A triangle has vertices `A_i(x_i , y_i)fori=1,2,3` If the orthocentre 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_1y_2-y_3y_2(y_3-y_1)+x_1(x_3-x_1)`
`x_1-x_2y_2-y_3y_3(y_1-y_2)+x_1(x_1-x_2)|=0`

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

If the join of (x_1,y_1) and (x_2,y_2) makes on obtuse angle at (x_3,y_3), then prove that (x_3-x_1)(x_3-x_2)+(y_3-y_1)(y_3-y_2)<0

If y=(a x+b)/(x^2+c), prove that (2x y_1+y)y_3=3(x y_2+y_1)y_2.

The value of |[2x_1y_1, x_1y_2+x_2y_1, x_1y_3+x_3y_1], [x_1y_2+x_2y_1, 2x_2y_2, x_2y_3+x_3y_2], [x_1y_3+x_3y_1, x_2y_3+x_3y_2, 2x_3y_3]| is.

If the tangent at (x_1,y_1) to the curve x^3+y^3=a^3 meets the curve again in (x_2,y_2), then prove that (x_2)/(x_1)+(y_2)/(y_1)=-1

If the tangent at (x_1,y_1) to the curve x^3+y^3=a^3 meets the curve again in (x_2,y_2), then prove that (x_2)/(x_1)+(y_2)/(y_1)=-1

If three points (x_1,y_1),(x_2, y_2),(x_3, y_3) lie on the same line, prove that (y_2-y_3)/(x_2x_3)+(y_3-y_1)/(x_3x_1)+(y_1-y_2)/(x_1x_2)=0

If the points (x_1, y_1),(x_2,y_2), and (x_3, y_3) are collinear show that (y_2-y_3)/(x_2x_3)+(y_3-y_1)/(x_3x_1)+(y_1-y_2)/(x_1x_2)=0

If (x_i ,y_i),i=1,2,3, are the vertices of an equilateral triangle such that (x_1+2)^2+(y_1-3)^2=(x_2+2)^2+(y_2-3)^2=(x_3+2)^2+(y_3-3)^2, then find the value of (x_1+x_2+x_3)/(y_1+y_2+y_3) .

If (x_i,y_i),i=1,2,3 are the vertices of an equilateral triangle such that (x_1+2)^2+(y_1-3)^2=(x_2+2)^2+(y_2-3)^2=(x_3+2)^2+(y_3-3)^2 , then find the value of (x_1+x_2+x_3)/(y_1+y_2+y_3)