Using properties of determinants, show that ΔABC is an isosceles if : |(1,1,1)(1+cosA,1+cosB,1+cosC)(cos^2A + cosA, cos^2B +cos B,cos^2C + cosC)|=0

Question

Using properties of determinants, show that ΔABC is an isosceles if: 

 

Answer 1

We have,

 

Applying C₁ \(\longrightarrow\)C₁ - C₃ and C₂ \(\longrightarrow\)C₂ - C₃

Taking common (cos A - cos C) from C₁ and (cos B - cos C) from C_2, we get

Applying C₁ \(\longrightarrow\)C₁ - C₂, we get

Expanding along R_1, we get

Hence,  ΔABC is an isosceles triangle.