50 CHAPTER 1 Advanced Euclidean Geometry
Example 1. Show that the medians of 4 ABCare concurrent and the
point of concurrency (the centroid) divides each median in a ratio of
2:1.
Solution. We assign mass 1 to each of the pointsA, B,andC, giving
rise to the following weighted triangle:
The pointG, begin the center of mass, is on the intersection of all three
medians—hence they are concurrent. The second statement is equally
obvious asAG:GD= 2 : 1; similarly for the other ratios.
Example 2. In 4 ABC,Dis the midpoint of[BC]andE is on[AC]
withAE :EC = 1 : 2. LettingG be the intersections of the Cevians
[AD]and[BE], findAG:GDandBG:GE.
Solution. The picture below tells the story:
From the above, one hasAG:GD= 1 : 1, andBG:GE= 3 : 1.