VII. CONGRUENT LINES IN A TRIANGLE 55
x
y
Figure 51
Figure 52
and N be the midpoints of BG and CG. Segment MN , which joins the midpoints
of two sides of triangle BCG, is parallel to BC and equal to half of it. But EF is
parallel to BC and equal to half of it. This means that EF MN is a parallelogram,
whose diagonals divide each other in half. Therefore EG = GM = MB and
F G = GN = NC.
Thus median BE passes through the point situated at one third the length of
CF . But the same reasoning can also be applied to show that the median AD
passes through the same point. The thereom is proved.
Remark. The point where the medians meet is also called the center of mass
of the triangle. The reason for this name is given in the study of mechanics.
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