SOLUTION: Let a, b and c be sides of a triangle in arithmetic progression in this order. Show that cos((A-C)/2)/cos((A+C)/2)=2

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Question 1089830: Let a, b and c be sides of a triangle in arithmetic progression in this order. Show that cos((A-C)/2)/cos((A+C)/2)=2
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
There's almost certainly a more elegant way than this, but here goes:
WLOG, assume a = 1-x, b = x, and c = 1+x. Let I be the incenter, and r be the inradius. We have semiperimeter 3/2. Additionally, if we let D, E, and F be the points of tangency with the incircle and sides BC, CA, AB respectively, we have that , , and . Finding the area of triangle ABC using A = rs and Heron's formula we have:


Squaring both sides and simplifying gives . (*)

Now we are trying to show . Using sum and difference formulas we have



and also:



It follows that if and only if . This simplifies to which we showed is true in (*). Therefore .