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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.

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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.

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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: Challenge Level:2 Challenge Level:2
This is not the typical format for a problem, but the opportunity for geometric reasoning is so rich that to make the problem accessible at this level we have bridged some of the harder parts of the investigation, for example using vectors to establish the partition ratio on the interior lines, and then just given these results as information.

The detail is included in the Hints section for those who are ready for it.