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 2 of 4: Measure the side spread

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D=\max(a,b,c)-\min(a,b,c)\quad\Longrightarrow\quad D_{k+1}=2D_k.

Detailed analysis

Assume a<=b<=c. The transformed sides, after the auxiliary factor 2, are 2(s-a), 2(s-b), 2(s-c). Their largest-minus-smallest difference is 2(c-a), so the spread doubles at each iteration.