Question

$A , B, C$ and $D$, are four points in a plane with position vectors $\overrightarrow{a},\overrightarrow{b}, \overrightarrow{c} $ and $\overrightarrow{d}$ respectively such that $(\overrightarrow{a}-\overrightarrow{d}).(\overrightarrow{b}-\overrightarrow{c})=(\overrightarrow{b}-\overrightarrow{d}).(\overrightarrow{c}-\overrightarrow{a})=0$ The point $D$, then, is the... of the $\Delta ABC$.

Answer 1

$(\overrightarrow{a}-\overrightarrow{d})\cdot (\overrightarrow{b}-\overrightarrow{c})=(\overrightarrow{b}-\overrightarrow{d}) \cdot (\overrightarrow{c}-\overrightarrow{a})=0$
$\Rightarrow AD\perp BC$ and $BD \perp CA$
which clearly represents from figure that $D$ is
orthocentre of $\Delta ABC$