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Kissing

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?

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Gold Again

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

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Golden Construction

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

Pentabuild

Age 16 to 18 Challenge Level:
The angles are all multiples of $36^o$ and the figure has many symmetries. Form a quadratic equation by equating ratios of corresponding sides in the similar triangles.

In explaining the construction look for $\sqrt 5$ appearing from triangle $YOS$ and the golden ratio appearing as the radius of circle $C_5$ (see Note). Concentrate on cosines which, in this case, are easier than the sines to find exactly.