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(D) Given that $(\overline{a}-\overline{d}) \cdot(\overline{b}-\overline{c})=0$ and $(\overline{b}-\overline{d}) \cdot(\overline{c}-\overline{a})=0$.

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orthocentre

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(D) Given that $(\overline{a}-\overline{d}) \cdot(\overline{b}-\overline{c})=0$ and $(\overline{b}-\overline{d}) \cdot(\overline{c}-\overline{a})=0$. This can be written in terms of vectors as $\overline{AD} \cdot \overline{BC} = 0$ and $\overline{BD} \cdot \overline{CA} = 0$. This implies that $\overline{AD} \perp \overline{BC}$ and $\overline{BD} \perp \overline{CA}$. Since $D$ is the point of intersection of the altitudes $AD$ and $BD$ of $	riangle ABC$,$D$ is the orthocentre of $	riangle ABC$.

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