You may recall that the point inside a triangle that
lies one-third of the way from each side toward the opposite vertex
is the point where the triangle’s three medians intersect. Show that
the centroid lies at the intersection of the medians by showing that
it too lies one-third of the way from each side toward the opposite
vertex. To do so, take the following steps.
i) Stand one side of the triangle on the x-axis as in part (b) of
the accompanying figure. Express dm in terms of L and dy.
ii) Use similar triangles to show that L = (b/h)(h - y). Substitute
this expression for L in your formula for dm.
iii) Show that y = h/3.
iv) Extend the argument to the other sides.
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