The vertices of triangle ABC are position vectors a, b and c relative to some origin O. Find the position vector of the centroid of triangle ABC relative to the same origin O
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The centroid of a triangle is the point where the three medians of the triangle intersect. A median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. Show more…
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Zhumagali S.
The position vectors $\bar{a}, \bar{b}, \bar{c}$ and $\bar{d}$ of four points $A, B, C, D$ on a plane are such that $\frac{|(\bar{d}-\bar{a}) \times(\bar{b}-\bar{a})|}{|\bar{b}-a|}=\frac{|(\bar{d}-\bar{b}) \times(\bar{c}-\bar{b})|}{|\bar{c}-\vec{b}|}$ $=\frac{\mid(\overline{\mathrm{d}}-\overline{\mathrm{c}}) \times(\overline{\mathrm{a}}-\overline{\mathrm{c})} \mid}{|\overline{\mathrm{a}}-\overline{\mathrm{c}}|} .$ Then $\mathrm{D}$ is the (a) centroid of $\triangle \mathrm{ABC}$ (b) circumcentre of $\triangle \mathrm{ABC}$ (c) incenter of $\triangle \mathrm{ABC}$ (d) orthocenter of $\Delta \mathrm{ABC}$
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