Special Points in Triangles

A variable line cuts the line $2y=x-2$ and $2y=-x+2$ in points $A$ and $B$ respectively. If $A$ lies in first quadrant, $B$ lies in $4^{\mathrm{th}}$ quadrant and area of $∆AOB$ is $4$ sq. units, then find locus of centroid of $∆AOB$

The locus of circumcentre of the triangle formed by vertices $A\left(\left(-pq-p-q\right),-\left(1+p\right)\left(1+q\right)\right), B\left(pq+p-q,\left(1+p\right)\left(1+q\right)\right), C\left(pq+q-p,\left(1+p\right)\left(1+q\right)\right)$ is

The $x$ -coordinates of the incentre of the triangle that has the coordinates of midpoints of its sides as$(0,1),$ $(1,1)$ and $(1,0)$ is

Let the orthocentre and centroid of a triangle be $A(-3,5)$ and $B(3,3)$. respectively. If $C$ is the circumcentre of this triangle, then the radius of the circle having line segment $AC$ as diameter, is

The number of complex numbers $z$ such that $|z-1|=|z+1|=|z-i|$ equals

The equations of perpendicular of the sides $AB \text{and} AC$ of triangle $ABC$ are $x-y-4=0$ and $2x-y-5=0$ respectively. If the vertex $A$ is $(-2,3)$ and circumcentre is $\left(\frac{3}{2},\frac{5}{2}\right),$ then which of the following is true.

Find locus of centroid of $\Delta ABC,$ if $B(1, 1), C(4, 2)$ and $A$ lies on the line $y=x+3$

A triangle $ABC$ with vertices  $A\left(-1,0\right), B\left(-2,\frac{3}{4}\right)$ and $C\left(-3,-\frac{7}{6}\right)$ has its orthocenter at $H$. Then, the orthocenter of triangle$△BCH$ will be

The line $L_1\equiv 4x+3y-12=0$ intersects the $x$ and the $y$-axis at $A$ and $B$, respectively. A variable line perpendicular to $L_1$ intersects the $x$ and the $y$-axis at $P$ and $Q$, respectively. Find the locus of the circumcentre of triangle $ABQ$.

The sides of a triangle are $L_r\equiv x\cos {\alpha }_r+y\sin {\alpha }_r-p_r=0$ for $r=1, 2, 3$. Show that its orthocentre is given by $L_1 \cos \left({\alpha }_2-{\alpha }_3\right)=L_2 \cos \left({\alpha }_3-{\alpha }_1\right)=L_3 \cos \left({\alpha }_1-{\alpha }_2\right)$.

Prove that the circumcentre, orthocentre, incentre & centroid of the triangle formed by the points $A(-1,11) ; B(-9,-8) ; C(15,-2)$ are collinear, without actually finding any of them.

Find the locus of the circumcentre of a triangle whose two sides are along the coordinate axes and third side passes through the point of intersection of the lines $ax+by+c=0$ and $lx+my+n=0$.

Find the locus of the centroid of a triangle whose vertices are $(a \cos t, a \sin t), (b \sin t, -b\cos t)$ and $\left(1, 0\right)$ where $\text{'} t \text{'}$ is the parameter.

For triangle whose vertices are $(0, 0), (5, 12)$ and $(16,12)$. Find coordinates of
$\left(i\right)$ Centroid
$\left(ii\right)$ Circumcentre
$\left(iii\right)$ Incentre 
$\left(iv\right)$ Excentre opposite to vertex $(5,12)$

Lines $L_1:y-x=0$ and $L_2:2x+y=0$ intersect the line $L_3:y+2=0$ at $P$ and $Q$ respectively. The bisector of the acute angle between $L_1$ and $L_2$ intersects $L_3$ at $R$.

Statement $\mathrm{I} :$The ratio $PR:RQ$ equals $2\sqrt{2}:\sqrt{5}$. 

Statement $\mathrm{II} :$ In any triangle, bisector of an angle divides the triangle into two similar triangles.

Find the locus of the middle points of chords of the circle $x^2+y^2=a^2$ which subtend a right angle at the point $\left(c,0\right).$

The orthocentre of the triangle $ABC$ is '$B$' and the circumcentre is '$S$' $\left(a, b\right)$. If $A$ is the origin then the co-ordinates of $C$ are:

A variable straight line passes through a fixed point $\left(a,b\right)$ intersecting the coordinate axes at $A$ and $B$. If $O$ is the origin, then locus of centroid of triangle $OAB$ is:

Let $O\left(0,0\right),P\left(3,4\right),Q\left(6,0\right)$ be the vertices of a triangle $OPQ.$ The point $R$ inside the triangle $OPQ$ is such that the triangles $OPR, PQR, OQR$ are of equal area. The coordinates of $R$ are