06/06/2020 (2011: National Team Round, Problem 9)

Q: In triangle ABC, point D is on segment AC with AD: DC = 2:1, and point E is on segment AB with AE: EB = 2:3. Segments EC and DB intersect at point K. What is DK: KB? Express your answer as a common fraction.

A: 2/9

We should draw a picture, because a) none is given, and b) this is geometry. It would look something like this:

So there's this cool 'trick,' which helps you find out what the ratio between one length in a triangle and another length in a triangle is, and it's shown below:

So we apply this to our situation and we get:

We get DK: KB = 2:9 = 2/9.