`(i)` Area of a triangle whose vertices are `(x_1, y_1), (x_2, y_2)` and `(x_3, y_3)`, is given by the expression `1/2 [x_1(y_2,y_3) + x_2 (y_3,y_1) + x_3 (y_1,y_2)]`.
Now this expression can be written in the form of a determinant as
`Delta = 1/2 | (x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1) |`
`(ii)` If points` (x_1, y_1), (x_2 , y_2)` and `(x_3 , y_3)` and collinear, then
` | (x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1) | = 0`
`(iii)` If `a_rx + b_ry + c_r = 0; r = 1, 2, 3` are the sides of a triangle, then the area of
the triangle is given by
`Delta = 1/ |2C_1C_2C_3| |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|^2`
where `C_1,C_2` and `C_3` are the cofactors of the elements `c_1 ,c_2` and `c_3` ,
respectively, in the determinant
`|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|`
(iv) Equation of straight line passing through two points `(x_1_, y_1)` and` (x_2 , y_2)` is
`|(x,y,1),(x_1,y_1,1),(x_2,y_2,1)|=0`
(v) If three lines `a_rx + b_ry + c_r = 0; r = 1, 2, 3` are concurrent, then
`|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|=0`
(vi) If `ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0` represents a pair of straight lines,
then
`|(a,h,g),(g,b,f),(g,f,c)|=0`
(vii) Equation of circle through three non-collinear points `(x_1, y_1 ), (x_2 , y_2)` and
`(x_3, y_3 )` is given by
`|(x^2+y^2 , x , y , 1),(x_1^2+y_1^2 , x_1 , y_1 , 1),(x_3^2+y_3^2 , x_3 , y_3 , 1)| = 0 `
Remarks
(i) Since area is a positive quantity, we always take the absolute value of the determinant in (1).
(ii) If area is given, use both positive and negative values of the determinant for calculation.
(iii) The area of the triangle formed by three collinear points is zero.
`(i)` Area of a triangle whose vertices are `(x_1, y_1), (x_2, y_2)` and `(x_3, y_3)`, is given by the expression `1/2 [x_1(y_2,y_3) + x_2 (y_3,y_1) + x_3 (y_1,y_2)]`.
Now this expression can be written in the form of a determinant as
`Delta = 1/2 | (x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1) |`
`(ii)` If points` (x_1, y_1), (x_2 , y_2)` and `(x_3 , y_3)` and collinear, then
` | (x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1) | = 0`
`(iii)` If `a_rx + b_ry + c_r = 0; r = 1, 2, 3` are the sides of a triangle, then the area of
the triangle is given by
`Delta = 1/ |2C_1C_2C_3| |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|^2`
where `C_1,C_2` and `C_3` are the cofactors of the elements `c_1 ,c_2` and `c_3` ,
respectively, in the determinant
`|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|`
(iv) Equation of straight line passing through two points `(x_1_, y_1)` and` (x_2 , y_2)` is
`|(x,y,1),(x_1,y_1,1),(x_2,y_2,1)|=0`
(v) If three lines `a_rx + b_ry + c_r = 0; r = 1, 2, 3` are concurrent, then
`|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|=0`
(vi) If `ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0` represents a pair of straight lines,
then
`|(a,h,g),(g,b,f),(g,f,c)|=0`
(vii) Equation of circle through three non-collinear points `(x_1, y_1 ), (x_2 , y_2)` and
`(x_3, y_3 )` is given by
`|(x^2+y^2 , x , y , 1),(x_1^2+y_1^2 , x_1 , y_1 , 1),(x_3^2+y_3^2 , x_3 , y_3 , 1)| = 0 `
Remarks
(i) Since area is a positive quantity, we always take the absolute value of the determinant in (1).
(ii) If area is given, use both positive and negative values of the determinant for calculation.
(iii) The area of the triangle formed by three collinear points is zero.