Why do this problem?
In this problem, constructing some lines on a triangle divides the area in a surprising way. We hope that by trying some examples and calculating the areas, students will be intrigued by the result they notice, and curious to prove whether it will always work. Along the way, they will have plenty of opportunity to work with similar triangles, ratios and enlargements.
Possible approach
Begin by showing students a rightangled triangle such as the one below, with points marked $\frac13$ of the way along each side and joined to make a new triangle.
Invite them to draw some triangles, perhaps on squared paper, and calculate the areas. Once they have created some examples, bring the class together and share results. Perhaps students could add their results to a table on the board:
Area of rightangled triangle 
Area of triangle inside 




"What do you notice?"
"The rightangled triangle's area is always three times the area of the inside triangle"
To prove the result, students might find it useful to add some extra lines to their diagram:
As the triangles are rightangled, students could use similar triangles to explain why the small triangles at A, B and C are each $\frac19$ of the area of the whole triangle. Students can then explore the triangles with the same length base that share an apex; this gives us another three triangles with area $\frac19$. Students could be encouraged to write up this argument as a proof.
Alternatively, students who are familiar with vectors may like to express the lengths in the diagram in terms of the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$, and then calculate the relevant areas.
Key questions
Which areas are easiest to calculate?
Can we use these areas to work out the shaded area?
Are there any extra points or lines we could add?
Possible extension
The problem uses a rightangled triangle for ease of area calculation, but the result is true for any triangle. For some students, it might be appropriate to skip straight to thinking about this general case:
Students could then move on to
Areas and Ratios, and
Another Triangle in a Triangle.
Possible support
Students could investigate
Two Ladders to explore similar triangles.