$$A, B, C$$ and $$D,$$ are four points in a plane with position vectors $$a, b, c$$ and $$d$$ respectively such that $$$\left( {\overrightarrow a - \overrightarrow d } \right)\left( {\overrightarrow b - \overrightarrow c } \right) = \left( {\overrightarrow b - \overrightarrow d } \right)\left( {\overrightarrow c - \overrightarrow a } \right) = 0$$$

The point $$D,$$ then, is the ................ of the triangle $$ABC.$$

The points with position vectors $$a+b,$$ $$a-b,$$ and $$a+kb$$ are collinear for all real values of $$k.$$

Four person K, L, M, N are initially at the four corners of a square of side d. Each person now moves with a uniform speed v in such a way that K always moves directly towards L, L directly towards M, M directly towards N, and N directly towards K. The four person will meet at a time _____________.

A projectile fired from the ground follows a parabolic path. The speed of the projectile is minimum at the top of its path.