Question

cotA + cotB + cotC = sqrt(3)
prove that ABC is equilateral triangle
Refused solution:
11 years ago

Answer 1

if this was an equalateral triangle then A B and C are all the same so you could just substutite one of the values such as A for the other two and you would be able to obtain your result.

Answer 2

This may show that its true, but it doesn't say that there are no other solutions.

Answer 3

instead of trying to find them, lets assume that two of the angles are the same and one is different

Answer 4

There is. A long two-pages solution.

Answer 5

hmm well I will try to give it a shot and see if I can figure it out. Sometimes these problems are more difficult then they seem, but the best way to approach them is to think logically one step at a time.

Answer 6

They do, at least this one. This problem was solved using Trigonometry.

Answer 7

yes this will most likely be a problem which involves several trigonometric identites.

Answer 8

I recommend for you to make yourself a forumla sheet with all the trig identies that you have learned, and dont just use an already made one. The key is making it yourself and using it to solve problems. If you do this by the time of the exam you will realize that you have many of the formulas commited to memory and you can then focus on the few that you need to memorize for your test.

Answer 9

Yeah, I tried to do it myself. The difficulty was to find from where to start.

Answer 10

ahh ok, was the solution telling you to prove by contradiction?

Answer 11

No clue.

Answer 12

Two cases, either all angles are equal or at least one is different from the others. Assume by way of contradiction that one of the angles, say, A is different from two other angles (B and C). Note also that all angles must equal \(\pi\) $$ A+2B=\pi\\ \cot A + 2\cot B=\sqrt{3} $$ Two equations, two unknowns. Solvable. Answer is \(\pi\over 3\) or 60 degrees. Since The conclusion contradicts our assumptions then our assumption must have been false. Therefore \(A=B=C=60^\circ\) http://www.wolframalpha.com/input/?i=cot%28x%29%2B2cot%28y%29%3Dsqrt%283%29%2Cx%2B2y%3Dpi%2Cx+%3C+pi%2Cy%3Cpi

QED

Answer 13

NICee!

Answer 14

found this too.

http://math.stackexchange.com/questions/574166/how-to-show-that-the-triangle-is-equilateral-triangle