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Prove that, given a triangle with sides $a,b,c$, there exists a triangle with sides $a+2b,b+2c,c+2a$ that has an area three times the original

I have used Heron's formula but got lost in algebra! Any one got other approach?

egreg
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Callie12
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    Please show the work you did (in using Heron's formula) to the point where you get stuck in algebra; you may have used the formula correctly, but made an algebraic error somewhere, which we can then identify, if that's the case. – amWhy Jul 09 '18 at 14:15
  • You don't say what the original triangle is. – John Jul 09 '18 at 14:24
  • Presumably a triangle with sides a, b, c. – Paul Jul 09 '18 at 14:26
  • Area**2 = s(s-a)(s-b)(s-c) for both cases .......and then it is too complex ... for me – Callie12 Jul 09 '18 at 18:37

1 Answers1

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Assume the original triangle is an equilateral triangle with side $s$, then the new triangle is also equilateral with sides $3s$.

The area of the new triangle is $9$ times the area of the original, not $3$ times.

Thus the statement is not true in general.

stressed out
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Mohammad Riazi-Kermani
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  • I tried setting $a=1$ and I found that there does not seem to exist any $b,c$ such that the original problem statement would be true. So, I believe the statement is never true, but I am not sure how to prove it. – SlipEternal Jul 09 '18 at 15:55
  • At last I have found the source of this problem ! I think it is perhaps a number theory type exercise. Thought it may be helpful to show original question.:http://kriukov-unam.no-ip.org/pub/olymp/rus-math-olymp-2007.pdf – Callie12 Aug 09 '18 at 11:36