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Philippine Mathematical Olympiad

Geometry Difficulty 5.1 AIME, harder Prove it Philippines

Problem:

In triangle ABCA B C, the medians ADA D and BEB E meet at the centroid GG. Determine the ratio of the area of quadrilateral CDGEC D G E to the area of triangle ABCA B C.

Solution

Solution:

Refer to the figure on the right.
{[CDGE]}=[CDE]+[GED]=14[ABC]+13[BED]=14[ABC]+13(14[ABC])=14[ABC]+112[ABC]=13[ABC] \begin{aligned} \{[C D G E] \} & =[C D E]+[G E D] \\ & =\frac{1}{4}[A B C]+\frac{1}{3}[B E D] \\ & =\frac{1}{4}[A B C]+\frac{1}{3}\left(\frac{1}{4}[A B C]\right) \\ & =\frac{1}{4}[A B C]+\frac{1}{12}[A B C] \\ & =\frac{1}{3}[A B C] \end{aligned}

Figure 1

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.