A student of mine brought the following question to my attention. I am currently not able to solve it, any help would be appreciated. It should be a simple circle theorem that I have now forgotten.
Question: Let $AB$ be a chord of a circle with centre $O$ and let $P$ be a point on $AB$. We are given that $OP$ has length $4$, $AP$ has length $11$, $BP$ has length $3$. We want to determine the radius of the circle.
Let $M$ be the midpoint of $AB$. $AB=14$ and hence $PM=\dfrac{14}{2}-3=4$, which is equal to $OP$. So, $M$ is the same point as $O$. The radius is $7$.
Alternatively, we can use power of a point.
We have $(r-4)(r+4)=3\times 11$.