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Question Number 13223 by Tinkutara last updated on 16/May/17
If a, b, c are sides of triangle show that  (1 + ((b−c)/a))^a (1 + ((c−a)/b))^b (1 + ((a−b)/c))^c  < 1
Ifa,b,caresidesoftriangleshowthat(1+bca)a(1+cab)b(1+abc)c<1
Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 17/May/17
if:  a=b=c⇒RHS=1≮LHS=1  1+((b−c)/a)=2×((p−c)/a),1+((c−a)/b)=2((p−a)/b),1+((a−b)/c)=2((p−b)/c)  (RHS)_1 =(1+((b−c)/a))(1+((c−a)/b))(1+((a−b)/c))⇒  (RHS)_1 =8×(((p−a)(p−b)(p−c))/(abc))=8×((S^2 /p)/(4RS))=  =((2S)/(pR))=((2r)/R)≤1  (according to Euler′s teorem)  d^2 =R^2 −2Rr=R(R−2r)≥0⇒R≥2r  abc=((2S)/(sinC)).2RsinC=4R.S,  S=p.r  (1+((b−c)/a))^a (1+((c−a)/b))^b (1+((a−b)/c))^c <  (1+((b−c)/a))^(abc) (1+((c−a)/b))^(abc) (1+((a−b)/c))^(abc) =  [(1+((b−c)/a))(1+((c−a)/b))(1+((a−b)/c))]^(abc) =  =(((2r)/R))^(abc) ≤1^(abc) ≤1     .■
if:a=b=cRHS=1LHS=11+bca=2×pca,1+cab=2pab,1+abc=2pbc(RHS)1=(1+bca)(1+cab)(1+abc)(RHS)1=8×(pa)(pb)(pc)abc=8×S2p4RS==2SpR=2rR1(accordingtoEulersteorem)d2=R22Rr=R(R2r)0R2rabc=2SsinC.2RsinC=4R.S,S=p.r(1+bca)a(1+cab)b(1+abc)c<(1+bca)abc(1+cab)abc(1+abc)abc=[(1+bca)(1+cab)(1+abc)]abc==(2rR)abc1abc1.◼

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