Is it possible to represent 3 positive numbers each as the difference of semiperimeter and one of the side of a triangle

Question

It is possible to represent three positive numbersas sum of two numbers from a set of 3 numbers. But is it possible represent 3 positive numbers let $a,b,c $ as $s-x,\ s-y,\ s-z$ where $x,y,z$ are the sides of a triangle and $s $ is the semiperimeter of the triangle

Answer 1

Yes it is possible Let

$a=s-x=\dfrac {y+z-x}{2}$, $b=s-y=\dfrac {z+x-y}{2}$, $c=s-z=\dfrac {x+y-z}{2}$ where $x,y,z$ are real positive numbers. If we can show that $x,y,z $ follows triangle inequality then we are done. As $a>0$ hence $y+z-x>0$ and this is true for all of them. Hence it is possible to represent 3 numbers as difference of semiperimeter of a triangle and one of the side of the triangle.

We can find $x,y,z $ in this way $$x=b+c $$ $$y=c+a $$ $$z=a+b $$