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

Let ABC be a ...

Question

Let $A B C$ be a triangle whose centroid is $G$ , orthocentre is $H$ and circumcentre is the origin $O$ . If $D$ is any point in the plane of the triangle such that no three of $O, A, C$ and $D$ are collinear satisfying the relation $- - \to A D + - - \to B D + - - \to C H + 3 - - \to H G = λ - - \to H D$ , then the value of the scalar $λ$ is

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Solution

$L.H.S. = \to d - \to a + \to d - \to b + \to h - \to c + 3 (\to g - \to h)$
$= 2 \to d - (\to a + \to b + \to c) + 3 \frac{(\to a + \to b + \to c)}{3} - 2 \to h$
$= 2 \to d - 2 \to h = 2 (\to d - \to h) = 2 - - \to H D$
$\Rightarrow λ = 2$

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