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 4 of 4: Verify the equilateral case
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a=b=c⟹s−a=s−b=s−c=a/2.a=b=c\Longrightarrow\quad s-a=s-b=s-c=a/2.
Detailed analysis

For an equilateral triangle, every new side is the same, namely half the old side. Thus every stage is again an equilateral triangle, and the construction can be continued indefinitely. These and only these are the required originals.