# If P is orthocentre, Q is the circumcentre and G is the centroid of a triangle ABC, then prove that QPQGQP¯=3QG¯. - Mathematics and Statistics

Sum

If P is orthocentre, Q is the circumcentre and G is the centroid of a triangle ABC, then prove that bar"QP" = 3bar"QG".

#### Solution

Let bar"p" and bar"g" be the position vectors of P and G w.r.t. the circumcentre Q.

i.e. bar"QR" = bar"p" and bar"QG" = bar"g"

We know that Q, G, P are collinear and G divides segment QP internally in the ratio 1 : 2.

∴ by section formula for internal division,

bar"g" = (1.bar"p" + 2bar"q")/(1 + 2) = bar"p"/3       .....[∵ bar"q" = 0]

∴ bar"p" = 3bar"g"

∴ bar"QP" = 3bar"QG".

Concept: Vectors and Their Types
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