تقارن در صفحه

قرینه نقطه $A\left(\alpha ,\beta \right)$ نسبت به خط مزبور، نقطه $A^{\text{'}}\left(x_{A^{\text{'}}},y_{A^{\text{'}}}\right)$ بوده و نقطه $H$ روی خط d وسط $A{A}^{\text{'}}$ است.

$\begin{array}{l}{m}_d=-\frac{a}{b}\Rightarrow m_{A{A}^{\text{'}}}=-\frac{1}{m_d}=\frac{b}{a}\\ \\ A{A}^{\text{'}}:y-y_A=m_{A{A}^{\text{'}}}\left(x-x_A\right)\\ \\ \Rightarrow y-\beta =\frac{b}{a}\left(x-\alpha \right)\\ \\ \Rightarrow \frac{y-\beta }{b}=\frac{x-\alpha }{a}=\lambda \\ \\ \Rightarrow \left\{\begin{array}{l}\frac{x_H-\alpha }{a}=\lambda \Rightarrow x_H=\alpha +a\lambda \\ \frac{y_H-\beta }{b}=\lambda \Rightarrow y_H=\beta +b\lambda \end{array}\right.\end{array}$

مختصات $H\left(x_H,y_H\right)\in d$ در معادله خط $d$ صادق است:

$\begin{array}{l}d:ax+by+c=0\\ \\ \Rightarrow a\left(\alpha +a\lambda \right)+b\left(\beta +b\lambda \right)+c=0\\ \\ \Rightarrow a\alpha +a^2\lambda +b\beta +b^2\lambda +c=0\\ \\ \Rightarrow \left(a\alpha +b\beta +c\right)=-a^2\lambda -b^2\lambda \\ \\ \Rightarrow \lambda =-\frac{a\alpha +b\beta +c}{a^2+b^2}\end{array}$

توجه کنید که:

$\left\{\begin{array}{l}{x}_H=\frac{\alpha +x_{A^{\text{'}}}}{2}\Rightarrow x_{A^{\text{'}}}=2x_H-\alpha =2\left(\alpha +a\lambda \right)-\alpha =2\alpha +2a\lambda -\alpha =\alpha +2a\lambda \\ \\ y_H=\frac{\beta +y_{A^{\text{'}}}}{2}\Rightarrow y_{A^{\text{'}}}=2y_H-\beta =2\left(\beta +b\lambda \right)-\beta =2\beta +2b\lambda -\beta =\beta +2b\lambda \end{array}\right.$