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Coke Machine

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At a Glance

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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: Challenge Level:3 Challenge Level:3 Challenge Level:3

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: $\begin{eqnarray} {AP^2\over AB.AC}+{PC^2\over AC.BC} &= 1 + {PB^2\over AB.BC}. \end{eqnarray}$