A triangle has vertices `A_(i) (x_(i),y_(i))` for i= 1,2,3,. If the orthocenter of triangle is (0,0) then prove that
`|{:(x_(2)-x_(3),,y_(2)-y_(3),,y_(1)(y_(2)-y_(3))+x_(1)(x_(2)-x_(3))),(x_(3)-x_(1) ,,y_(3)-y_(1),,y_(2)(y_(3)-y_(1))+x_(2)(x_(3)-x_(1))),( x_(1)-x_(2),,y_(1)-y_(2),,y_(3)(y_(1)-y_(2))+x_(3)(x_(1)-x_(2))):}|=0`

Explanation

Altitudes of triangle are concurrent at `H(0,0)` Then `A_(1) H bot A_(2)A_(3)`
`rArr (y_(1)-0)/(x_(1)-0).(y_(2)-y_(3))/(x^(2)-x_(3))=-1`
`" or " y_(1)(y_(2)-y_(3))+x_(1)(x_(2)-x_(3))=0`
Similarly `A_(2)H bot A_(3)A_(1) " and " A_(3) H bot A_(1)A_(2)`
`rArr y_(2)(y_(3)-y_(1)) +x_(2)(x_(3)-x_(1))=0`
`" and " y_(3)(y_(1)-y_(2))+x_(3)(x_(1)-x_(2))=0`
`Dela =|{:(x_(2)-x_(3),,y_(2)-y_(3),,y_(1)(y_(2)-y_(3))+x_(1)(x_(2)-x_(3))),(x_(3)-x_(1) ,,y_(3)-y_(1),,y_(2)(y_(3)-y_(1))+x_(2)(x_(3)-x_(1))),( x_(1)-x_(2),,y_(1)-y_(2),,y_(3)(y_(1)-y_(2))+x_(3)(x_(1)-x_(2))):}|`
`= |{:(x_(2)-x_(3),,y_(2)-y_(3),,0),(x_(3)-x_(1),,y_(3)-y_(1),,0),(x_(1)-x_(2),,y_(1)-y_(2),,0):}|=0`