combinepdf-min-0393.pdf - Example 5 Prove that the three...

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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 (1—A)a—%aa=(1—a)b—%Ab. ‘ ' ' 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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