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 3 of 4: Rule out a non-equilateral triangle
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D0>0⟹Dk=2kD0 eventually exceeds the fixed perimeter bound.D_0>0\quad\Longrightarrow\quad D_k=2^kD_0\text{ eventually exceeds the fixed perimeter bound}.
Detailed analysis

If a!=b or b!=c, then D_0>0. Since the auxiliary perimeter is fixed and a valid triangle must have its largest side less than half that perimeter, its side spread is bounded. The doubling rule eventually breaks this bound, so the process must stop.