Special Points in a Triangle

The points $\mathrm{A}(–1, 4), \mathrm{B}(5, 2)$ are the vertices of a triangle of which$\mathrm{O}(0,–3)$ is centroid, then the third vertex C is_____.

Let $(\alpha ,\beta )$ be the centroid of the triangle formed by the lines $15x-y=82, 6x-5y=-4$ and $9x+4y=17$. Then $\alpha +2\beta$ and $2\alpha -\beta$ are the roots of the equation

If $\left(\alpha , \beta \right)$ is the orthocenter of the triangle $ABC$ with vertices $A\left(3, –7\right), B\left(–1, 2\right)$ and $C\left(4, 5\right)$, then $9\alpha -6\beta +60$ is equal to

Let $C(\alpha , \beta )$ be the circumcentre of the triangle formed by the lines $4x+3y=69$, $4y-3x=17$, and $x+7y=61$. Then $(\alpha -\beta {)}^2+\alpha +\beta$ is equal to

The orthocentre of the triangle having vertices $A\left(1,2\right), B\left(3,-4\right)$ and $C\left(0,6\right)$ is

The equations of perpendicular bisectors of sides $AB$ and $AC$ of a triangle $ABC$ are $x+y+1=0$ and $x-y+1=0$ respectively. If circumradius of $∆ABC$ is $2$ units and locus of vertex $A$ is $x^2+y^2+gx+c=0$, then $\left(g^2+c^2\right)$ is equal to

Let in triangle $ABC$, $A=45°, B=60°, C=75°$ then the ratio in which the orthocentre divides the altitude $AD$ is

The circumcentre of the triangle formed by the points $\left(a\cos \alpha ,a\sin \alpha \right), \left(a\cos \beta ,a\sin \beta \right) \& \left(a\cos \gamma ,a\sin \gamma \right)$ is 

A triangle $ABC$ is formed by the lines $x+y+2=0, x-2y+5=0 \& 7x+y-10=0$. $P$ is a point inside the triangle $ABC$ such that area of the triangle $PAB, PBC \& PCA$ are equal. If the co-ordinates of the point $P$ are $\left(a,b\right)$ and the area of the triangle $ABC$ is $\delta$, then find $\left(a+b+\delta \right)$.

Find the orthocentre of triangle with vertices $\left(3,4\right),\left(-4,-3\right)$ and $\left(5\cos \theta , -5\sin \theta \right).$

The centroid of the triangle formed by the lines $x-3y+3=0,x+3y+3=0,x+y-1=0$ is

The base of a triangle is the axis of $x$ and its other two sides are given by the equation $y=\left(\frac{1+\alpha }{\alpha }\right)x+\left(1+\alpha \right)$ and $y=\left(\frac{1+\beta }{\beta }\right)x+\left(1+\beta \right)$. Locus of its orthocenter is

Suppose $∆ABC$ is an isosceles triangle with $\angle C=90°,A=\left(2,3\right)$ and $B=\left(4,5\right)$. Then the centroid of the triangle is

Let the equation $x^3+y^3+3xy=1$ represents the coordinates of one vertex $A$ and the equation of side $BC$ of the triangle $ABC$. If $B$ is the orthocentre of the triangle $ABC$, then the equation of side $AB$ is $y=mx+c$. Then absolute value ' of $\left(4-m-c\right)$ is

A right triangle has sides '$a$' and '$b$' where $a>b$. If the right angle is bisected then find the distance between orthocentres of the smaller triangles using coordinate geometry.

Number of right isosceles triangles that can be formed with points lying on the curve $8x^3+y^3+6xy=1$ is

The centroid of a triangle whose vertices are $\left(-8,4\right), \left(P,6\right)$ and $\left(-3,9\right)$ is $\left(\frac{-17}{3},\frac{19}{3}\right)$, then find the value of $P$.

Centroid of a triangle divides its median in the ratio

If the orthocentre and circumcentre of a triangle are $\left(0,0\right) \& \left(3,6\right)$ respectively then the centroid of the triangle is