Question

in a triangle ABC if angleC=60degree prove that 1/a+c +1/b+c =3/a+b+c

Answer 1

it is simply Prove Prove that in triangle abc of angles is 60 degree then it is sure to prove that all the angles are of 60 degree because this sum of triangle in an angle is 20 degree so handset is prove that 3 divided by a + b + c equal to a + b + c over one

Answer 2

Try this formula to prove reverse

a^2 = b^2 + c^2 − 2bc Cos A
b^2 = c^2 + a^2 − 2ca Cos B
c^2 = a^2 + b^2 − 2ab Cos C

u know the angle of C = 60

So c^2 = a^2 + b^2 − 2ab cosC
=> c^2 = a^2 + b^2 − 2ab cos 60 => c^2 = a^2 + b^2 − ab

Simplify

1/(a+c) + 1/(b+c) = 3 / ( a+b+c )

=> (a+b+2c) / (a+c) (b+c) = 3 / (a+b+c)

=> (a+b+2c) (a+b+c) = 3(a+c) (b+c)

=> a^2 + b^2 + 2ab + 2c^2 + 3ac + 3bc = 3ab + 3ac + 3bc + 3c^2

=> a^2 + b^2 + 2ab - 3ab = 3 c^2 - 2 c^2 (cancel 3ac & 3bc)


=> a^2 + b^2 – ab = c^2

so both matched hence the proof