Here we will discuss a proof of the fact that the centroid of a triangle ( The point of concurrency of medians) separates the medians into two segements in a 1:2 ratio:
The proof is quite intuitive once we add segment FD into the picture. The simple observation that the triangles FDG and ABG are similar is all we must have to complete the proof. Segments FD and AB are parallel we can determine that Angles EAG and GFD are congruent along with angles EBG and FDG. Therefore the two triangles are similar, further we know AB is twice the length of FD, and therefore the ratio between the side lengths of the triangles is 1:2.
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