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Method D

TASK M-A M-B1 M-B2 M-C1 M-C2 M-D PS PPS

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