Geometry

Find the coordinates of the center of the circle.

\(A\{22,15\}\\ B\{-25,0\}\\ C\{22,-29\} \)

\(f_{mAC}(x)=\dfrac{y_ A+y_B}{2}=\frac{15-29}{2}\\ f_{mAC}(x)=-7\\ P_{mAB}\{\frac{x_A+x_B}{2},\frac{y_ A+y_B}{2}\}=\{\frac{22-25}{2},\frac{15+0}{2}\}\\ P_{mAB}\{-1.5,7.5\} \)

\(m_{AB}=\dfrac{y_A-y_B}{x_A-x_B}=\dfrac{15-0}{22+25}=0.3191\\ m_{mAB}=-\dfrac{1}{m_{AB}}=-3.13\overline 3\\ f_{mAB}(x)=m_{mAB}(x-x_{P_{mAB}})+y_{P_{mAB}}\\ f_{mAB}(x)=-3.13\overline 3(x+1.5)+7.5\\ f_{mAB}(x)=-3.13\overline 3x+2.8\)

\(f_{mAB}(x)=f_{mAC}(x)\\ -3.13\overline 3x+2.8=-7\\ -3.13\overline 3x=-9.8\\ \color{blue}x_{\circ}=3.12766\\ y_\circ=f_{m_{AC}}(x)\\ \color{blue}y_\circ=-7\)

The coordinates of the center of the circle are \(\color{blue}P_0\ (3.12766, -7).\)

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