判断点是否在矩形内

如图判断点$P$P是否在矩形$P_{1}P_{2}P_{3}P_{4}$P1​P2​P3​P4​内部？

从上图可以看出：
点 ${P}$P 位于矩形内部 $\Leftrightarrow$⇔ $\left\{\begin{aligned} \measuredangle PP_{1}P_{4}\leq90\\ \measuredangle PP_{1}P_{2}\leq90\\ \measuredangle PP_{3}P_{4}\leq90 \\ \measuredangle PP_{3}P_{2}\leq90 \end{aligned}\right.$⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧​∡PP1​P4​≤90∡PP1​P2​≤90∡PP3​P4​≤90∡PP3​P2​≤90​ $\Leftrightarrow$⇔ $\left\{\begin{aligned} \vec{P_{1}P} \cdot \vec{P_{1}P_{4}} \geq 0\\ \vec{P_{1}P} \cdot \vec{P_{1}P_{2}} \geq 0\\ \vec{P_{3}P} \cdot \vec{P_{3}P_{4}} \geq 0\\ \vec{P_{3}P} \cdot \vec{P_{3}P_{2}} \geq 0\\ \end{aligned}\right.$⎩⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎧​P1​P​⋅P1​P4​​≥0P1​P​⋅P1​P2​​≥0P3​P​⋅P3​P4​​≥0P3​P​⋅P3​P2​​≥0​
所以，点$P$P在矩形$P_{1}P_{2}P_{3}P_{4}$P1​P2​P3​P4​内部的条件为：

$\vec{P_{1}P} \cdot \vec{P_{1}P_{4}} \geq 0$P1​P​⋅P1​P4​​≥0 && $\vec{P_{1}P} \cdot \vec{P_{1}P_{2}} \geq 0$P1​P​⋅P1​P2​​≥0 && $\vec{P_{3}P} \cdot \vec{P_{3}P_{4}} \geq 0$P3​P​⋅P3​P4​​≥0 && $\vec{P_{3}P} \cdot \vec{P_{3}P_{2}} \geq 0$P3​P​⋅P3​P2​​≥0