Matematické Fórum

stuart clark · 16. 11. 2017 09:16

In a $\triangle ABC$ , If $2(ab^2+bc^2+ca^2)=a^2b+b^2c+c^2a+3abc,$ Then nature of $\triangle$ is

vanok · 19. 11. 2017 17:10

Hi,
Your problem spells in way equivalent
$2(ab^2+bc^2+ca^2)=a^2b+b^2c+c^2a+3abc,$
$a \ge b \ge c >0$
$2b \ge a$

A trivial solution is $a=b=c$
(Triangle ABC is equilateral)

And for example WA confirms that it is the unique solution.

Marian · 20. 11. 2017 11:42

↑ vanok:

I don't understand the argumentation of the kind you used above: "And for example WA confirms...".

In my opinion, the solutions in the section "Zajímavé úlohy z ..." should always be complete. (But may be, I'm wrong.)

vanok · 20. 11. 2017 13:40 — Editoval vanok (22. 11. 2017 09:14)

↑ Marian:
Hi,
Yes, it is a question since was published the demonstration of 4 colors which was made by means of a computer.
Here also, I contented myself with answer by software of calculation.
Here, the situation is relatively simple. For a, b authorized I formed an equation of second degree in c, and I verified that such an equation does not admit other solutions that that already mentioned.
But I am not a robot to write kilometric formulae and if somebody want to do it I shall leave with him this pleasure.

Marian · 20. 11. 2017 14:17

↑ vanok:

Thank you for your clarification.

stuart clark · 14. 12. 2017 13:22

Thanks vanok

Matematické Fórum

#1 16. 11. 2017 09:16

Triangle

#2 19. 11. 2017 17:10

Re: Triangle

#3 20. 11. 2017 11:42

Re: Triangle

#4 20. 11. 2017 13:40 — Editoval vanok (22. 11. 2017 09:14)

Re: Triangle

#5 20. 11. 2017 14:17

Re: Triangle

#6 14. 12. 2017 13:22

Re: Triangle

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