Dot Sums in a Disc

★ 7/10plane geometry Extrema Completing the Square vectors

Problem

Points $A, B, C$ lie on a circle $O$ of radius $1$ , with $AB$ a diameter, and $P$ is a point in the closed disc. Find the range of possible values of

\overrightarrow{PA}\cdot\overrightarrow{PB} + \overrightarrow{PB}\cdot\overrightarrow{PC} + \overrightarrow{PC}\cdot\overrightarrow{PA}.

Answer

Solution

Difficulty	7/10
Topics	plane geometry, Extrema, Completing the Square, vectors

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