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**Method D**

Here we return to the specific multiplier 1.2. In the diagram, E and F are the positions of P when BP is a minimum and a maximum. Can we prove that the locus of P is a circle with diameter EF?

It is worth stopping the movie at any convenient point (when P is *not* at E or F). Can we show that the angle EPF is 90˚ (and hence, using our circle theorems, that P is on a circle with diameter EF)? One way to proceed is suggested below:

We know that E cuts AB in the ratio AP/BP. What does this tell us about PE and the angle APB? (This involves a theorem explored in GEOdd.) Similarly, F cuts AB *externally* in the ratio AP/BP. What does this tell us about PF and the *exterior* angle BPQ? (This involves an extension of the theorem explored in GEOdd.) Can we combine these results to show that angle EPF = 90˚?

Finally, we consider an algebraic approach: PS