Question

Prove that points `P(a,b+c),Q(b,c+a)` and `R(c,a+b)` lie on the same line?

Answer 1

Please see below.

Explanation:

The three points `P(x_1,y_1)`, `Q(x_2,y_2)` and `R(x_3,y_3)` are collinear if slope of `PQ` and that of `QR` is equal.

We are given the points as `P(a,b+c),Q(b,c+a)`and `R(c,a+b)`

Slope of `PQ` is `(c+a-(b+c))/(b-a)=(a-b)/(b-a)=-1`

and slope of `QR` is `(a+b-(c+a))/(c-b)=(b-c)/(c-b)=-1`

As the slopes are equal, `P,Q` and `R` are collinear.