Construct a triangle and its medians. Construct a second triangle with the three sides having the lenths of the three medians from your first triangle. Find some relationship between the two triangles.
Next, use the lengths of the three medians to construct the
sides of a second triangle.
Remark It will suffice to examine only one triangle drawn using the lengths of the medians of triangle ABC since all such triangles are congruent. We know this because SSS = SSS or because the ratio of the areas is one.
The chart is set up to compare what had appeared to be the
corresonding sides and angles. Now we can easily see that the
two triangles are not congruent nor are they similar.
One way that we know that the triangles are not congruent is by
noticing that none of the measurements of the sides nor of the
angles are equal. Another way is to compare the areas. As stated
earlier, the ratio of the areas of two congruent triangles is
one. Well, Area GJL / Area BAC = 0.75 (close, but not congruent).