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Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

Gold Again

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.


Explain how to construct a regular pentagon accurately using a straight edge and compass.

Golden Construction

Age 16 to 18 Challenge Level:

Just do part 1 if you like where you will construct a golden rectangle and find an approximate value of the golden ratio by measurement.

If you want to go further there are many hints within this question to take you through it. You will find the value of $\phi$ first by drawing and measurement, then by using geometry and calculating with surds (square roots), then by taking a reading from a graph and finally by rearranging a formula into a quadratic equation and solving it.