Problem
ALG-B2-M08-P004 Triangle substitution
#4
★★☆☆☆ Level 2 of 5
Let \(a,b,c\) be the sides of a triangle. Prove that there exist \(x,y,z>0\) such that \(a=y+z\), \(b=z+x\), \(c=x+y\).
Hint. Take halves of \(b+c-a\), \(c+a-b\), \(a+b-c\).
Set \(x=(b+c-a)/2\), \(y=(c+a-b)/2\), \(z=(a+b-c)/2\). Triangle inequalities give \(x,y,z>0\). Direct checking gives \(y+z=a\), \(z+x=b\), \(x+y=c\).
Teaching goal: recognize the appropriate substitution and check the domain.