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Question Number 39458 by MJS last updated on 06/Jul/18
find the greatest possible square insribed in  a triangle with sides a b c
findthegreatestpossiblesquareinsribedinatrianglewithsidesabc
Commented by MJS last updated on 06/Jul/18
the side s of the square not necessarily on  one of the sides of the triangle
thesidesofthesquarenotnecessarilyononeofthesidesofthetriangle
Commented by ajfour last updated on 06/Jul/18
In △PCR       ((s(√2))/(sin γ)) = ((b−y)/(sin (θ+(π/4))))        ....(i)  In △SRA        (s/(sin α)) = (y/(sin (β−θ)))       .....(ii)  eliminating y amongst (i), (ii)      ((s(√2)sin (θ+(π/4)))/(sin γ)) =b− ((s sin (β−θ))/(sin α))  ⇒  s=(b/((((√2)sin (𝛉+(𝛑/4)))/(sin 𝛄))+((sin (𝛃−𝛉))/(sin 𝛂))))      =(b/(((sin θ+cos θ)/(sin γ))+((sin βcos θ−cos βsin θ)/(sin α))))  =((bsin γ sin α)/(sin θ(sin α−cos β sin γ)+cos θ(sin α+sin β sin γ)))         s=((bsin γ sin α)/(Asin (θ+φ)))  where     tan 𝛗 = ((sin α+sin β sin γ)/(sin α−cos β sin γ))   A=(√((sin α−cos β sin γ)^2 +(sin α+sin β sin γ)^2 ))  for s to be maximum      sin (θ+φ) has to be minimum.  For many of the cases(though  not all),      sin (θ+φ) is minimum if θ=0  Then        s_(max)  = ((bsin 𝛄 sin 𝛂)/(Asin 𝛗))            = ((bsin γ sin α)/(sin α+sin β sin γ))  and since        ((sin α)/a)= ((sin β)/b) = ((sin γ)/c) = (1/(2R))     s_(max) =  ((abc)/(2aR+bc)) = (a/(((2aR)/(bc))+1)) .   This formula will have to be  amended if sin (θ+φ)< sin φ  for nonzero θ .
InPCRs2sinγ=bysin(θ+π4).(i)InSRAssinα=ysin(βθ)..(ii)eliminatingyamongst(i),(ii)s2sin(θ+π4)sinγ=bssin(βθ)sinαs=b2sin(θ+π4)sinγ+sin(βθ)sinα=bsinθ+cosθsinγ+sinβcosθcosβsinθsinα=bsinγsinαsinθ(sinαcosβsinγ)+cosθ(sinα+sinβsinγ)s=bsinγsinαAsin(θ+ϕ)wheretanϕ=sinα+sinβsinγsinαcosβsinγA=(sinαcosβsinγ)2+(sinα+sinβsinγ)2forstobemaximumsin(θ+ϕ)hastobeminimum.Formanyofthecases(thoughnotall),sin(θ+ϕ)isminimumifθ=0Thensmax=bsinγsinαAsinϕ=bsinγsinαsinα+sinβsinγandsincesinαa=sinβb=sinγc=12Rsmax=abc2aR+bc=a2aRbc+1.Thisformulawillhavetobeamendedifsin(θ+ϕ)<sinϕfornonzeroθ.
Commented by ajfour last updated on 06/Jul/18
Commented by MJS last updated on 06/Jul/18
thanks. I already thought there would be  several different cases...
thanks.Ialreadythoughttherewouldbeseveraldifferentcases

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