What is the ratio of the corresponding side lengths between triangle ACB and the smaller triangles?
The examination of the triangle centers of any triangle and its medial triangle reveals some interesting relationships. For a quick review of the four triangle centers, Click Here. Below are some conjectures regarding the centers of medial triangles. Conjecture #1: The centroid of a triangle is also the centroid of its medial triangle. To begin considering this hypothesis, examine the figure below. It certainly appears that this is the case. Since appearances can often be deceiving in geometry, a proof of this conjecture is offered below. Given: Triangle DEF is the medial triangle of Triangle ABC; N is the centroid of Triangle ABC. Prove: N is the centroid of Triangle DEF. If N is the centroid of Triangle DEF, then Segements EL, DM, and FK are medians of Triangle DEF. If this is the case, then K, L, and M must be the midpoints of Segments DE, DF, and EF, respectively. By verifying each of these midpoints, the proof will be complete. In order for L to be the midpoint of Segement DF, Seg
