Determine all triangles ABC for which tan(A-B)+tan(B-C)+tan(C-A)=0.

Question

There's a hint: "Can you relate A-B to B-C and C-A?"

Should I apply the tangent difference formula (tan(x-y))? Help would be appreciated, thanks.

Answer 1

You will find a nice discussion at

http://answers.tutorvista.com/1101359/trigonometry-application-rule.html

showing that if A+B+C=180,

tan(A-B)+tan(B-C)+tan(C-A) = tan(A-B)*tan(B-C)*tan(C-A)

Given that, one of the factors must be zero. So, A=B or B=C or A=C. That is, the triangle is isosceles.

Answer 2

Thanks steve!