How does the solution get BP = 6? Should it not be 14 since AP = 4 and AB = 10, where A is closer to P than B?
Geom IIA 8.18
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To answer your question, the point $P$ is inside the circle for this problem, so it is in between $A$ and $B$. So if $AB=10$, and $AP=4$, we get $BP=6$.