (c^10)+(c^12))((a-b+c) (a+b+c) poly2+(a+c) ((a^12)-4 (a^10) (b^2)+6 (a^8) (b^4)-4 (a^6) (b^6)+(a^4) (b^8)+(a^11) c-(a^9) (b^2) c-2 (a^7) (b^4) c+2 (a^5) (b^6) c+(a^3) (b^8) c-a (b^10) c-3 (a^10) (c^2)+7 (a^8) (b^2) (c^2)-3 (a^6) (b^4) (c^2)-3 (a^4) (b^6) (c^2)+2 (a^2) (b^8) (c^2)-3 (a^9) (c^3)+2 (a^7) (b^2) (c^3)+(a^5) (b^4) (c^3)-(a^3) (b^6) (c^3)+a (b^8) (c^3)+3 (a^8) (c^4)-3 (a^6) (b^2) (c^4)+2 (a^4) (b^4) (c^4)-3 (a^2) (b^6) (c^4)+(b^8) (c^4)+2 (a^7) (c^5)-2 (a^5) (b^2) (c^5)+(a^3) (b^4) (c^5)+2 a (b^6) (c^5)-2 (a^6) (c^6)-3 (a^4) (b^2) (c^6)-3 (a^2) (b^4) (c^6)-4 (b^6) (c^6)+2 (a^5) (c^7)+2 (a^3) (b^2) (c^7)-2 a (b^4) (c^7)+3 (a^4) (c^8)+7 (a^2) (b^2) (c^8)+6 (b^4) (c^8)-3 (a^3) (c^9)-a (b^2) (c^9)-3 (a^2) (c^10)-4 (b^2) (c^10)+a (c^11)+(c^12))poly);d=
((((a^2)-a b+(b^2)-(c^2)))^2) ((((a^2)+a b+(b^2)-(c^2)))^2) ((((a^2)-(b^2)-b c-(c^2)))^2) ((((a^2)-(b^2)+b c-(c^2)))^2) ((((a^2)-(b^2)-a c+(c^2)))^2) ((((a^2)-(b^2)+a c+(c^2)))^2) 2 (a-b-c) (a+b-c) (a-b+c) (a+b+c) ((a+c) ((a^12)-4 (a^10) (b^2)+6 (a^8) (b^4)-4 (a^6) (b^6)+(a^4) (b^8)+(a^11) c-(a^9) (b^2) c-2 (a^7) (b^4) c+2 (a^5) (b^6) c+(a^3) (b^8) c-a (b^10) c-3 (a^10) (c^2)+7 (a^8) (b^2) (c^2)-3 (a^6) (b^4) (c^2)-3 (a^4) (b^6) (c^2)+2 (a^2) (b^8) (c^2)-3 (a^9) (c^3)+2 (a^7) (b^2) (c^3)+(a^5) (b^4) (c^3)-(a^3) (b^6) (c^3)+a (b^8) (c^3)+3 (a^8) (c^4)-3 (a^6) (b^2) (c^4)+2 (a^4) (b^4) (c^4)-3 (a^2) (b^6) (c^4)+(b^8) (c^4)+2 (a^7) (c^5)-2 (a^5) (b^2) (c^5)+(a^3) (b^4) (c^5)+2 a (b^6) (c^5)-2 (a^6) (c^6)-3 (a^4) (b^2) (c^6)-3 (a^2) (b^4) (c^6)-4 (b^6) (c^6)+2 (a^5) (c^7)+2 (a^3) (b^2) (c^7)-2 a (b^4) (c^7)+3 (a^4) (c^8)+7 (a^2) (b^2) (c^8)+6 (b^4) (c^8)-3 (a^3) (c^9)-a (b^2) (c^9)-3 (a^2) (c^10)-4 (b^2) (c^10)+a (c^11)+(c^12))poly+(a-b+c) (a+b+c) poly2);

n/d

Refine[Sqrt[Factor[ToAngles[(n/d)]]],TriangleQ[]]

%/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]/.Thread[{A,B,C}->Angles[t]]

(√(((a^12)-4 (a^10) (b^2)+7 (a^8) (b^4)-8 (a^6) (b^6)+7 (a^4) (b^8)-4 (a^2) (b^10)+(b^12)-4 (a^10) (c^2)+8 (a^8) (b^2) (c^2)-4 (a^6) (b^4) (c^2)-4 (a^4) (b^6) (c^2)+8 (a^2) (b^8) (c^2)-4 (b^10) (c^2)+7 (a^8) (c^4)-4 (a^6) (b^2) (c^4)+3 (a^4) (b^4) (c^4)-4 (a^2) (b^6) (c^4)+7 (b^8) (c^4)-8 (a^6) (c^6)-4 (a^4) (b^2) (c^6)-4 (a^2) (b^4) (c^6)-8 (b^6) (c^6)+7 (a^4) (c^8)+8 (a^2) (b^2) (c^8)+7 (b^4) (c^8)-4 (a^2) (c^10)-4 (b^2) (c^10)+(c^12))/(a^2b^2c^2/Circumradius^2)))/(a b c Abs[-1+2 Cos[A]] Abs[1+2 Cos[A]] Abs[-1+2 Cos[B]] Abs[1+2 Cos[B]] Abs[-1+2 Cos[C]] Abs[1+2 Cos[C]])

ToSidelengths[%]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]/.Thread[{A,B,C}->Angles[t]]

Refine[%148,TriangleQ[]&&Circumradius>0]

(√((a^12)-4 (a^10) (b^2)+7 (a^8) (b^4)-8 (a^6) (b^6)+7 (a^4) (b^8)-4 (a^2) (b^10)+(b^12)-4 (a^10) (c^2)+8 (a^8) (b^2) (c^2)-4 (a^6) (b^4) (c^2)-4 (a^4) (b^6) (c^2)+8 (a^2) (b^8) (c^2)-4 (b^10) (c^2)+7 (a^8) (c^4)-4 (a^6) (b^2) (c^4)+3 (a^4) (b^4) (c^4)-4 (a^2) (b^6) (c^4)+7 (b^8) (c^4)-8 (a^6) (c^6)-4 (a^4) (b^2) (c^6)-4 (a^2) (b^4) (c^6)-8 (b^6) (c^6)+7 (a^4) (c^8)+8 (a^2) (b^2) (c^8)+7 (b^4) (c^8)-4 (a^2) (c^10)-4 (b^2) (c^10)+(c^12)) Circumradius)/((a^2) (b^2) (c^2) Abs[(-1+2 Cos[A])(1+2 Cos[A])(-1+2 Cos[B])(1+2 Cos[B])(-1+2 Cos[C])(1+2 Cos[C])])

(Circumradius/((a^2)(b^2)(c^2)Abs[(4Cos[A]^2-1)(4Cos[B]^2-1)(4Cos[C]^2-1)])((((a^12)-4 (a^10) (b^2)+7 (a^8) (b^4)-8 (a^6) (b^6)+7 (a^4) (b^8)-4 (a^2) (b^10)+(b^12)-4 (a^10) (c^2)+8 (a^8) (b^2) (c^2)-4 (a^6) (b^4) (c^2)-4 (a^4) (b^6) (c^2)+8 (a^2) (b^8) (c^2)-4 (b^10) (c^2)+7 (a^8) (c^4)-4 (a^6) (b^2) (c^4)+3 (a^4) (b^4) (c^4)-4 (a^2) (b^6) (c^4)+7 (b^8) (c^4)-8 (a^6) (c^6)-4 (a^4) (b^2) (c^6)-4 (a^2) (b^4) (c^6)-8 (b^6) (c^6)+7 (a^4) (c^8)+8 (a^2) (b^2) (c^8)+7 (b^4) (c^8)-4 (a^2) (c^10)-4 (b^2) (c^10)+(c^12)) )^(1/2))//ToSidelengths)/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]/.Thread[{A,B,C}->Angles[t]]

InputForm[(Circumradius/((a^2)(b^2)(c^2)Abs[(4Cos[A]^2-1)(4Cos[B]^2-1)(4Cos[C]^2-1)])((((a^12)-4 (a^10) (b^2)+7 (a^8) (b^4)-8 (a^6) (b^6)+7 (a^4) (b^8)-4 (a^2) (b^10)+(b^12)-4 (a^10) (c^2)+8 (a^8) (b^2) (c^2)-4 (a^6) (b^4) (c^2)-4 (a^4) (b^6) (c^2)+8 (a^2) (b^8) (c^2)-4 (b^10) (c^2)+7 (a^8) (c^4)-4 (a^6) (b^2) (c^4)+3 (a^4) (b^4) (c^4)-4 (a^2) (b^6) (c^4)+7 (b^8) (c^4)-8 (a^6) (c^6)-4 (a^4) (b^2) (c^6)-4 (a^2) (b^4) (c^6)-8 (b^6) (c^6)+7 (a^4) (c^8)+8 (a^2) (b^2) (c^8)+7 (b^4) (c^8)-4 (a^2) (c^10)-4 (b^2) (c^10)+(c^12)) )^(1/2)))]

CircleRadius[YiuCircle]//TraditionalForm