### Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

### From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

### Star Gazing

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

# Triangle in a Triangle

##### Stage: 4 Challenge Level:

Take any triangle. Construct a point on each side so that the side length is divided in the ratio $1:2$

Now form a triangle by joining up these points.

Can you work out the fraction of the original triangle that is covered by this new inner triangle?

You may wish to try the follow-up problem Another Triangle in a Triangle.