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Question Number 216270 by MATHEMATICSAM last updated on 02/Feb/25

If (b − c)x + (c − a)y + (a − b)z = 0 then prove that ((b − c)/(y − z)) = ((c − a)/(z − x)) = ((a − b)/(x − y)) .

If(b−c)x+(c−a)y+(a−b)z=0thenprovethatb−cy−z=c−az−x=a−bx−y.

Answered by Rasheed.Sindhi last updated on 02/Feb/25

If (b − c)x + (c − a)y + (a − b)z = 0 then prove that ((b − c)/(y − z)) = ((c − a)/(z − x)) = ((a − b)/(x − y)) . (b − c)x + (c − a)y + (a − b)z = 0 Dividing by a−b (( (b − c)x + (c − a)y)/(a−b))+z=0; a≠b (( bx − cx + cy − ay)/(a−b))+x+z=x (( bx − cx + cy − ay+ax−bx)/(a−b))+z=x (( − cx + cy − ay+ax)/(a−b))+z=x z−x=(( cx − cy + ay−ax)/(a−b)) z−x=(( (x − y) (c− a))/(a−b)) ((z−x)/(c−a))=(( x − y)/(a−b)) ((c−a)/(z−x))=((a−b)/(x−y)) Similarly, ((b − c)/(y − z)) = ((c − a)/(z − x)) And finally, ((b − c)/(y − z)) = ((c − a)/(z − x)) = ((a − b)/(x − y))

If(b−c)x+(c−a)y+(a−b)z=0thenprovethatb−cy−z=c−az−x=a−bx−y.(b−c)x+(c−a)y+(a−b)z=0Dividingbya−b(b−c)x+(c−a)ya−b+z=0;a≠bbx−cx+cy−aya−b+x+z=xbx−cx+cy−ay+ax−bxa−b+z=x−cx+cy−ay+axa−b+z=xz−x=cx−cy+ay−axa−bz−x=(x−y)(c−a)a−bz−xc−a=x−ya−bc−az−x=a−bx−ySimilarly,b−cy−z=c−az−xAndfinally,b−cy−z=c−az−x=a−bx−y

Answered by Rasheed.Sindhi last updated on 02/Feb/25

(b − c)x + (c − a)y + (a − b)z = 0 prove that ((b − c)/(y − z)) = ((c − a)/(z − x)) = ((a − b)/(x − y)) (b − c)x + (c − a)y + (a − b)z = 0 ⇒z=((−(b − c)x − (c − a)y)/(a−b)) z−x=((−bx + cx −cy + ay)/(a−b))−x =(( −bx+cx −cy + ay−ax+bx)/(a−b)) =(( cx −cy + ay−ax)/(a−b)) =((c(x−y)−a(x−y))/(a−b)) =(((c−a)(x−y))/(a−b)) ((z−x)/(c−a))=((x−y)/(a−b)) ((c−a)/(z−x))=((a−b)/(x−y)) Similarly, ((b − c)/(y − z)) = ((c − a)/(z − x)) and finally, ((b − c)/(y − z)) = ((c − a)/(z − x)) = ((a − b)/(x − y))

(b−c)x+(c−a)y+(a−b)z=0provethatb−cy−z=c−az−x=a−bx−y(b−c)x+(c−a)y+(a−b)z=0⇒z=−(b−c)x−(c−a)ya−bz−x=−bx+cx−cy+aya−b−x=−bx+cx−cy+ay−ax+bxa−b=cx−cy+ay−axa−b=c(x−y)−a(x−y)a−b=(c−a)(x−y)a−bz−xc−a=x−ya−bc−az−x=a−bx−ySimilarly,b−cy−z=c−az−xandfinally,b−cy−z=c−az−x=a−bx−y

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