prove that if in triangle ABC tan(A/2)=5/6 tan(B/2)=2/5 then a,b,c will be in AP
We obtain the cosines from Pythagoras' Theorem.
From the Law of Sines
If a, b, and c are in Arithmetic progression,
then the sum of the 3 sides is 3 times the "middle side".
Hence, the sum of the 3 sines must be 3 times the sum of the "middle sine".
Therefore
a, c, b are in Arithmetic Progression.
b, c, a are in Arithmetic Progression.