PROVE THAT THE POINTS(a,b,c),(b,c+a)and(c,a+b) are collinear

Asked by Shambhu Nath Tiwary | 15th Nov, 2015, 10:44: PM

Expert Answer:

The pointsare said to be collinear if the area of the triangle formed by these three points (a, b+c), (b, c+a) and (c, a+b) is zero.
 
The space area space of space straight a space triangle space having space co minus ordinates space left parenthesis straight x subscript 1 comma straight y subscript 1 right parenthesis comma space space space left parenthesis straight x subscript 2 comma straight y subscript 2 right parenthesis comma space space space left parenthesis straight x subscript 3 comma straight y subscript 3 right parenthesis space is space given space by 1 half open vertical bar straight x subscript 1 open parentheses straight y subscript 2 minus straight y subscript 3 close parentheses plus straight x subscript 2 open parentheses straight y subscript 3 minus straight y subscript 1 close parentheses plus straight x subscript 3 open parentheses straight y subscript 1 minus straight y subscript 2 close parentheses close vertical bar  the space area space of space the space triangle space formed space by space these space three space points space left parenthesis straight a comma space straight b plus straight c right parenthesis comma space left parenthesis straight b comma space straight c plus straight a right parenthesis space and space left parenthesis straight c comma space straight a plus straight b right parenthesis equals 1 half open vertical bar straight a open parentheses straight c plus straight a minus straight a minus straight b close parentheses plus straight b open parentheses straight a plus straight b minus straight b minus straight c close parentheses plus straight c open parentheses straight b plus straight c minus straight c minus straight a close parentheses close vertical bar equals 1 half open vertical bar straight a open parentheses straight c minus straight b close parentheses plus straight b open parentheses straight a minus straight c close parentheses plus straight c open parentheses straight b minus straight a close parentheses close vertical bar equals 1 half open vertical bar space ac minus ab plus ab minus bc plus cb minus ac close vertical bar equals 0  Hence space points left parenthesis space straight a comma space straight b plus straight c right parenthesis comma space left parenthesis straight b comma space straight c plus straight a right parenthesis space and space left parenthesis straight c comma space straight a plus straight b right parenthesis space are space collinear. 

Answered by Vijaykumar Wani | 16th Nov, 2015, 11:32: AM