Question

prove that the points (a,b+c) ,(b,c+a) and (c,a+b) arecollinear

Answer 1

Hey
Here's your answer !!

The three points are (a,b+c) ,(b,c+a) and (c,a+b) .

If the points are collinear then the area of the triangle formed by them will have the area 0 .

So , we just need to prove that the triangle that they form have the area as 0.


Area of finding the area of the triangle is ----->

1/2 | x1 (y2 - y3) + x2 ( y3 - y1 ) + x3 ( y1 - y2 ) |

so putting the values in the equation , we get ,

1/2 | a ( c + a - a - b ) + b ( a + b - b - c ) + c ( b + c - c - a ) | .

= 1/2 | a ( c - b ) + b ( a - c ) + c ( b - a ) | .

= 1/2 | ac - ab + ab - ac + bc - ac | .

= 1/2 | 0 | .

= 0 .

Hence , the points (a,b+c) ,(b,c+a) and (c,a+b) are collinear .

Hope it helps !!