Tag: area of triangle and collinearity of three points

What is the area of the triangle formed by the points $(a,c+a), \displaystyle \left ( a^{2},c^{2} \right )$ and $(-a, c-a)$?

What is the area of the triangle formed by the points $(a,b+c), (b,c+a)$ and $(c,a+b)$?

The area of a triangle whose vertices are (-2,-2), (-1,-3) and (p,0) is 3 sq.units what is the value of p?

The area of a triangle, whose vertices are $(3, 2), (5, 2)$ and the point of intersection of the lines $x = a$ and $y = 5$, is $3$ square units. What is the value of $a$?

If $P=(x _{1}, y _{1}), Q=(x _{2}, y _{2})$ and $R=(x _{3}, y _{3})$ are three points of a triangle in $\mathbb{R}^{2}$. Then, area of a $\triangle PQR$ in terms of determinant of matrix $M=\begin{bmatrix} 1& 1 & 1 \ x _{1} & x _{2} & x _{3} \ y _{1} & y _{2} & y _{3}\end{bmatrix}$ is

If $\triangle _1,\triangle _2$ be the areas of two triangles with vertices $(b,c), (c,a), (a,b)$, and $ (ac-b^2, ab-c^2),(ba-c^2, bc-a^2), (cb-a^2, ca-b^2)$, then $\ \dfrac{\triangle _1}{\triangle _2}=(a+b+c)^2$

If ${ \left( { x } _{ 1 }-{ { x } _{ 2 } } \right)  }^{ 2 }+{ \left( { y } _{ 1 }-{ y } _{ 2 } \right)  }^{ 2 }={ a }^{ 2 }$, ${ \left( x _{ 2 }-{ x } _{ 3 } \right)  }^{ 2 }+{ \left( { y } _{ 2 }-{ y } _{ 3 } \right)  }^{ 2 }={ b }^{ 2 }$, ${ \left( { x } _{ 3 }-{ x } _{ 1 } \right)  }^{ 2 }+{ \left( { y } _{ 3 }-{ y } _{ 1 } \right)  }^{ 2 }={ c }^{ 2 }$ and $k\begin{vmatrix} { x } _{ 1 } & { y } _{ 1 } & 1 \ { x } _{ 2 } & { y } _{ 2 } & 1 \ { x } _{ 3 } & { y } _{ 3 } & 1 \end{vmatrix}=(a+b+c)(b+c-a)(c+a-b)\times (a+b-c)$, then the value of $k$ is

$(x _1 - x _2)^2 + (y _1 - y _2)^2 = a^2$;
$(x _2 - x _3)^2 + (y _2 - y _3)^2 = b^2$;
$(x _3 - x _1)^2 + (y _3 - y _1)^2 = c^2$;
then find $4 \begin{vmatrix}x _1 & y _1 & 1\ x _2 & y _2 & 1\ x _3 & y _3 & 1\end{vmatrix}^2 = $