MathLabs

Problem 1

A triangle with sides a,b,c is given. Let s=(a+b+c)/2 be its semiperimeter. Construct a triangle with sides s-a,s-b,s-c, and repeat this process while possible. For which original triangles can the process be repeated indefinitely?
Step 1 of 4: Track the perimeter
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(s−a)+(s−b)+(s−c)=s=a+b+c2.(s-a)+(s-b)+(s-c)=s=\frac{a+b+c}{2}.
Detailed analysis

The constructed side lengths are positive whenever the original triple is a triangle, and their sum is s, exactly half the old perimeter. At every stage, rescale the new triangle by 2; this auxiliary similar triangle has the same perimeter as the original stage.