$A, B, C, D$ are four points in a plane with position vectors $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ r

Question

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

Professor PC Thomas Classes · Answer

Given that\ $(\vec{a}-\vec{d}) \cdot(\vec{b}-\vec{c})=(\vec{b}-\vec{d}) \cdot(\vec{c}-\vec{a})=0$\ which implies that $A D \perp B C, B D \perp C A$.\ $	herefore \quad D$ is the orthocentre of triangle $A B C$.\