Question

Show that points A(a, b + c), B(b, c + a), C(c, a + b) are collinear.

Answer 1

It is known that the area of the triangle formed by points A(a, b + c), B(b, c + a) and C(c, a + b) is given by,

Δ = `1/2 |(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|`

x₁ = a, y₁ = b + c, x₂ = b, y₂ = c + a, x₃ = c, y₃ = a + b

= `1/2 |(a, b + c, 1),(b, c + a,1),(c, a + b, 1)|`  ...(C₁ → C₁ + C₂)

= `1/2 |(a + b + c, b + c, 1),(a + b + c, c + a, 1),(a + b + c, a + b, 1)|`

= `1/2 (a + b + c) |(1, b + c, 1),(1, c + a, 1),(1, a + b, 1)|`

= `1/2 (a + b + c) xx 0`  ...(C₁ and C₂ are the same.)

= 0

Hence, points A, B, C are collinear.