I haven't posted a puzzle for a long time. So here's one.
I start with any old triangle. I construct the three red lines shown above by the following:
1. Take one side of the triangle and divide it into thirds.
2. Draw a line from the one-third point to the opposite corner.
3. Repeat for the other sides of the triangle. Make sure to pick the appropriate one-third point so that it resembles the above figure (which is to scale).
Find the ratio of the area of the pink inner triangle to the area of the large black triangle. Prove it.
See the solution