Question 1149672: Medians AD, BE, and CF of triangle ABC meet at G, EF intersects AD at H, and AD=18. Find GH.
Diagram: https://imgur.com/5pOXsWz
Answer by ikleyn(52787)   (Show Source):
You can put this solution on YOUR website! .
Segment FE is the mid line in triangle ABC; therefore, line FE is parallel to BC.
Hence, triangles ABC and AFE are similar with the similarity coefficient 2 (from larger to smaller).
From it, you easily deduce that the length of AG is half of the length of AD; thus the length of AG is 18/2 = 9.
The intersection point divides the medians in proportion 2:1; therefore, the length of GD is 1/3 of the length AD;
thus the length of GD is 6 units.
It implies that the length of GF is 9-6 = 3 units. ANSWER
Solved.
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