Triangle Line Picking

Author

Eric W. Weisstein
September 27, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComputationalGeometry/TriangleLinePicking.nb.

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Closed form

lbar[a_,b_,c_]:=With[{s=(a+b+c)/2},((4s(s-a)(s-b)(s-c))/15)((1/(a^3))Log[(s/(s-a))]+(1/(b^3))Log[(s/(s-b))]+(1/(c^3))Log[(s/(s-c))])+((a+b+c)/15)+(((b+c)(((b-c))^2))/(30(a^2)))+(((c+a)(((c-a))^2))/(30(b^2)))+(((a+b)(((a-b))^2))/(30(c^2)))]

Equal@@lbar@@@Permutations[{a,b,c}]//FullSimplify

(3, 4, 5)

lbar[3,4,5]

Expand[LogExpand[%]]

N[%]

30-30-120

EuclideanDistance@@@Partition[AASTriangle[30Degree,30Degree,1][[1]],2,1,1]

lbar[1,1,(Sqrt[3])]

FullSimplify[%]

LogSimplify[%]

N[%]

Collect[LogSimplify[(1/180) (18 (Log[(Sqrt[3])+2]-Log[3]/2)+(Sqrt[3]) (36+Log[7+4 (Sqrt[3])]))],Log[2+(Sqrt[3])]]

((Sqrt[3])/5)-((Log[3])/20)+Simplify[((1/10)+(1/(30 (Sqrt[3]))))] Log[2+(Sqrt[3])]

30-60-90

EuclideanDistance@@@Partition[AASTriangle[30Degree,60Degree,1][[1]],2,1,1]

lbar[(1/2),((Sqrt[3])/2),1]

Simplify[%]

N[%]

Collect[LogSimplify[%],Log[2+(Sqrt[3])]]

(17/120)+((Sqrt[3])/40)+((9 Log[3])/160)+FullSimplify[((1/80)+(1/(30 (Sqrt[3])))) ]Log[2+(Sqrt[3])]

N[%]

45-60-75

EuclideanDistance@@@Partition[AASTriangle[45Degree,60Degree,2][[1]],2,1,1]//FullSimplify

lbar[2,(Sqrt[6]),1+(Sqrt[3])]

FullSimplify[%]

N[%]

Equilateral

lbar[1,1,1]

N[%]

Golden

lbar[1,GoldenRatio,GoldenRatio]//FullSimplify

TrigToExp[%]

N[%]

(7/30)+(1/(6 (Sqrt[5])))+(1/12) Log[1+(2/(Sqrt[5]))]+((Log[1+(2/(Sqrt[5]))])/(6 (Sqrt[5])))+((Log[(Sqrt[5])]+Log[((2/(Sqrt[5]))+1)])/(6 (Sqrt[5])))//N

(7/30)+(1/(6 (Sqrt[5])))+Log[1+(2/(Sqrt[5]))]((1/12)+(1/(6 (Sqrt[5])))+(1/(6 (Sqrt[5]))))+((Log[(Sqrt[5])])/(6 (Sqrt[5])))//N

FullSimplify[(7/30)+(1/(6 (Sqrt[5])))+Log[1+(2/(Sqrt[5]))]((1/12)+(1/(6 (Sqrt[5])))+(1/(6 (Sqrt[5]))))+((Log[(Sqrt[5])])/(6 (Sqrt[5])))]

Log[(Sqrt[5]),1+(2/(Sqrt[5]))]//N

Log[1+(2/(Sqrt[5])),(Sqrt[5])]//N

FullSimplify[(1/60) (2 (7+(Sqrt[5]))+(Sqrt[5]) Log[5]+(5+4 (Sqrt[5])) Log[1+(2/(Sqrt[5]))])-(1/60) (2 (7+(Sqrt[5])+(Sqrt[5]) ArcSinh[2])+(5+2 (Sqrt[5])) Log[1+(2/(Sqrt[5]))])]

ByteCount/@{(1/60) (2 (7+(Sqrt[5]))+(Sqrt[5]) Log[5]+(5+4 (Sqrt[5])) Log[1+(2/(Sqrt[5]))]),(1/60) (2 (7+(Sqrt[5])+(Sqrt[5]) ArcSinh[2])+(5+2 (Sqrt[5])) Log[1+(2/(Sqrt[5]))])}

(1/60) (2 (7+(Sqrt[5]))+Log[(5^(Sqrt[5]))(((1+(2/(Sqrt[5]))))^((5+4 (Sqrt[5])) ))])//N

(5^(Sqrt[5]))(((1+(2/(Sqrt[5]))))^(5 ))(((1+(2/(Sqrt[5]))))^(4 (Sqrt[5])))

Expand[(((1+(2/(Sqrt[5]))))^(5 ))]

FullSimplify[(5^(Sqrt[5]))((61/5)+(682/(25 (Sqrt[5]))))(((1+(2/(Sqrt[5]))))^(4 (Sqrt[5])))]

Isosceles

Assuming[a>0&&h>0,lbar[a,(Sqrt[(h^2)+((((a/2)))^2)]),(Sqrt[(h^2)+((((a/2)))^2)])]//FullSimplify]

Assuming[a>0&&c>0,lbar[a,c,c]//FullSimplify]

FindInstance[Im[((-(a^5)+8 (a^3) (h^2)+48 a (h^4)+8 (a^4) (Sqrt[(a^2)+4 (h^2)])+2 (h^2) (16 (a^3) Log[((a+(Sqrt[(a^2)+4 (h^2)]))/a)]+((((a^2)+4 (h^2)))^(3/2)) Log[1+((a (a+(Sqrt[(a^2)+4 (h^2)])))/(2 (h^2)))]))/(30 a ((((a^2)+4 (h^2)))^(3/2))))]≠0,{a,h},Reals]

IsoscelesRight

lbar[1,1,(Sqrt[2])]//FullSimplify

N[%]

(1/60) (4+8 (Sqrt[2])+8 ArcSinh[1]+(Sqrt[2]) Log[3+2 (Sqrt[2])])//ExpToTrig//Factor

N[{(1/60) (4+8 (Sqrt[2])+8 ArcSinh[1]+(Sqrt[2]) Log[3+2 (Sqrt[2])]),
(1/30)(2+4(Sqrt[2])+(4+(Sqrt[2]))ArcSinh[1]),
(1/60) (4+8 (Sqrt[2])+(Sqrt[2]) ArcCosh[3]+8 ArcSinh[1])},20]

LogSimplify[FullSimplify@lbar[1,1,(Sqrt[2])]]

((2 (Sqrt[2])+1)/15)+FullSimplify[(((2 )/15)+(1/(15 (Sqrt[2]))))]ArcSinh[1]

N[%]

Kepler

lbar[Sqrt[GoldenRatio], GoldenRatio,1]

FullSimplify[%]

LogSimplify[%]

lbar[Sqrt[GoldenRatio], GoldenRatio,1]//FunctionExpand//FullSimplify

LogSimplify[%]

Right

Assuming[a>0&&b>0,lbar[a,b,(Sqrt[(a^2)+(b^2)])]//FullSimplify]