Area of a Triangle

The incentre of the triangle formed by $x=0, y=0$ and $3x+4y=12$ is at ...........

The mid points of the sides of a triangle are $\left(5,0\right),\left(5,12\right)$ and $\left(0,12\right)$. Then orthocentre of this triangle is

$A(a,b),B\left(x_1,y_1\right)$ and $C\left(x_2,y_2\right)$ are the vertices of a triangle. If $a,x_1,x_2$ are in $\mathrm{GP}$ with common ratio $r$ and $b,y_1,y_2$ are in $\mathrm{GP}$ with common ratio $s,$ then area of $∆ABC$ is :

An equilateral triangle has each side equal to $a$. If the co-ordinates of its vertices are $\left(x_1,y_1\right),\left(x_2,y_2\right)$ and $\left(x_3,y_3\right)$ then the square of the determinant $\left|\begin{array}{lll}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$ equals :

If each of the vertices of a triangle has integral co-ordinates then the triangle may be:

If the co-ordinates of the vertices of a triangle are rational numbers, then which of the following points of the triangle will always have rational co-ordinates :

Find the area of the hexagon whose vertices taken in order are $(5,0),(4,2),(1,3),(-2,2),(-3,-1)$ and $(0,-4)$.

The co-ordinates of points $A,B,C$ and $D$ are $(-3,5),(4,-2),(x, 3x)$ and $(6,3)$ respectively and $\frac{\Delta ABC}{\Delta BCD}=\frac{2}{3}$, find $x$

Show that the area of the triangle with vertices $(\lambda ,\lambda -2),(\lambda +3,\lambda )$ and $(\lambda +2,\lambda +2)$ is independent of $\lambda$

$A,B,C$ are the points $(-1,5),(3,1)$ and $(5,7)$ respectively. $D,E,F$ are the middle points of $BC,CA$ and $AB$ respectively. Prove that $∆ABC=4∆DEF$.

If the area of the quadrilateral whose angular points taken in order are $(1,2),(-5,6),(7,-4)$ and $(\lambda ,-2)$ be zero, show that $\lambda -3=0$.

The area of a triangle is $5 \mathrm{sq}.\mathrm{units}$. Two of its vertices are $(2,1)$ and $(3,-2)$ The third Vertex is $(x, y)$ where $y=x+3.$ Find the coordinates of the third vertex.

If ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\beta }_1,{\beta }_2,{\beta }_3$ are the values of $n$ for which $\sum _{r=0}^{n-1}{x}^{2r}$ is divisible by $\sum _{r=0}^{n-1}{x}^r$ then prove that the triangle having vertices $\left({\alpha }_1,{\beta }_1\right),\left({\alpha }_2,{\beta }_2\right)$ and $\left({\alpha }_3,{\beta }_3\right)$ cannot be an equilateral triangle.

Let $\alpha =\underset{m\to \infty }{\lim }\underset{n\to \infty }{\lim }{\cos }^{2m}\left(n!\pi x\right)$, where $x$ is rational, $\beta =\underset{m\to \infty }{\lim }\underset{n\to \infty }{\lim }{\cos }^{2m}(n!\pi x)$, where $X$ is irrational then find the area of the triangle having vertices $(\alpha ,\beta ),(-2,1)$ and $(2,1)$.

If the area of the triangle whose vertices are $(b, c),(c, a)$ and $(a, b)$ is $\Delta$ , then find the area of triangle whose vertices are
$\left(ac-b^2,ab-c^2\right),\left(ba-c^2,bc-a^2\right)\&\left(cb-a^2,ca-b^2\right)$

The points with the co-ordinates $\left(2a,3a\right),\left(3b,2b\right)$ and $\left(c,c\right)$ are collinear $\left(a,b,c\ne 0\right)$:

Show that the area of a triangle is four times the area of the triangle formed by joining the middle points of the sides of the former.

If $x_1,x_2,x_3$ as well as $y_1,y_2,y_3$ are in $\mathrm{GP}$ with same common ratio, then the points $\left(x_1,y_1\right),\left(x_2,y_2\right)$ and $\left(x_3,y_3\right)$

If $a,b,c$ are distinct real numbers, show that the points $\left(a,a^2\right),\left(b,b^2\right)$ and $\left(c,c^2\right)$ are not collinear.

If $x_1,x_2,x_3$ are in $A.P.$ and $y_1,y_2,y_3$ are also in $A.P.$, prove that the points$\left(x_1,y_1\right),\left(x_2,y_2\right)$ $\left(x_3,y_3\right)$ are collinear.