if `(x_(1),x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2), (x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=b^(2) (x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=c^(2).` where a,b,c are positive then prove that `4 |{:(x_(1),,y_(1),,1),(x_(2) ,,y_(2),,1),( x_(3),, y_(3),,1):}| = (a+b+c) (b+c-a) (c+a-b)(a+b-c)`

if `(x_(1),x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2), (x_(2)-x_(3))^(2)+(y_(2

1.	if `(x_(1),x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2), (x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=b^(2) (x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=c^(2).` where a,b,c are positive then prove that `4 \|{:(x_(1),,y_(1),,1),(x_(2) ,,y_(2),,1),( x_(3),, y_(3),,1):}\| = (a+b+c) (b+c-a) (c+a-b)(a+b-c)`
Answer» we have `sqrt((x_(1)-x_(2))^(2)+(y_(1)-y_(2))^(2))=a` `sqrt((x^(2)-x_(3))^(2)+(y_(2)-y_(3))^(2))=b` `sqrt((x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2))=c` Consider triangle having vertices `A(x_(1),y_(1)),B(x_(2),y_(2))` and So , `AB=a,BC=b " and "AC= c.Also ,2s=a+b+c` where s is semi-perimetre. Area of triangle ABC `Delta=(1)/(2) \|\|{:(x_(1),,y_(1),,1),(x_(2),,y_(2),,1),(x_(3),,y_(3),,1):}\|\|` Also `Delta= sqrt(s(s-a)(s-b)(s-c))` `:. \|\|{:(x_(1),,y_(1),,1),(x_(2),,y_(2),,1),(x_(3),,y_(3),,1):}\|\|=sqrt(s(s-a)(s-b)(s-c))` Squaring and simplifying we get `4 \|{:(x_(1),,y_(1),,1),(x_(2),,y_(2),,1),(x_(3),,y_(3),,1):}\|^(2)` =`(a+b+c)(b+c-a)(c+a-b)(a+b-c)`