1. From Pythagoras theorem
c2=a2+b2
The area of the triangle
A=21βab=21βpc Then
p=cabβ
p=p(a,b)=a2+b2βabβ Differentiate by a
βaβpβ=a2+b2βbββa2+b2β(a2+b2)a2bβ
=a2+b2β(a2+b2)a2b+b3βa2bβSubstitute c=a2+b2β
βaβpβ=c3b3β
2.
A=const=>21βab=const Let a=kb,k>0. Then c2=a2+b2=k2b2+b2=b2(1+k2).
cbβ=k21ββ=const
p=cbβ(a) Use that cbβ=const, if A=const. Then
(βaβpβ)Aβ=cbβ
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