Unformatted text preview: Example 5. Prove that the three medians of a triangle are concurrent. Prooﬁ Name the vertices of a triangle by O, A and B. Let D, E, F be the midpoints of OB, 0A, AB.
Suppose that AD and BE intersect at G. Let (ﬂ = a and (ﬂ = b. Hence B
CE=§a and Cﬁ=%b.
Since G lies on both AD and BE and inside the triangle,
from Exam le 4 there exist real numbers A and a such that D O =(l—A)a+/\(%b)=(l—n)b+u(%a). F By rearranging terms, we get
' ' O A Since a cannot be a non-zero scalar multlple of b, we have (l—A)—%u=0 and (1—,1)_%A=0_ B
By solving the above simultaneous equations, we have A = g ' 2 —G>_F1gure 10.
and it: So we have 0 —(a + 1)) Since F is the midpoint of AB, so (ﬂ: (a + 1)) Thus (ﬂ and (W are in the same direction.
Hence G lies on OF and therefore the three medians of the triangle OAB are concurrent. El Many calculations with vector quantities in the plane can be done geometrically using scale ...
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