# In any Δ ABC, then prove that  Note: In any ΔABC we define ‘a’ as the length of the side opposite to A, ‘b’ as the length of the side opposite to B and ‘c’ as the length of the side opposite to C.

The key point to solve the problem:

The idea of cosine formula in ΔABC

Cos A = Cos B = Cos C = As we have to prove under given conditions.

Only cos terms are involved so we will apply cosine formula to find cos A , cos B, and cos C and we will take their ratio. b + c = 12k ….eqn 1

c + a = 13k ….eqn 2

a + b = 15k ….eqn 3

But only above relation is not sufficient to find cosines as k is unknown, either we need to express k in terms of a , b or c or express a , b , c in terms of k. Later part is easier.

we will find a,b,c in terms of k

Adding eqn 1,2 and 3 we have –

2 (a + b + c) = 40k

a + b+ c = 20k

a = 20k – (b + c) = 20k – 12k = 8k

Similarly, b = 20k – (c + a) = 20k – 13k = 7k

And c = 20k – (a + b) = 20k – 15k = 5k

Hence,

Cos A = Cos B = cos C =   ….Hence proved.

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