### Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..

### At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

### Six Discs

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

# Cosines Rule

##### Stage: 4 Challenge Level:

The Cosine Rule for $\Delta APC$ and $\Delta BPC$, where $\angle ACP=\theta$, gives $\begin{eqnarray} AP^2 &= AC^2+PC^2-2AC.PC \cos\theta,\cr PB^2&= BC^2+PC^2-2BC.PC \cos \theta. \end{eqnarray}$

Hence $\begin{eqnarray} \frac{BC^2+PC^2-PB^2}{ 2BC.PC}&= \frac{AC^2+PC^2-AP^2}{ 2AC.PC} = \cos\theta. \end{eqnarray}$
Hence, multiplying both sides by $2PC/AB$, we find that $\begin{eqnarray} {AP^2\over AC.AB} +{PC^2\over AB}\left({AC-BC\over BC.AC}\right) &= {PB^2\over AB.BC} +{AC-BC\over AB}.\end{eqnarray}$ As $AB+BC=AC$, we get the result: $\begin{eqnarray} {AP^2\over AB.AC}+{PC^2\over AC.BC} &= 1 + {PB^2\over AB.BC}. \end{eqnarray}$