Six Points with a Fixed Dot Product

★ 7/10plane geometry Circles Casework vectors

Problem

In the figure, square $ABCD$ has side length $6$ . Points $E$ and $F$ lie on sides $AD$ and $BC$ respectively, with $DE = 2EA$ and $CF = 2FB$ . Suppose $\lambda$ is a constant for which there are exactly six distinct points $P$ on the sides of square $ABCD$ satisfying

\overrightarrow{PE} \cdot \overrightarrow{PF} = \lambda.

Find the range of possible values of $\lambda$ .

Answer

Solution

Difficulty	7/10
Topics	plane geometry, Circles, Casework, vectors

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