Vectors, anyone?

Question

There's a hexagon ABCDEF with midpoints MNOPQR on their respective sides. Prove that both triangle MQO and triangle RNP have the same centroid. Nothing mentioned bout the hexagon being a regular one or not.
Your help is greatly appreciated!

Answer 1

The centroid of a triangle is the same as the centroid of particles of equal mass placed at each vertex of the triangle.

Consider particles of equal mass placed at M, Q and O.
The one at M can be replaced by equal particles of half its mass at A and B.
The one at Q can similarly be replaced by particles at E and F.
The one at O can similarly be replaced by particles at C and D.
The centroid of these replacement particles is the centroid of the hexagon.

The same argument applies in respect of particles placed at R, N and P.

The centroids of the triangles MQO and RNP are therefore both the centroid of the hexagon.

Answer 2

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Answer 3

Assume the Hexagon to be regular and take the incircle (which will pass through the midpoints).