Lucas Central Circle

Author

Eric W. Weisstein
January 22, 2005

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

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

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

Figure

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Circle Center

Center[LucasCentralCircle][[1]]//TraditionalForm

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

Circle Function

CircleFunction[LucasCentralCircle]//TraditionalForm

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[LucasCentralCircle]]]//Timing

Radius

CircleRadius[LucasCentralCircle]//TraditionalForm

Incident Centers

(ctrs=IncidentCenters[KimberlingCenters,LucasCentralCircle])//Timing

Derivation

v=TrilinearVertexMatrix[LucasCentralTriangle]

eqn=Factor[TrilinearEquation@@TrilinearVertexMatrix[LucasCentralTriangle]]

eqn=(-8 a (b^3) (c^3) (S^3) SB (α^2)-4 (a^3) (b^3) c (S^2) SA SB (α^2)-4 a (b^3) (c^3) (S^2) SA SB (α^2)-2 (a^3) (b^3) c S (SA^2) SB (α^2)-4 a (b^5) c (S^2) (SB^2) (α^2)-8 a (b^3) (c^3) (S^2) (SB^2) (α^2)-4 (a^3) (b^3) c S SA (SB^2) (α^2)-2 a (b^5) c S SA (SB^2) (α^2)-2 a (b^3) (c^3) S SA (SB^2) (α^2)-(a^3) (b^3) c (SA^2) (SB^2) (α^2)-4 a (b^5) c S (SB^3) (α^2)-2 a (b^3) (c^3) S (SB^3) (α^2)-(a^3) (b^3) c SA (SB^3) (α^2)-a (b^5) c SA (SB^3) (α^2)-a (b^5) c (SB^4) (α^2)-8 a (b^3) (c^3) (S^3) SC (α^2)-4 (a^3) b (c^3) (S^2) SA SC (α^2)-4 a (b^3) (c^3) (S^2) SA SC (α^2)-2 (a^3) b (c^3) S (SA^2) SC (α^2)+4 (a^3) (b^3) c (S^2) SB SC (α^2)-4 a (b^5) c (S^2) SB SC (α^2)+4 (a^3) b (c^3) (S^2) SB SC (α^2)-20 a (b^3) (c^3) (S^2) SB SC (α^2)-4 a b (c^5) (S^2) SB SC (α^2)+4 (a^5) b c S SA SB SC (α^2)-4 (a^3) (b^3) c S SA SB SC (α^2)-4 (a^3) b (c^3) S SA SB SC (α^2)-8 a (b^3) (c^3) S SA SB SC (α^2)+(a^5) b c (SA^2) SB SC (α^2)-(a^3) (b^3) c (SA^2) SB SC (α^2)-(a^3) b (c^3) (SA^2) SB SC (α^2)+4 (a^3) (b^3) c S (SB^2) SC (α^2)-6 a (b^5) c S (SB^2) SC (α^2)+2 (a^3) b (c^3) S (SB^2) SC (α^2)-12 a (b^3) (c^3) S (SB^2) SC (α^2)-2 a b (c^5) S (SB^2) SC (α^2)+(a^5) b c SA (SB^2) SC (α^2)-(a^3) (b^3) c SA (SB^2) SC (α^2)-a (b^5) c SA (SB^2) SC (α^2)-(a^3) b (c^3) SA (SB^2) SC (α^2)-2 a (b^3) (c^3) SA (SB^2) SC (α^2)+(a^3) (b^3) c (SB^3) SC (α^2)-2 a (b^5) c (SB^3) SC (α^2)-2 a (b^3) (c^3) (SB^3) SC (α^2)-8 a (b^3) (c^3) (S^2) (SC^2) (α^2)-4 a b (c^5) (S^2) (SC^2) (α^2)-4 (a^3) b (c^3) S SA (SC^2) (α^2)-2 a (b^3) (c^3) S SA (SC^2) (α^2)-2 a b (c^5) S SA (SC^2) (α^2)-(a^3) b (c^3) (SA^2) (SC^2) (α^2)+2 (a^3) (b^3) c S SB (SC^2) (α^2)-2 a (b^5) c S SB (SC^2) (α^2)+4 (a^3) b (c^3) S SB (SC^2) (α^2)-12 a (b^3) (c^3) S SB (SC^2) (α^2)-6 a b (c^5) S SB (SC^2) (α^2)+(a^5) b c SA SB (SC^2) (α^2)-(a^3) (b^3) c SA SB (SC^2) (α^2)-(a^3) b (c^3) SA SB (SC^2) (α^2)-2 a (b^3) (c^3) SA SB (SC^2) (α^2)-a b (c^5) SA SB (SC^2) (α^2)+(a^3) (b^3) c (SB^2) (SC^2) (α^2)-a (b^5) c (SB^2) (SC^2) (α^2)+(a^3) b (c^3) (SB^2) (SC^2) (α^2)-4 a (b^3) (c^3) (SB^2) (SC^2) (α^2)-a b (c^5) (SB^2) (SC^2) (α^2)-2 a (b^3) (c^3) S (SC^3) (α^2)-4 a b (c^5) S (SC^3) (α^2)-(a^3) b (c^3) SA (SC^3) (α^2)-a b (c^5) SA (SC^3) (α^2)+(a^3) b (c^3) SB (SC^3) (α^2)-2 a (b^3) (c^3) SB (SC^3) (α^2)-2 a b (c^5) SB (SC^3) (α^2)-a b (c^5) (SC^4) (α^2)+16 (a^2) (b^2) (c^3) (S^4) α β+8 (a^4) (b^2) c (S^3) SA α β+16 (a^2) (b^2) (c^3) (S^3) SA α β+8 (a^4) (b^2) c (S^2) (SA^2) α β+4 (a^2) (b^2) (c^3) (S^2) (SA^2) α β+2 (a^4) (b^2) c S (SA^3) α β+8 (a^2) (b^4) c (S^3) SB α β+16 (a^2) (b^2) (c^3) (S^3) SB α β+8 (a^4) (b^2) c (S^2) SA SB α β+8 (a^2) (b^4) c (S^2) SA SB α β+16 (a^2) (b^2) (c^3) (S^2) SA SB α β+8 (a^4) (b^2) c S (SA^2) SB α β+2 (a^2) (b^4) c S (SA^2) SB α β+4 (a^2) (b^2) (c^3) S (SA^2) SB α β+2 (a^4) (b^2) c (SA^3) SB α β+8 (a^2) (b^4) c (S^2) (SB^2) α β+4 (a^2) (b^2) (c^3) (S^2) (SB^2) α β+2 (a^4) (b^2) c S SA (SB^2) α β+8 (a^2) (b^4) c S SA (SB^2) α β+4 (a^2) (b^2) (c^3) S SA (SB^2) α β+2 (a^4) (b^2) c (SA^2) (SB^2) α β+2 (a^2) (b^4) c (SA^2) (SB^2) α β+2 (a^2) (b^4) c S (SB^3) α β+2 (a^2) (b^4) c SA (SB^3) α β-8 (a^4) (c^3) (S^3) SC α β+16 (a^2) (b^2) (c^3) (S^3) SC α β-8 (b^4) (c^3) (S^3) SC α β+8 (a^2) (c^5) (S^3) SC α β+8 (b^2) (c^5) (S^3) SC α β-4 (a^6) c (S^2) SA SC α β+8 (a^4) (b^2) c (S^2) SA SC α β-8 (a^4) (c^3) (S^2) SA SC α β+16 (a^2) (b^2) (c^3) (S^2) SA SC α β-4 (b^4) (c^3) (S^2) SA SC α β+8 (a^2) (c^5) (S^2) SA SC α β+4 (b^2) (c^5) (S^2) SA SC α β-4 (a^6) c S (SA^2) SC α β+6 (a^4) (b^2) c S (SA^2) SC α β+4 (a^2) (b^2) (c^3) S (SA^2) SC α β+2 (a^2) (c^5) S (SA^2) SC α β-(a^6) c (SA^3) SC α β+(a^4) (b^2) c (SA^3) SC α β+(a^4) (c^3) (SA^3) SC α β+8 (a^2) (b^4) c (S^2) SB SC α β-4 (b^6) c (S^2) SB SC α β-4 (a^4) (c^3) (S^2) SB SC α β+16 (a^2) (b^2) (c^3) (S^2) SB SC α β-8 (b^4) (c^3) (S^2) SB SC α β+4 (a^2) (c^5) (S^2) SB SC α β+8 (b^2) (c^5) (S^2) SB SC α β+4 (a^4) (b^2) c S SA SB SC α β+4 (a^2) (b^4) c S SA SB SC α β-4 (a^4) (c^3) S SA SB SC α β+16 (a^2) (b^2) (c^3) S SA SB SC α β-4 (b^4) (c^3) S SA SB SC α β+4 (a^2) (c^5) S SA SB SC α β+4 (b^2) (c^5) S SA SB SC α β-(a^6) c (SA^2) SB SC α β+3 (a^4) (b^2) c (SA^2) SB SC α β+(a^4) (c^3) (SA^2) SB SC α β+4 (a^2) (b^2) (c^3) (SA^2) SB SC α β+6 (a^2) (b^4) c S (SB^2) SC α β-4 (b^6) c S (SB^2) SC α β+4 (a^2) (b^2) (c^3) S (SB^2) SC α β+2 (b^2) (c^5) S (SB^2) SC α β+3 (a^2) (b^4) c SA (SB^2) SC α β-(b^6) c SA (SB^2) SC α β+4 (a^2) (b^2) (c^3) SA (SB^2) SC α β+(b^4) (c^3) SA (SB^2) SC α β+(a^2) (b^4) c (SB^3) SC α β-(b^6) c (SB^3) SC α β+(b^4) (c^3) (SB^3) SC α β-8 (a^4) (c^3) (S^2) (SC^2) α β+4 (a^2) (b^2) (c^3) (S^2) (SC^2) α β-8 (b^4) (c^3) (S^2) (SC^2) α β+4 (a^2) (c^5) (S^2) (SC^2) α β+4 (b^2) (c^5) (S^2) (SC^2) α β+4 (c^7) (S^2) (SC^2) α β-2 (a^6) c S SA (SC^2) α β+2 (a^4) (b^2) c S SA (SC^2) α β-8 (a^4) (c^3) S SA (SC^2) α β+4 (a^2) (b^2) (c^3) S SA (SC^2) α β-2 (b^4) (c^3) S SA (SC^2) α β+6 (a^2) (c^5) S SA (SC^2) α β+2 (c^7) S SA (SC^2) α β-(a^6) c (SA^2) (SC^2) α β+(a^4) (b^2) c (SA^2) (SC^2) α β-(a^4) (c^3) (SA^2) (SC^2) α β+2 (a^2) (c^5) (SA^2) (SC^2) α β+2 (a^2) (b^4) c S SB (SC^2) α β-2 (b^6) c S SB (SC^2) α β-2 (a^4) (c^3) S SB (SC^2) α β+4 (a^2) (b^2) (c^3) S SB (SC^2) α β-8 (b^4) (c^3) S SB (SC^2) α β+6 (b^2) (c^5) S SB (SC^2) α β+2 (c^7) S SB (SC^2) α β-2 (a^4) (c^3) SA SB (SC^2) α β+4 (a^2) (b^2) (c^3) SA SB (SC^2) α β-2 (b^4) (c^3) SA SB (SC^2) α β+2 (a^2) (c^5) SA SB (SC^2) α β+2 (b^2) (c^5) SA SB (SC^2) α β+(a^2) (b^4) c (SB^2) (SC^2) α β-(b^6) c (SB^2) (SC^2) α β-(b^4) (c^3) (SB^2) (SC^2) α β+2 (b^2) (c^5) (SB^2) (SC^2) α β-2 (a^4) (c^3) S (SC^3) α β-2 (b^4) (c^3) S (SC^3) α β-2 (a^2) (c^5) S (SC^3) α β-2 (b^2) (c^5) S (SC^3) α β+4 (c^7) S (SC^3) α β-2 (a^4) (c^3) SA (SC^3) α β+(a^2) (c^5) SA (SC^3) α β-(b^2) (c^5) SA (SC^3) α β+(c^7) SA (SC^3) α β-2 (b^4) (c^3) SB (SC^3) α β-(a^2) (c^5) SB (SC^3) α β+(b^2) (c^5) SB (SC^3) α β+(c^7) SB (SC^3) α β-(a^2) (c^5) (SC^4) α β-(b^2) (c^5) (SC^4) α β+(c^7) (SC^4) α β-8 (a^3) b (c^3) (S^3) SA (β^2)-4 (a^5) b c (S^2) (SA^2) (β^2)-8 (a^3) b (c^3) (S^2) (SA^2) (β^2)-4 (a^5) b c S (SA^3) (β^2)-2 (a^3) b (c^3) S (SA^3) (β^2)-(a^5) b c (SA^4) (β^2)-4 (a^3) (b^3) c (S^2) SA SB (β^2)-4 (a^3) b (c^3) (S^2) SA SB (β^2)-2 (a^5) b c S (SA^2) SB (β^2)-4 (a^3) (b^3) c S (SA^2) SB (β^2)-2 (a^3) b (c^3) S (SA^2) SB (β^2)-(a^5) b c (SA^3) SB (β^2)-(a^3) (b^3) c (SA^3) SB (β^2)-2 (a^3) (b^3) c S SA (SB^2) (β^2)-(a^3) (b^3) c (SA^2) (SB^2) (β^2)-8 (a^3) b (c^3) (S^3) SC (β^2)-4 (a^5) b c (S^2) SA SC (β^2)+4 (a^3) (b^3) c (S^2) SA SC (β^2)-20 (a^3) b (c^3) (S^2) SA SC (β^2)+4 a (b^3) (c^3) (S^2) SA SC (β^2)-4 a b (c^5) (S^2) SA SC (β^2)-6 (a^5) b c S (SA^2) SC (β^2)+4 (a^3) (b^3) c S (SA^2) SC (β^2)-12 (a^3) b (c^3) S (SA^2) SC (β^2)+2 a (b^3) (c^3) S (SA^2) SC (β^2)-2 a b (c^5) S (SA^2) SC (β^2)-2 (a^5) b c (SA^3) SC (β^2)+(a^3) (b^3) c (SA^3) SC (β^2)-2 (a^3) b (c^3) (SA^3) SC (β^2)-4 (a^3) b (c^3) (S^2) SB SC (β^2)-4 a (b^3) (c^3) (S^2) SB SC (β^2)-4 (a^3) (b^3) c S SA SB SC (β^2)+4 a (b^5) c S SA SB SC (β^2)-8 (a^3) b (c^3) S SA SB SC (β^2)-4 a (b^3) (c^3) S SA SB SC (β^2)-(a^5) b c (SA^2) SB SC (β^2)-(a^3) (b^3) c (SA^2) SB SC (β^2)+a (b^5) c (SA^2) SB SC (β^2)-2 (a^3) b (c^3) (SA^2) SB SC (β^2)-a (b^3) (c^3) (SA^2) SB SC (β^2)-2 a (b^3) (c^3) S (SB^2) SC (β^2)-(a^3) (b^3) c SA (SB^2) SC (β^2)+a (b^5) c SA (SB^2) SC (β^2)-a (b^3) (c^3) SA (SB^2) SC (β^2)-8 (a^3) b (c^3) (S^2) (SC^2) (β^2)-4 a b (c^5) (S^2) (SC^2) (β^2)-2 (a^5) b c S SA (SC^2) (β^2)+2 (a^3) (b^3) c S SA (SC^2) (β^2)-12 (a^3) b (c^3) S SA (SC^2) (β^2)+4 a (b^3) (c^3) S SA (SC^2) (β^2)-6 a b (c^5) S SA (SC^2) (β^2)-(a^5) b c (SA^2) (SC^2) (β^2)+(a^3) (b^3) c (SA^2) (SC^2) (β^2)-4 (a^3) b (c^3) (SA^2) (SC^2) (β^2)+a (b^3) (c^3) (SA^2) (SC^2) (β^2)-a b (c^5) (SA^2) (SC^2) (β^2)-2 (a^3) b (c^3) S SB (SC^2) (β^2)-4 a (b^3) (c^3) S SB (SC^2) (β^2)-2 a b (c^5) S SB (SC^2) (β^2)-(a^3) (b^3) c SA SB (SC^2) (β^2)+a (b^5) c SA SB (SC^2) (β^2)-2 (a^3) b (c^3) SA SB (SC^2) (β^2)-a (b^3) (c^3) SA SB (SC^2) (β^2)-a b (c^5) SA SB (SC^2) (β^2)-a (b^3) (c^3) (SB^2) (SC^2) (β^2)-2 (a^3) b (c^3) S (SC^3) (β^2)-4 a b (c^5) S (SC^3) (β^2)-2 (a^3) b (c^3) SA (SC^3) (β^2)+a (b^3) (c^3) SA (SC^3) (β^2)-2 a b (c^5) SA (SC^3) (β^2)-a (b^3) (c^3) SB (SC^3) (β^2)-a b (c^5) SB (SC^3) (β^2)-a b (c^5) (SC^4) (β^2)+16 (a^2) (b^3) (c^2) (S^4) α γ+8 (a^4) b (c^2) (S^3) SA α γ+16 (a^2) (b^3) (c^2) (S^3) SA α γ+8 (a^4) b (c^2) (S^2) (SA^2) α γ+4 (a^2) (b^3) (c^2) (S^2) (SA^2) α γ+2 (a^4) b (c^2) S (SA^3) α γ-8 (a^4) (b^3) (S^3) SB α γ+8 (a^2) (b^5) (S^3) SB α γ+16 (a^2) (b^3) (c^2) (S^3) SB α γ+8 (b^5) (c^2) (S^3) SB α γ-8 (b^3) (c^4) (S^3) SB α γ-4 (a^6) b (S^2) SA SB α γ-8 (a^4) (b^3) (S^2) SA SB α γ+8 (a^2) (b^5) (S^2) SA SB α γ+8 (a^4) b (c^2) (S^2) SA SB α γ+16 (a^2) (b^3) (c^2) (S^2) SA SB α γ+4 (b^5) (c^2) (S^2) SA SB α γ-4 (b^3) (c^4) (S^2) SA SB α γ-4 (a^6) b S (SA^2) SB α γ+2 (a^2) (b^5) S (SA^2) SB α γ+6 (a^4) b (c^2) S (SA^2) SB α γ+4 (a^2) (b^3) (c^2) S (SA^2) SB α γ-(a^6) b (SA^3) SB α γ+(a^4) (b^3) (SA^3) SB α γ+(a^4) b (c^2) (SA^3) SB α γ-8 (a^4) (b^3) (S^2) (SB^2) α γ+4 (a^2) (b^5) (S^2) (SB^2) α γ+4 (b^7) (S^2) (SB^2) α γ+4 (a^2) (b^3) (c^2) (S^2) (SB^2) α γ+4 (b^5) (c^2) (S^2) (SB^2) α γ-8 (b^3) (c^4) (S^2) (SB^2) α γ-2 (a^6) b S SA (SB^2) α γ-8 (a^4) (b^3) S SA (SB^2) α γ+6 (a^2) (b^5) S SA (SB^2) α γ+2 (b^7) S SA (SB^2) α γ+2 (a^4) b (c^2) S SA (SB^2) α γ+4 (a^2) (b^3) (c^2) S SA (SB^2) α γ-2 (b^3) (c^4) S SA (SB^2) α γ-(a^6) b (SA^2) (SB^2) α γ-(a^4) (b^3) (SA^2) (SB^2) α γ+2 (a^2) (b^5) (SA^2) (SB^2) α γ+(a^4) b (c^2) (SA^2) (SB^2) α γ-2 (a^4) (b^3) S (SB^3) α γ-2 (a^2) (b^5) S (SB^3) α γ+4 (b^7) S (SB^3) α γ-2 (b^5) (c^2) S (SB^3) α γ-2 (b^3) (c^4) S (SB^3) α γ-2 (a^4) (b^3) SA (SB^3) α γ+(a^2) (b^5) SA (SB^3) α γ+(b^7) SA (SB^3) α γ-(b^5) (c^2) SA (SB^3) α γ-(a^2) (b^5) (SB^4) α γ+(b^7) (SB^4) α γ-(b^5) (c^2) (SB^4) α γ+16 (a^2) (b^3) (c^2) (S^3) SC α γ+8 (a^2) b (c^4) (S^3) SC α γ+8 (a^4) b (c^2) (S^2) SA SC α γ+16 (a^2) (b^3) (c^2) (S^2) SA SC α γ+8 (a^2) b (c^4) (S^2) SA SC α γ+8 (a^4) b (c^2) S (SA^2) SC α γ+4 (a^2) (b^3) (c^2) S (SA^2) SC α γ+2 (a^2) b (c^4) S (SA^2) SC α γ+2 (a^4) b (c^2) (SA^3) SC α γ-4 (a^4) (b^3) (S^2) SB SC α γ+4 (a^2) (b^5) (S^2) SB SC α γ+16 (a^2) (b^3) (c^2) (S^2) SB SC α γ+8 (b^5) (c^2) (S^2) SB SC α γ+8 (a^2) b (c^4) (S^2) SB SC α γ-8 (b^3) (c^4) (S^2) SB SC α γ-4 b (c^6) (S^2) SB SC α γ-4 (a^4) (b^3) S SA SB SC α γ+4 (a^2) (b^5) S SA SB SC α γ+4 (a^4) b (c^2) S SA SB SC α γ+16 (a^2) (b^3) (c^2) S SA SB SC α γ+4 (b^5) (c^2) S SA SB SC α γ+4 (a^2) b (c^4) S SA SB SC α γ-4 (b^3) (c^4) S SA SB SC α γ-(a^6) b (SA^2) SB SC α γ+(a^4) (b^3) (SA^2) SB SC α γ+3 (a^4) b (c^2) (SA^2) SB SC α γ+4 (a^2) (b^3) (c^2) (SA^2) SB SC α γ-2 (a^4) (b^3) S (SB^2) SC α γ+2 (b^7) S (SB^2) SC α γ+4 (a^2) (b^3) (c^2) S (SB^2) SC α γ+6 (b^5) (c^2) S (SB^2) SC α γ+2 (a^2) b (c^4) S (SB^2) SC α γ-8 (b^3) (c^4) S (SB^2) SC α γ-2 b (c^6) S (SB^2) SC α γ-2 (a^4) (b^3) SA (SB^2) SC α γ+2 (a^2) (b^5) SA (SB^2) SC α γ+4 (a^2) (b^3) (c^2) SA (SB^2) SC α γ+2 (b^5) (c^2) SA (SB^2) SC α γ-2 (b^3) (c^4) SA (SB^2) SC α γ-(a^2) (b^5) (SB^3) SC α γ+(b^7) (SB^3) SC α γ+(b^5) (c^2) (SB^3) SC α γ-2 (b^3) (c^4) (SB^3) SC α γ+4 (a^2) (b^3) (c^2) (S^2) (SC^2) α γ+8 (a^2) b (c^4) (S^2) (SC^2) α γ+2 (a^4) b (c^2) S SA (SC^2) α γ+4 (a^2) (b^3) (c^2) S SA (SC^2) α γ+8 (a^2) b (c^4) S SA (SC^2) α γ+2 (a^4) b (c^2) (SA^2) (SC^2) α γ+2 (a^2) b (c^4) (SA^2) (SC^2) α γ+4 (a^2) (b^3) (c^2) S SB (SC^2) α γ+2 (b^5) (c^2) S SB (SC^2) α γ+6 (a^2) b (c^4) S SB (SC^2) α γ-4 b (c^6) S SB (SC^2) α γ+4 (a^2) (b^3) (c^2) SA SB (SC^2) α γ+3 (a^2) b (c^4) SA SB (SC^2) α γ+(b^3) (c^4) SA SB (SC^2) α γ-b (c^6) SA SB (SC^2) α γ+2 (b^5) (c^2) (SB^2) (SC^2) α γ+(a^2) b (c^4) (SB^2) (SC^2) α γ-(b^3) (c^4) (SB^2) (SC^2) α γ-b (c^6) (SB^2) (SC^2) α γ+2 (a^2) b (c^4) S (SC^3) α γ+2 (a^2) b (c^4) SA (SC^3) α γ+(a^2) b (c^4) SB (SC^3) α γ+(b^3) (c^4) SB (SC^3) α γ-b (c^6) SB (SC^3) α γ+16 (a^3) (b^2) (c^2) (S^4) β γ+8 (a^5) (b^2) (S^3) SA β γ-8 (a^3) (b^4) (S^3) SA β γ+8 (a^5) (c^2) (S^3) SA β γ+16 (a^3) (b^2) (c^2) (S^3) SA β γ-8 (a^3) (c^4) (S^3) SA β γ+4 (a^7) (S^2) (SA^2) β γ+4 (a^5) (b^2) (S^2) (SA^2) β γ-8 (a^3) (b^4) (S^2) (SA^2) β γ+4 (a^5) (c^2) (S^2) (SA^2) β γ+4 (a^3) (b^2) (c^2) (S^2) (SA^2) β γ-8 (a^3) (c^4) (S^2) (SA^2) β γ+4 (a^7) S (SA^3) β γ-2 (a^5) (b^2) S (SA^3) β γ-2 (a^3) (b^4) S (SA^3) β γ-2 (a^5) (c^2) S (SA^3) β γ-2 (a^3) (c^4) S (SA^3) β γ+(a^7) (SA^4) β γ-(a^5) (b^2) (SA^4) β γ-(a^5) (c^2) (SA^4) β γ+16 (a^3) (b^2) (c^2) (S^3) SB β γ+8 a (b^4) (c^2) (S^3) SB β γ+8 (a^5) (b^2) (S^2) SA SB β γ-8 (a^3) (b^4) (S^2) SA SB β γ-4 a (b^6) (S^2) SA SB β γ+4 (a^5) (c^2) (S^2) SA SB β γ+16 (a^3) (b^2) (c^2) (S^2) SA SB β γ+8 a (b^4) (c^2) (S^2) SA SB β γ-4 (a^3) (c^4) (S^2) SA SB β γ+2 (a^7) S (SA^2) SB β γ+6 (a^5) (b^2) S (SA^2) SB β γ-8 (a^3) (b^4) S (SA^2) SB β γ-2 a (b^6) S (SA^2) SB β γ+4 (a^3) (b^2) (c^2) S (SA^2) SB β γ+2 a (b^4) (c^2) S (SA^2) SB β γ-2 (a^3) (c^4) S (SA^2) SB β γ+(a^7) (SA^3) SB β γ+(a^5) (b^2) (SA^3) SB β γ-2 (a^3) (b^4) (SA^3) SB β γ-(a^5) (c^2) (SA^3) SB β γ+4 (a^3) (b^2) (c^2) (S^2) (SB^2) β γ+8 a (b^4) (c^2) (S^2) (SB^2) β γ+2 (a^5) (b^2) S SA (SB^2) β γ-4 a (b^6) S SA (SB^2) β γ+4 (a^3) (b^2) (c^2) S SA (SB^2) β γ+6 a (b^4) (c^2) S SA (SB^2) β γ+2 (a^5) (b^2) (SA^2) (SB^2) β γ-(a^3) (b^4) (SA^2) (SB^2) β γ-a (b^6) (SA^2) (SB^2) β γ+a (b^4) (c^2) (SA^2) (SB^2) β γ+2 a (b^4) (c^2) S (SB^3) β γ+(a^3) (b^4) SA (SB^3) β γ-a (b^6) SA (SB^3) β γ+a (b^4) (c^2) SA (SB^3) β γ+16 (a^3) (b^2) (c^2) (S^3) SC β γ+8 a (b^2) (c^4) (S^3) SC β γ+4 (a^5) (b^2) (S^2) SA SC β γ-4 (a^3) (b^4) (S^2) SA SC β γ+8 (a^5) (c^2) (S^2) SA SC β γ+16 (a^3) (b^2) (c^2) (S^2) SA SC β γ-8 (a^3) (c^4) (S^2) SA SC β γ+8 a (b^2) (c^4) (S^2) SA SC β γ-4 a (c^6) (S^2) SA SC β γ+2 (a^7) S (SA^2) SC β γ-2 (a^3) (b^4) S (SA^2) SC β γ+6 (a^5) (c^2) S (SA^2) SC β γ+4 (a^3) (b^2) (c^2) S (SA^2) SC β γ-8 (a^3) (c^4) S (SA^2) SC β γ+2 a (b^2) (c^4) S (SA^2) SC β γ-2 a (c^6) S (SA^2) SC β γ+(a^7) (SA^3) SC β γ-(a^5) (b^2) (SA^3) SC β γ+(a^5) (c^2) (SA^3) SC β γ-2 (a^3) (c^4) (SA^3) SC β γ+16 (a^3) (b^2) (c^2) (S^2) SB SC β γ+8 a (b^4) (c^2) (S^2) SB SC β γ+8 a (b^2) (c^4) (S^2) SB SC β γ+4 (a^5) (b^2) S SA SB SC β γ-4 (a^3) (b^4) S SA SB SC β γ+4 (a^5) (c^2) S SA SB SC β γ+16 (a^3) (b^2) (c^2) S SA SB SC β γ+4 a (b^4) (c^2) S SA SB SC β γ-4 (a^3) (c^4) S SA SB SC β γ+4 a (b^2) (c^4) S SA SB SC β γ+2 (a^5) (b^2) (SA^2) SB SC β γ-2 (a^3) (b^4) (SA^2) SB SC β γ+2 (a^5) (c^2) (SA^2) SB SC β γ+4 (a^3) (b^2) (c^2) (SA^2) SB SC β γ-2 (a^3) (c^4) (SA^2) SB SC β γ+4 (a^3) (b^2) (c^2) S (SB^2) SC β γ+8 a (b^4) (c^2) S (SB^2) SC β γ+2 a (b^2) (c^4) S (SB^2) SC β γ+(a^3) (b^4) SA (SB^2) SC β γ-a (b^6) SA (SB^2) SC β γ+4 (a^3) (b^2) (c^2) SA (SB^2) SC β γ+3 a (b^4) (c^2) SA (SB^2) SC β γ+2 a (b^4) (c^2) (SB^3) SC β γ+4 (a^3) (b^2) (c^2) (S^2) (SC^2) β γ+8 a (b^2) (c^4) (S^2) (SC^2) β γ+2 (a^5) (c^2) S SA (SC^2) β γ+4 (a^3) (b^2) (c^2) S SA (SC^2) β γ+6 a (b^2) (c^4) S SA (SC^2) β γ-4 a (c^6) S SA (SC^2) β γ+2 (a^5) (c^2) (SA^2) (SC^2) β γ-(a^3) (c^4) (SA^2) (SC^2) β γ+a (b^2) (c^4) (SA^2) (SC^2) β γ-a (c^6) (SA^2) (SC^2) β γ+4 (a^3) (b^2) (c^2) S SB (SC^2) β γ+2 a (b^4) (c^2) S SB (SC^2) β γ+8 a (b^2) (c^4) S SB (SC^2) β γ+4 (a^3) (b^2) (c^2) SA SB (SC^2) β γ+(a^3) (c^4) SA SB (SC^2) β γ+3 a (b^2) (c^4) SA SB (SC^2) β γ-a (c^6) SA SB (SC^2) β γ+2 a (b^4) (c^2) (SB^2) (SC^2) β γ+2 a (b^2) (c^4) (SB^2) (SC^2) β γ+2 a (b^2) (c^4) S (SC^3) β γ+(a^3) (c^4) SA (SC^3) β γ+a (b^2) (c^4) SA (SC^3) β γ-a (c^6) SA (SC^3) β γ+2 a (b^2) (c^4) SB (SC^3) β γ-8 (a^3) (b^3) c (S^3) SA (γ^2)-4 (a^5) b c (S^2) (SA^2) (γ^2)-8 (a^3) (b^3) c (S^2) (SA^2) (γ^2)-4 (a^5) b c S (SA^3) (γ^2)-2 (a^3) (b^3) c S (SA^3) (γ^2)-(a^5) b c (SA^4) (γ^2)-8 (a^3) (b^3) c (S^3) SB (γ^2)-4 (a^5) b c (S^2) SA SB (γ^2)-20 (a^3) (b^3) c (S^2) SA SB (γ^2)-4 a (b^5) c (S^2) SA SB (γ^2)+4 (a^3) b (c^3) (S^2) SA SB (γ^2)+4 a (b^3) (c^3) (S^2) SA SB (γ^2)-6 (a^5) b c S (SA^2) SB (γ^2)-12 (a^3) (b^3) c S (SA^2) SB (γ^2)-2 a (b^5) c S (SA^2) SB (γ^2)+4 (a^3) b (c^3) S (SA^2) SB (γ^2)+2 a (b^3) (c^3) S (SA^2) SB (γ^2)-2 (a^5) b c (SA^3) SB (γ^2)-2 (a^3) (b^3) c (SA^3) SB (γ^2)+(a^3) b (c^3) (SA^3) SB (γ^2)-8 (a^3) (b^3) c (S^2) (SB^2) (γ^2)-4 a (b^5) c (S^2) (SB^2) (γ^2)-2 (a^5) b c S SA (SB^2) (γ^2)-12 (a^3) (b^3) c S SA (SB^2) (γ^2)-6 a (b^5) c S SA (SB^2) (γ^2)+2 (a^3) b (c^3) S SA (SB^2) (γ^2)+4 a (b^3) (c^3) S SA (SB^2) (γ^2)-(a^5) b c (SA^2) (SB^2) (γ^2)-4 (a^3) (b^3) c (SA^2) (SB^2) (γ^2)-a (b^5) c (SA^2) (SB^2) (γ^2)+(a^3) b (c^3) (SA^2) (SB^2) (γ^2)+a (b^3) (c^3) (SA^2) (SB^2) (γ^2)-2 (a^3) (b^3) c S (SB^3) (γ^2)-4 a (b^5) c S (SB^3) (γ^2)-2 (a^3) (b^3) c SA (SB^3) (γ^2)-2 a (b^5) c SA (SB^3) (γ^2)+a (b^3) (c^3) SA (SB^3) (γ^2)-a (b^5) c (SB^4) (γ^2)-4 (a^3) (b^3) c (S^2) SA SC (γ^2)-4 (a^3) b (c^3) (S^2) SA SC (γ^2)-2 (a^5) b c S (SA^2) SC (γ^2)-2 (a^3) (b^3) c S (SA^2) SC (γ^2)-4 (a^3) b (c^3) S (SA^2) SC (γ^2)-(a^5) b c (SA^3) SC (γ^2)-(a^3) b (c^3) (SA^3) SC (γ^2)-4 (a^3) (b^3) c (S^2) SB SC (γ^2)-4 a (b^3) (c^3) (S^2) SB SC (γ^2)-8 (a^3) (b^3) c S SA SB SC (γ^2)-4 (a^3) b (c^3) S SA SB SC (γ^2)-4 a (b^3) (c^3) S SA SB SC (γ^2)+4 a b (c^5) S SA SB SC (γ^2)-(a^5) b c (SA^2) SB SC (γ^2)-2 (a^3) (b^3) c (SA^2) SB SC (γ^2)-(a^3) b (c^3) (SA^2) SB SC (γ^2)-a (b^3) (c^3) (SA^2) SB SC (γ^2)+a b (c^5) (SA^2) SB SC (γ^2)-2 (a^3) (b^3) c S (SB^2) SC (γ^2)-2 a (b^5) c S (SB^2) SC (γ^2)-4 a (b^3) (c^3) S (SB^2) SC (γ^2)-2 (a^3) (b^3) c SA (SB^2) SC (γ^2)-a (b^5) c SA (SB^2) SC (γ^2)-(a^3) b (c^3) SA (SB^2) SC (γ^2)-a (b^3) (c^3) SA (SB^2) SC (γ^2)+a b (c^5) SA (SB^2) SC (γ^2)-a (b^5) c (SB^3) SC (γ^2)-a (b^3) (c^3) (SB^3) SC (γ^2)-2 (a^3) b (c^3) S SA (SC^2) (γ^2)-(a^3) b (c^3) (SA^2) (SC^2) (γ^2)-2 a (b^3) (c^3) S SB (SC^2) (γ^2)-(a^3) b (c^3) SA SB (SC^2) (γ^2)-a (b^3) (c^3) SA SB (SC^2) (γ^2)+a b (c^5) SA SB (SC^2) (γ^2)-a (b^3) (c^3) (SB^2) (SC^2) (γ^2));

Factor[Factor[ToSidelengths[eqn]]/.x_^(3/2):>x Sqrt[x]]

eqn2=((a^9) b c (α^2)+7 (a^7) (b^3) c (α^2)-17 (a^5) (b^5) c (α^2)+9 (a^3) (b^7) c (α^2)+7 (a^7) b (c^3) (α^2)-34 (a^5) (b^3) (c^3) (α^2)-17 (a^3) (b^5) (c^3) (α^2)+4 a (b^7) (c^3) (α^2)-17 (a^5) b (c^5) (α^2)-17 (a^3) (b^3) (c^5) (α^2)-8 a (b^5) (c^5) (α^2)+9 (a^3) b (c^7) (α^2)+4 a (b^3) (c^7) (α^2)+4 (a^7) b c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)-8 (a^5) (b^3) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)-2 (a^3) (b^5) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)+2 a (b^7) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)-8 (a^5) b (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)-24 (a^3) (b^3) (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)-2 a (b^5) (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)-2 (a^3) b (c^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)-2 a (b^3) (c^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)+2 a b (c^7) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)+5 (a^10) c α β-9 (a^8) (b^2) c α β+4 (a^6) (b^4) c α β+4 (a^4) (b^6) c α β-9 (a^2) (b^8) c α β+5 (b^10) c α β-15 (a^8) (c^3) α β-42 (a^6) (b^2) (c^3) α β+66 (a^4) (b^4) (c^3) α β-42 (a^2) (b^6) (c^3) α β-15 (b^8) (c^3) α β+6 (a^6) (c^5) α β+100 (a^4) (b^2) (c^5) α β+100 (a^2) (b^4) (c^5) α β+6 (b^6) (c^5) α β+18 (a^4) (c^7) α β-14 (a^2) (b^2) (c^7) α β+18 (b^4) (c^7) α β-19 (a^2) (c^9) α β-19 (b^2) (c^9) α β+5 (c^11) α β+2 (a^8) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β-14 (a^6) (b^2) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β+16 (a^4) (b^4) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β-14 (a^2) (b^6) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β+2 (b^8) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β-18 (a^6) (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β+24 (a^4) (b^2) (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β+24 (a^2) (b^4) (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β-18 (b^6) (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β+30 (a^4) (c^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β+52 (a^2) (b^2) (c^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β+30 (b^4) (c^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β-14 (a^2) (c^7) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β-14 (b^2) (c^7) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β+9 (a^7) (b^3) c (β^2)-17 (a^5) (b^5) c (β^2)+7 (a^3) (b^7) c (β^2)+a (b^9) c (β^2)+4 (a^7) b (c^3) (β^2)-17 (a^5) (b^3) (c^3) (β^2)-34 (a^3) (b^5) (c^3) (β^2)+7 a (b^7) (c^3) (β^2)-8 (a^5) b (c^5) (β^2)-17 (a^3) (b^3) (c^5) (β^2)-17 a (b^5) (c^5) (β^2)+4 (a^3) b (c^7) (β^2)+9 a (b^3) (c^7) (β^2)+2 (a^7) b c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)-2 (a^5) (b^3) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)-8 (a^3) (b^5) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)+4 a (b^7) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)-2 (a^5) b (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)-24 (a^3) (b^3) (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)-8 a (b^5) (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)-2 (a^3) b (c^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)-2 a (b^3) (c^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)+2 a b (c^7) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)+5 (a^10) b α γ-15 (a^8) (b^3) α γ+6 (a^6) (b^5) α γ+18 (a^4) (b^7) α γ-19 (a^2) (b^9) α γ+5 (b^11) α γ-9 (a^8) b (c^2) α γ-42 (a^6) (b^3) (c^2) α γ+100 (a^4) (b^5) (c^2) α γ-14 (a^2) (b^7) (c^2) α γ-19 (b^9) (c^2) α γ+4 (a^6) b (c^4) α γ+66 (a^4) (b^3) (c^4) α γ+100 (a^2) (b^5) (c^4) α γ+18 (b^7) (c^4) α γ+4 (a^4) b (c^6) α γ-42 (a^2) (b^3) (c^6) α γ+6 (b^5) (c^6) α γ-9 (a^2) b (c^8) α γ-15 (b^3) (c^8) α γ+5 b (c^10) α γ+2 (a^8) b (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ-18 (a^6) (b^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ+30 (a^4) (b^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ-14 (a^2) (b^7) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ-14 (a^6) b (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ+24 (a^4) (b^3) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ+52 (a^2) (b^5) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ-14 (b^7) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ+16 (a^4) b (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ+24 (a^2) (b^3) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ+30 (b^5) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ-14 (a^2) b (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ-18 (b^3) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ+2 b (c^8) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ+5 (a^11) β γ-19 (a^9) (b^2) β γ+18 (a^7) (b^4) β γ+6 (a^5) (b^6) β γ-15 (a^3) (b^8) β γ+5 a (b^10) β γ-19 (a^9) (c^2) β γ-14 (a^7) (b^2) (c^2) β γ+100 (a^5) (b^4) (c^2) β γ-42 (a^3) (b^6) (c^2) β γ-9 a (b^8) (c^2) β γ+18 (a^7) (c^4) β γ+100 (a^5) (b^2) (c^4) β γ+66 (a^3) (b^4) (c^4) β γ+4 a (b^6) (c^4) β γ+6 (a^5) (c^6) β γ-42 (a^3) (b^2) (c^6) β γ+4 a (b^4) (c^6) β γ-15 (a^3) (c^8) β γ-9 a (b^2) (c^8) β γ+5 a (c^10) β γ-14 (a^7) (b^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ+30 (a^5) (b^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ-18 (a^3) (b^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ+2 a (b^8) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ-14 (a^7) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ+52 (a^5) (b^2) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ+24 (a^3) (b^4) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ-14 a (b^6) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ+30 (a^5) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ+24 (a^3) (b^2) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ+16 a (b^4) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ-18 (a^3) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ-14 a (b^2) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ+2 a (c^8) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ+4 (a^7) (b^3) c (γ^2)-8 (a^5) (b^5) c (γ^2)+4 (a^3) (b^7) c (γ^2)+9 (a^7) b (c^3) (γ^2)-17 (a^5) (b^3) (c^3) (γ^2)-17 (a^3) (b^5) (c^3) (γ^2)+9 a (b^7) (c^3) (γ^2)-17 (a^5) b (c^5) (γ^2)-34 (a^3) (b^3) (c^5) (γ^2)-17 a (b^5) (c^5) (γ^2)+7 (a^3) b (c^7) (γ^2)+7 a (b^3) (c^7) (γ^2)+a b (c^9) (γ^2)+2 (a^7) b c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2)-2 (a^5) (b^3) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2)-2 (a^3) (b^5) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2)+2 a (b^7) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2)-2 (a^5) b (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2)-24 (a^3) (b^3) (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2)-2 a (b^5) (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2)-8 (a^3) b (c^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2)-8 a (b^3) (c^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2)+4 a b (c^7) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2));

circ=Circumcircle[eqn2];

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

Symmetrize[Center[circ][[1]]]

-a (Factor[5 (a^8)-14 (a^6) (b^2)+4 (a^4) (b^4)+10 (a^2) (b^6)-5 (b^8)-14 (a^6) (c^2)-24 (a^4) (b^2) (c^2)+54 (a^2) (b^4) (c^2)-4 (b^6) (c^2)+4 (a^4) (c^4)+54 (a^2) (b^2) (c^4)+26 (b^4) (c^4)+10 (a^2) (c^6)-4 (b^2) (c^6)-5 (c^8)]+4Area Factor[-14 (a^4) (b^2) +16 (a^2) (b^4) -2 (b^6) -14 (a^4) (c^2) +24 (a^2) (b^2) (c^2) +16 (b^4) (c^2) +16 (a^2) (c^4) +16 (b^2) (c^4) -2 (c^6)])

InputForm[%]

t=Triangle[{{0,0},{1,0},{.3,.7}}];

Circumcircle[LucasCentralTriangle[t]]

(circ/.Thread[{a,b,c}->SideLengths[Evaluate[t]]])[t]

Cartesian[CyclicTrilinears[-a (5 (a^8)-14 (a^6) (b^2)+4 (a^4) (b^4)+10 (a^2) (b^6)-5 (b^8)-14 (a^6) (c^2)-24 (a^4) (b^2) (c^2)+54 (a^2) (b^4) (c^2)-4 (b^6) (c^2)+4 (a^4) (c^4)+54 (a^2) (b^2) (c^4)+26 (b^4) (c^4)+10 (a^2) (c^6)-4 (b^2) (c^6)-5 (c^8)-2 (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (7 (a^4) (b^2)-8 (a^2) (b^4)+(b^6)+7 (a^4) (c^2)-12 (a^2) (b^2) (c^2)-8 (b^4) (c^2)-8 (a^2) (c^4)-8 (b^2) (c^4)+(c^6)))][t]]

IncidentCenters[KimberlingCenters,CircleFunction[eqn2]]//Timing

FullSimplify[Together//@CircleFunction[eqn2][[1]],TriangleQ[]]

Factor[Refine[ToAngles[(b c ((a^8)+(a^6) (7 (b^2)+7 (c^2)+4 (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]))-(a^4) ((b^2)+(c^2)) (17 (b^2)+17 (c^2)+8 (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]))+2 (((b-c))^2) (((b+c))^2) ((c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+(b^2) (2 (c^2)+(Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])))-(a^2) (-9 (b^6)-9 (c^6)+2 (c^4) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+(b^4) (17 (c^2)+2 (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]))+(b^2) (17 (c^4)+24 (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])))))/(5 (a^10)-19 (a^8) ((b^2)+(c^2))-2 (a^6) (-7 (b^4)-7 (c^4)+8 (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+8 (b^2) (2 (c^2)+(Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])))+(((b-c))^2) (((b+c))^2) (5 (b^6)-9 (b^4) (c^2)+5 (c^6)-(b^2) (9 (c^4)+16 (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])))+2 (a^4) (7 (b^6)+7 (c^6)+16 (c^4) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+(b^4) (67 (c^2)+16 (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]))+(b^2) (67 (c^4)+28 (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])))-(a^2) (19 (b^8)+19 (c^8)+16 (c^6) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+16 (b^6) (2 (c^2)+(Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]))-2 (b^4) (67 (c^4)+28 (c^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]))-8 (b^2) (-4 (c^6)+7 (c^4) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]))))],Area>0]]

f=Factor[Coefficient[poly=(a^8)+16 (a^6) Area+7 (a^6) (b^2)-32 (a^4) Area (b^2)-17 (a^4) (b^4)-8 (a^2) Area (b^4)+9 (a^2) (b^6)+8 Area (b^6)+7 (a^6) (c^2)-32 (a^4) Area (c^2)-34 (a^4) (b^2) (c^2)-96 (a^2) Area (b^2) (c^2)-17 (a^2) (b^4) (c^2)-8 Area (b^4) (c^2)+4 (b^6) (c^2)-17 (a^4) (c^4)-8 (a^2) Area (c^4)-17 (a^2) (b^2) (c^4)-8 Area (b^2) (c^4)-8 (b^4) (c^4)+9 (a^2) (c^6)+8 Area (c^6)+4 (b^2) (c^6),Area]]

ToAngles[Factor[poly-f Area]+f Area]

f=Factor[Coefficient[poly=(5 (a^10)-19 (a^8) (b^2)-64 (a^6) Area (b^2)+14 (a^6) (b^4)+128 (a^4) Area (b^4)+14 (a^4) (b^6)-64 (a^2) Area (b^6)-19 (a^2) (b^8)+5 (b^10)-19 (a^8) (c^2)-64 (a^6) Area (c^2)-32 (a^6) (b^2) (c^2)+224 (a^4) Area (b^2) (c^2)+134 (a^4) (b^4) (c^2)+224 (a^2) Area (b^4) (c^2)-32 (a^2) (b^6) (c^2)-64 Area (b^6) (c^2)-19 (b^8) (c^2)+14 (a^6) (c^4)+128 (a^4) Area (c^4)+134 (a^4) (b^2) (c^4)+224 (a^2) Area (b^2) (c^4)+134 (a^2) (b^4) (c^4)+128 Area (b^4) (c^4)+14 (b^6) (c^4)+14 (a^4) (c^6)-64 (a^2) Area (c^6)-32 (a^2) (b^2) (c^6)-64 Area (b^2) (c^6)+14 (b^4) (c^6)-19 (a^2) (c^8)-19 (b^2) (c^8)+5 (c^10)),Area]]

ToAngles[Factor[poly-f Area]+f Area]

(-2Area ((a^4)+9 (a^2) (b^2)+9 (a^2) (c^2)+4 (b^2) (c^2))+ (2 (a^6)-4 (a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-12 (a^2) (b^2) (c^2)-(b^4) (c^2)-(a^2) (c^4)-(b^2) (c^4)+(c^6)))/(-2 Area (5 (a^6)-9 (a^4) (b^2)-9 (a^2) (b^4)+5 (b^6)-9 (a^4) (c^2)-58 (a^2) (b^2) (c^2)-9 (b^4) (c^2)-9 (a^2) (c^4)-9 (b^2) (c^4)+5 (c^6))-4 (2 (a^6) (b^2)-4 (a^4) (b^4)+2 (a^2) (b^6)+2 (a^6) (c^2)-7 (a^4) (b^2) (c^2)-7 (a^2) (b^4) (c^2)+2 (b^6) (c^2)-4 (a^4) (c^4)-7 (a^2) (b^2) (c^4)-4 (b^4) (c^4)+2 (a^2) (c^6)+2 (b^2) (c^6)))

(b c(-2Area ((a^4)+9 (a^2) (b^2)+9 (a^2) (c^2)+4 (b^2) (c^2))+ (2 (a^6)-4 (a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-12 (a^2) (b^2) (c^2)-(b^4) (c^2)-(a^2) (c^4)-(b^2) (c^4)+(c^6)))/(-2 Area (5 (a^6)-9 (a^4) (b^2)-9 (a^2) (b^4)+5 (b^6)-9 (a^4) (c^2)-58 (a^2) (b^2) (c^2)-9 (b^4) (c^2)-9 (a^2) (c^4)-9 (b^2) (c^4)+5 (c^6))-4 (2 (a^6) (b^2)-4 (a^4) (b^4)+2 (a^2) (b^6)+2 (a^6) (c^2)-7 (a^4) (b^2) (c^2)-7 (a^2) (b^4) (c^2)+2 (b^6) (c^2)-4 (a^4) (c^4)-7 (a^2) (b^2) (c^4)-4 (b^4) (c^4)+2 (a^2) (c^6)+2 (b^2) (c^6)))//ToSidelengths)/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

((b c ((a^8)+16 (a^6) Area+7 (a^6) (b^2)-32 (a^4) Area (b^2)-17 (a^4) (b^4)-8 (a^2) Area (b^4)+9 (a^2) (b^6)+8 Area (b^6)+7 (a^6) (c^2)-32 (a^4) Area (c^2)-34 (a^4) (b^2) (c^2)-96 (a^2) Area (b^2) (c^2)-17 (a^2) (b^4) (c^2)-8 Area (b^4) (c^2)+4 (b^6) (c^2)-17 (a^4) (c^4)-8 (a^2) Area (c^4)-17 (a^2) (b^2) (c^4)-8 Area (b^2) (c^4)-8 (b^4) (c^4)+9 (a^2) (c^6)+8 Area (c^6)+4 (b^2) (c^6)))/(5 (a^10)-19 (a^8) (b^2)-64 (a^6) Area (b^2)+14 (a^6) (b^4)+128 (a^4) Area (b^4)+14 (a^4) (b^6)-64 (a^2) Area (b^6)-19 (a^2) (b^8)+5 (b^10)-19 (a^8) (c^2)-64 (a^6) Area (c^2)-32 (a^6) (b^2) (c^2)+224 (a^4) Area (b^2) (c^2)+134 (a^4) (b^4) (c^2)+224 (a^2) Area (b^4) (c^2)-32 (a^2) (b^6) (c^2)-64 Area (b^6) (c^2)-19 (b^8) (c^2)+14 (a^6) (c^4)+128 (a^4) Area (c^4)+134 (a^4) (b^2) (c^4)+224 (a^2) Area (b^2) (c^4)+134 (a^2) (b^4) (c^4)+128 Area (b^4) (c^4)+14 (b^6) (c^4)+14 (a^4) (c^6)-64 (a^2) Area (c^6)-32 (a^2) (b^2) (c^6)-64 Area (b^2) (c^6)+14 (b^4) (c^6)-19 (a^2) (c^8)-19 (b^2) (c^8)+5 (c^10))//ToSidelengths)/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

TraditionalForm[(b c(-2Area ((a^4)+9 (a^2) (b^2)+9 (a^2) (c^2)+4 (b^2) (c^2))+ (2 (a^6)-4 (a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-12 (a^2) (b^2) (c^2)-(b^4) (c^2)-(a^2) (c^4)-(b^2) (c^4)+(c^6)))/(-2 Area (5 (a^6)-9 (a^4) (b^2)-9 (a^2) (b^4)+5 (b^6)-9 (a^4) (c^2)-58 (a^2) (b^2) (c^2)-9 (b^4) (c^2)-9 (a^2) (c^4)-9 (b^2) (c^4)+5 (c^6))-4 (2 (a^6) (b^2)-4 (a^4) (b^4)+2 (a^2) (b^6)+2 (a^6) (c^2)-7 (a^4) (b^2) (c^2)-7 (a^2) (b^4) (c^2)+2 (b^6) (c^2)-4 (a^4) (c^4)-7 (a^2) (b^2) (c^4)-4 (b^4) (c^4)+2 (a^2) (c^6)+2 (b^2) (c^6))))]

InputForm[(b c(-2Area ((a^4)+9 (a^2) (b^2)+9 (a^2) (c^2)+4 (b^2) (c^2))+ (2 (a^6)-4 (a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-12 (a^2) (b^2) (c^2)-(b^4) (c^2)-(a^2) (c^4)-(b^2) (c^4)+(c^6)))/(-2 Area (5 (a^6)-9 (a^4) (b^2)-9 (a^2) (b^4)+5 (b^6)-9 (a^4) (c^2)-58 (a^2) (b^2) (c^2)-9 (b^4) (c^2)-9 (a^2) (c^4)-9 (b^2) (c^4)+5 (c^6))-4 (2 (a^6) (b^2)-4 (a^4) (b^4)+2 (a^2) (b^6)+2 (a^6) (c^2)-7 (a^4) (b^2) (c^2)-7 (a^2) (b^4) (c^2)+2 (b^6) (c^2)-4 (a^4) (c^4)-7 (a^2) (b^2) (c^4)-4 (b^4) (c^4)+2 (a^2) (c^6)+2 (b^2) (c^6))))]

r=Refine[Sqrt[Factor[ToSidelengths[Factor[Distance[Center[LucasCentralCircle],TrilinearVertexMatrix[LucasCentralTriangle][[1]]]^2]]]],TriangleQ[]]

f=Factor[Coefficient[poly=(12 (a^16)+52 (a^14) (b^2)-392 (a^12) (b^4)+384 (a^10) (b^6)+296 (a^8) (b^8)-448 (a^6) (b^10)+96 (a^4) (b^12)+20 (a^2) (b^14)-4 (b^16)+52 (a^14) (c^2)-784 (a^12) (b^2) (c^2)+6 (a^10) (b^4) (c^2)+2894 (a^8) (b^6) (c^2)-298 (a^6) (b^8) (c^2)-962 (a^4) (b^10) (c^2)+128 (a^2) (b^12) (c^2)+20 (b^14) (c^2)-392 (a^12) (c^4)+6 (a^10) (b^2) (c^4)+4545 (a^8) (b^4) (c^4)+4494 (a^6) (b^6) (c^4)-267 (a^4) (b^8) (c^4)-714 (a^2) (b^10) (c^4)+384 (a^10) (c^6)+2894 (a^8) (b^2) (c^6)+4494 (a^6) (b^4) (c^6)+3318 (a^4) (b^6) (c^6)+574 (a^2) (b^8) (c^6)-120 (b^10) (c^6)+296 (a^8) (c^8)-298 (a^6) (b^2) (c^8)-267 (a^4) (b^4) (c^8)+574 (a^2) (b^6) (c^8)+208 (b^8) (c^8)-448 (a^6) (c^10)-962 (a^4) (b^2) (c^10)-714 (a^2) (b^4) (c^10)-120 (b^6) (c^10)+96 (a^4) (c^12)+128 (a^2) (b^2) (c^12)+20 (a^2) (c^14)+20 (b^2) (c^14)-4 (c^16)+16 (a^14) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-108 (a^12) (b^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-62 (a^10) (b^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+474 (a^8) (b^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-142 (a^6) (b^8) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-118 (a^4) (b^10) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+36 (a^2) (b^12) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-108 (a^12) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-312 (a^10) (b^2) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+1480 (a^8) (b^4) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+1360 (a^6) (b^6) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-512 (a^4) (b^8) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-168 (a^2) (b^10) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+20 (b^12) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-62 (a^10) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+1480 (a^8) (b^2) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+3320 (a^6) (b^4) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+1500 (a^4) (b^6) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-192 (a^2) (b^8) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-70 (b^10) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+474 (a^8) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+1360 (a^6) (b^2) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+1500 (a^4) (b^4) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+744 (a^2) (b^6) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+50 (b^8) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-142 (a^6) (c^8) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-512 (a^4) (b^2) (c^8) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-192 (a^2) (b^4) (c^8) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+50 (b^6) (c^8) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-118 (a^4) (c^10) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-168 (a^2) (b^2) (c^10) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-70 (b^4) (c^10) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+36 (a^2) (c^12) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+20 (b^2) (c^12) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])),(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])]]

f 4Area+Factor[poly-f (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])]

f=Factor[Coefficient[poly=(5 (a^10)-19 (a^8) (b^2)+14 (a^6) (b^4)+14 (a^4) (b^6)-19 (a^2) (b^8)+5 (b^10)-19 (a^8) (c^2)-32 (a^6) (b^2) (c^2)+134 (a^4) (b^4) (c^2)-32 (a^2) (b^6) (c^2)-19 (b^8) (c^2)+14 (a^6) (c^4)+134 (a^4) (b^2) (c^4)+134 (a^2) (b^4) (c^4)+14 (b^6) (c^4)+14 (a^4) (c^6)-32 (a^2) (b^2) (c^6)+14 (b^4) (c^6)-19 (a^2) (c^8)-19 (b^2) (c^8)+5 (c^10)-16 (a^6) (b^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+32 (a^4) (b^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-16 (a^2) (b^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-16 (a^6) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+56 (a^4) (b^2) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+56 (a^2) (b^4) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-16 (b^6) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+32 (a^4) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+56 (a^2) (b^2) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+32 (b^4) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-16 (a^2) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-16 (b^2) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])) ,(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])]]

f 4Area+Factor[poly-f (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])]

r

t=Triangle[{{0,0},{1,0},{.3,.7}}];

ToSidelengths[r]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

InputForm[(a b c √(12 (a^16)+52 (a^14) (b^2)-392 (a^12) (b^4)+384 (a^10) (b^6)+296 (a^8) (b^8)-448 (a^6) (b^10)+96 (a^4) (b^12)+20 (a^2) (b^14)-4 (b^16)+52 (a^14) (c^2)-784 (a^12) (b^2) (c^2)+6 (a^10) (b^4) (c^2)+2894 (a^8) (b^6) (c^2)-298 (a^6) (b^8) (c^2)-962 (a^4) (b^10) (c^2)+128 (a^2) (b^12) (c^2)+20 (b^14) (c^2)-392 (a^12) (c^4)+6 (a^10) (b^2) (c^4)+4545 (a^8) (b^4) (c^4)+4494 (a^6) (b^6) (c^4)-267 (a^4) (b^8) (c^4)-714 (a^2) (b^10) (c^4)+384 (a^10) (c^6)+2894 (a^8) (b^2) (c^6)+4494 (a^6) (b^4) (c^6)+3318 (a^4) (b^6) (c^6)+574 (a^2) (b^8) (c^6)-120 (b^10) (c^6)+296 (a^8) (c^8)-298 (a^6) (b^2) (c^8)-267 (a^4) (b^4) (c^8)+574 (a^2) (b^6) (c^8)+208 (b^8) (c^8)-448 (a^6) (c^10)-962 (a^4) (b^2) (c^10)-714 (a^2) (b^4) (c^10)-120 (b^6) (c^10)+96 (a^4) (c^12)+128 (a^2) (b^2) (c^12)+20 (a^2) (c^14)+20 (b^2) (c^14)-4 (c^16)+8 Area (2 (a^2)+(b^2)+(c^2)) ((a^4)-7 (a^2) (b^2)+2 (b^4)-7 (a^2) (c^2)-5 (b^2) (c^2)+2 (c^4)) (4 (a^8)-(a^6) (b^2)-16 (a^4) (b^4)+9 (a^2) (b^6)-(a^6) (c^2)-43 (a^4) (b^2) (c^2)-21 (a^2) (b^4) (c^2)+5 (b^6) (c^2)-16 (a^4) (c^4)-21 (a^2) (b^2) (c^4)-10 (b^4) (c^4)+9 (a^2) (c^6)+5 (b^2) (c^6))))/4/((-Area(5 (a^6)-9 (a^4) (b^2)-9 (a^2) (b^4)+5 (b^6)-9 (a^4) (c^2)-58 (a^2) (b^2) (c^2)-9 (b^4) (c^2)-9 (a^2) (c^4)-9 (b^2) (c^4)+5 (c^6))-2  (2 (a^6) (b^2)-4 (a^4) (b^4)+2 (a^2) (b^6)+2 (a^6) (c^2)-7 (a^4) (b^2) (c^2)-7 (a^2) (b^4) (c^2)+2 (b^6) (c^2)-4 (a^4) (c^4)-7 (a^2) (b^2) (c^4)-4 (b^4) (c^4)+2 (a^2) (c^6)+2 (b^2) (c^6))) (2Area+(a^2)))]

ToSidelengths[(a b c √(12 (a^16)+52 (a^14) (b^2)-392 (a^12) (b^4)+384 (a^10) (b^6)+296 (a^8) (b^8)-448 (a^6) (b^10)+96 (a^4) (b^12)+20 (a^2) (b^14)-4 (b^16)+52 (a^14) (c^2)-784 (a^12) (b^2) (c^2)+6 (a^10) (b^4) (c^2)+2894 (a^8) (b^6) (c^2)-298 (a^6) (b^8) (c^2)-962 (a^4) (b^10) (c^2)+128 (a^2) (b^12) (c^2)+20 (b^14) (c^2)-392 (a^12) (c^4)+6 (a^10) (b^2) (c^4)+4545 (a^8) (b^4) (c^4)+4494 (a^6) (b^6) (c^4)-267 (a^4) (b^8) (c^4)-714 (a^2) (b^10) (c^4)+384 (a^10) (c^6)+2894 (a^8) (b^2) (c^6)+4494 (a^6) (b^4) (c^6)+3318 (a^4) (b^6) (c^6)+574 (a^2) (b^8) (c^6)-120 (b^10) (c^6)+296 (a^8) (c^8)-298 (a^6) (b^2) (c^8)-267 (a^4) (b^4) (c^8)+574 (a^2) (b^6) (c^8)+208 (b^8) (c^8)-448 (a^6) (c^10)-962 (a^4) (b^2) (c^10)-714 (a^2) (b^4) (c^10)-120 (b^6) (c^10)+96 (a^4) (c^12)+128 (a^2) (b^2) (c^12)+20 (a^2) (c^14)+20 (b^2) (c^14)-4 (c^16)+8 Area (2 (a^2)+(b^2)+(c^2)) ((a^4)-7 (a^2) (b^2)+2 (b^4)-7 (a^2) (c^2)-5 (b^2) (c^2)+2 (c^4)) (4 (a^8)-(a^6) (b^2)-16 (a^4) (b^4)+9 (a^2) (b^6)-(a^6) (c^2)-43 (a^4) (b^2) (c^2)-21 (a^2) (b^4) (c^2)+5 (b^6) (c^2)-16 (a^4) (c^4)-21 (a^2) (b^2) (c^4)-10 (b^4) (c^4)+9 (a^2) (c^6)+5 (b^2) (c^6))))/4/((-Area(5 (a^6)-9 (a^4) (b^2)-9 (a^2) (b^4)+5 (b^6)-9 (a^4) (c^2)-58 (a^2) (b^2) (c^2)-9 (b^4) (c^2)-9 (a^2) (c^4)-9 (b^2) (c^4)+5 (c^6))-2  (2 (a^6) (b^2)-4 (a^4) (b^4)+2 (a^2) (b^6)+2 (a^6) (c^2)-7 (a^4) (b^2) (c^2)-7 (a^2) (b^4) (c^2)+2 (b^6) (c^2)-4 (a^4) (c^4)-7 (a^2) (b^2) (c^4)-4 (b^4) (c^4)+2 (a^2) (c^6)+2 (b^2) (c^6))) (2Area+(a^2)))]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

ToSidelengths[(2 a b c 16 √(12 (a^16)+52 (a^14) (b^2)-392 (a^12) (b^4)+384 (a^10) (b^6)+296 (a^8) (b^8)-448 (a^6) (b^10)+96 (a^4) (b^12)+20 (a^2) (b^14)-4 (b^16)+52 (a^14) (c^2)-784 (a^12) (b^2) (c^2)+6 (a^10) (b^4) (c^2)+2894 (a^8) (b^6) (c^2)-298 (a^6) (b^8) (c^2)-962 (a^4) (b^10) (c^2)+128 (a^2) (b^12) (c^2)+20 (b^14) (c^2)-392 (a^12) (c^4)+6 (a^10) (b^2) (c^4)+4545 (a^8) (b^4) (c^4)+4494 (a^6) (b^6) (c^4)-267 (a^4) (b^8) (c^4)-714 (a^2) (b^10) (c^4)+384 (a^10) (c^6)+2894 (a^8) (b^2) (c^6)+4494 (a^6) (b^4) (c^6)+3318 (a^4) (b^6) (c^6)+574 (a^2) (b^8) (c^6)-120 (b^10) (c^6)+296 (a^8) (c^8)-298 (a^6) (b^2) (c^8)-267 (a^4) (b^4) (c^8)+574 (a^2) (b^6) (c^8)+208 (b^8) (c^8)-448 (a^6) (c^10)-962 (a^4) (b^2) (c^10)-714 (a^2) (b^4) (c^10)-120 (b^6) (c^10)+96 (a^4) (c^12)+128 (a^2) (b^2) (c^12)+20 (a^2) (c^14)+20 (b^2) (c^14)-4 (c^16)+8 Area (2 (a^2)+(b^2)+(c^2)) ((a^4)-7 (a^2) (b^2)+2 (b^4)-7 (a^2) (c^2)-5 (b^2) (c^2)+2 (c^4)) (4 (a^8)-(a^6) (b^2)-16 (a^4) (b^4)+9 (a^2) (b^6)-(a^6) (c^2)-43 (a^4) (b^2) (c^2)-21 (a^2) (b^4) (c^2)+5 (b^6) (c^2)-16 (a^4) (c^4)-21 (a^2) (b^2) (c^4)-10 (b^4) (c^4)+9 (a^2) (c^6)+5 (b^2) (c^6))))/((-16Area(5 (a^6)-9 (a^4) (b^2)-9 (a^2) (b^4)+5 (b^6)-9 (a^4) (c^2)-58 (a^2) (b^2) (c^2)-9 (b^4) (c^2)-9 (a^2) (c^4)-9 (b^2) (c^4)+5 (c^6))-32  (2 (a^6) (b^2)-4 (a^4) (b^4)+2 (a^2) (b^6)+2 (a^6) (c^2)-7 (a^4) (b^2) (c^2)-7 (a^2) (b^4) (c^2)+2 (b^6) (c^2)-4 (a^4) (c^4)-7 (a^2) (b^2) (c^4)-4 (b^4) (c^4)+2 (a^2) (c^6)+2 (b^2) (c^6))) (16Area+2 (a^2) 4 ))]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

ToSidelengths[(2 a b c 16 Area^2√(12 (a^16)+52 (a^14) (b^2)-392 (a^12) (b^4)+384 (a^10) (b^6)+296 (a^8) (b^8)-448 (a^6) (b^10)+96 (a^4) (b^12)+20 (a^2) (b^14)-4 (b^16)+52 (a^14) (c^2)-784 (a^12) (b^2) (c^2)+6 (a^10) (b^4) (c^2)+2894 (a^8) (b^6) (c^2)-298 (a^6) (b^8) (c^2)-962 (a^4) (b^10) (c^2)+128 (a^2) (b^12) (c^2)+20 (b^14) (c^2)-392 (a^12) (c^4)+6 (a^10) (b^2) (c^4)+4545 (a^8) (b^4) (c^4)+4494 (a^6) (b^6) (c^4)-267 (a^4) (b^8) (c^4)-714 (a^2) (b^10) (c^4)+384 (a^10) (c^6)+2894 (a^8) (b^2) (c^6)+4494 (a^6) (b^4) (c^6)+3318 (a^4) (b^6) (c^6)+574 (a^2) (b^8) (c^6)-120 (b^10) (c^6)+296 (a^8) (c^8)-298 (a^6) (b^2) (c^8)-267 (a^4) (b^4) (c^8)+574 (a^2) (b^6) (c^8)+208 (b^8) (c^8)-448 (a^6) (c^10)-962 (a^4) (b^2) (c^10)-714 (a^2) (b^4) (c^10)-120 (b^6) (c^10)+96 (a^4) (c^12)+128 (a^2) (b^2) (c^12)+20 (a^2) (c^14)+20 (b^2) (c^14)-4 (c^16)+8 Area (2 (a^2)+(b^2)+(c^2)) ((a^4)-7 (a^2) (b^2)+2 (b^4)-7 (a^2) (c^2)-5 (b^2) (c^2)+2 (c^4)) (4 (a^8)-(a^6) (b^2)-16 (a^4) (b^4)+9 (a^2) (b^6)-(a^6) (c^2)-43 (a^4) (b^2) (c^2)-21 (a^2) (b^4) (c^2)+5 (b^6) (c^2)-16 (a^4) (c^4)-21 (a^2) (b^2) (c^4)-10 (b^4) (c^4)+9 (a^2) (c^6)+5 (b^2) (c^6))))/((-16Area^2(5 (a^6)-9 (a^4) (b^2)-9 (a^2) (b^4)+5 (b^6)-9 (a^4) (c^2)-58 (a^2) (b^2) (c^2)-9 (b^4) (c^2)-9 (a^2) (c^4)-9 (b^2) (c^4)+5 (c^6))-32 Area (2 (a^6) (b^2)-4 (a^4) (b^4)+2 (a^2) (b^6)+2 (a^6) (c^2)-7 (a^4) (b^2) (c^2)-7 (a^2) (b^4) (c^2)+2 (b^6) (c^2)-4 (a^4) (c^4)-7 (a^2) (b^2) (c^4)-4 (b^4) (c^4)+2 (a^2) (c^6)+2 (b^2) (c^6))) (16Area^2+2 (a^2) 4 Area))]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

ToSidelengths[(2 a b (a+b-c) c (a-b+c) (-a+b+c) (a+b+c) √(12 (a^16)+52 (a^14) (b^2)-392 (a^12) (b^4)+384 (a^10) (b^6)+296 (a^8) (b^8)-448 (a^6) (b^10)+96 (a^4) (b^12)+20 (a^2) (b^14)-4 (b^16)+52 (a^14) (c^2)-784 (a^12) (b^2) (c^2)+6 (a^10) (b^4) (c^2)+2894 (a^8) (b^6) (c^2)-298 (a^6) (b^8) (c^2)-962 (a^4) (b^10) (c^2)+128 (a^2) (b^12) (c^2)+20 (b^14) (c^2)-392 (a^12) (c^4)+6 (a^10) (b^2) (c^4)+4545 (a^8) (b^4) (c^4)+4494 (a^6) (b^6) (c^4)-267 (a^4) (b^8) (c^4)-714 (a^2) (b^10) (c^4)+384 (a^10) (c^6)+2894 (a^8) (b^2) (c^6)+4494 (a^6) (b^4) (c^6)+3318 (a^4) (b^6) (c^6)+574 (a^2) (b^8) (c^6)-120 (b^10) (c^6)+296 (a^8) (c^8)-298 (a^6) (b^2) (c^8)-267 (a^4) (b^4) (c^8)+574 (a^2) (b^6) (c^8)+208 (b^8) (c^8)-448 (a^6) (c^10)-962 (a^4) (b^2) (c^10)-714 (a^2) (b^4) (c^10)-120 (b^6) (c^10)+96 (a^4) (c^12)+128 (a^2) (b^2) (c^12)+20 (a^2) (c^14)+20 (b^2) (c^14)-4 (c^16)+8 Area (2 (a^2)+(b^2)+(c^2)) ((a^4)-7 (a^2) (b^2)+2 (b^4)-7 (a^2) (c^2)-5 (b^2) (c^2)+2 (c^4)) (4 (a^8)-(a^6) (b^2)-16 (a^4) (b^4)+9 (a^2) (b^6)-(a^6) (c^2)-43 (a^4) (b^2) (c^2)-21 (a^2) (b^4) (c^2)+5 (b^6) (c^2)-16 (a^4) (c^4)-21 (a^2) (b^2) (c^4)-10 (b^4) (c^4)+9 (a^2) (c^6)+5 (b^2) (c^6))))/((5 (a^10)-19 (a^8) (b^2)+14 (a^6) (b^4)+14 (a^4) (b^6)-19 (a^2) (b^8)+5 (b^10)-19 (a^8) (c^2)-32 (a^6) (b^2) (c^2)+134 (a^4) (b^4) (c^2)-32 (a^2) (b^6) (c^2)-19 (b^8) (c^2)+14 (a^6) (c^4)+134 (a^4) (b^2) (c^4)+134 (a^2) (b^4) (c^4)+14 (b^6) (c^4)+14 (a^4) (c^6)-32 (a^2) (b^2) (c^6)+14 (b^4) (c^6)-19 (a^2) (c^8)-19 (b^2) (c^8)+5 (c^10)-16 (a^6) (b^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+32 (a^4) (b^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-16 (a^2) (b^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-16 (a^6) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+56 (a^4) (b^2) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+56 (a^2) (b^4) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-16 (b^6) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+32 (a^4) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+56 (a^2) (b^2) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+32 (b^4) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-16 (a^2) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-16 (b^2) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])) ((a+b-c) (a-b+c) (-a+b+c) (a+b+c)+2 (a^2) (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)])))]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

ToSidelengths[r]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]