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Ratios of Distances between Centroid, Circumcenter, Incenter and Orthocenter of Triangle

If a, b, c ar...

Question

If a, b, c are the position vectors of the vertices of an equilateral triangle whose orthocenter is at the origin, then

\to a + \to b + \to c = \to 0

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{\to a}^{2} = {\to b}^{2} + {\to c}^{2}

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\to a + \to b = \to c

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Solution

The correct option is A $\to a + \to b + \to c = \to 0$

The position vector of the centroid of the triangle is $\frac{\to a + \to b + \to c}{3}$
Since, the triangle is an equilateral, therefore the orthocenter coincides with the centroid and hence $\frac{\to a + \to b + \to c}{3} = \to 0 \Rightarrow \to a + \to b + \to c = 0$

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