If the vertices of a triangle are $(a, 1), (b, 3),$ and $(4, c),$ then the centroid of the triangle will lie on the $x$-axis if

The centroid of a triangle is $(2, 3)$ and two of its vertices are $(5, 6)$ and $(-1, 4)$. Find the third vertex of the triangle.

Two vertices of a triangle are $(5, -1)$ and $(-2, 3)$. If the origin $(0, 0)$ is the orthocentre of this triangle,then the coordinates of the third vertex of that triangle are

The incentre of the triangle formed by the straight lines $y=\sqrt{3}x$,$y=-\sqrt{3}x$ and $y=3$ is

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

The orthocentre of the triangle with vertices $\left( 2, \frac{\sqrt{3} - 1}{2} \right)$,$\left( \frac{1}{2}, -\frac{1}{2} \right)$,and $\left( 2, -\frac{1}{2} \right)$ is: