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

If the midpoints of the sides of a triangle are $(5, 0)$,$(5, 12)$,and $(0, 12)$,then what is the orthocenter of this triangle?

Let $ABC$ be an equilateral triangle with orthocenter at the origin and the side $BC$ on the line $x+2\sqrt{2}y=4$. If the coordinates of the vertex $A$ are $(\alpha, \beta)$, then the greatest integer less than or equal to $|\alpha+\sqrt{2}\beta|$ is

In $\triangle ABC$,if $A=(1,2)$ and the equations of the medians through $B$ and $C$ are $x+y=5$ and $x=4$ respectively,then the area of $\triangle ABC$ is

The orthocentre of the triangle with vertices $(0, 0)$,$(3, 4)$,and $(4, 0)$ is

Assertion $(A)$: If the centroid and circumcenter of a triangle are known,its orthocenter can be found.
Reason $(R)$: The centroid,orthocenter,and circumcenter of a triangle are collinear.