¿como se halla la ecuación de la circunferencia conocidos tres puntos?

Pongamos que los 3puntos sean : <img src="https://tex.z-dn.net/?f=P_1%3D%28x_1%2Cy_1%29%2C%20P_2%3D%28x_2%2Cy_2%29%2C%20P_3%3D%28x_3%2Cy_3%29" />

Ahora traslademos al punto<img src="https://tex.z-dn.net/?f=P_1" /> al origen de coordenadas conjuntamente con los demás puntos, y tendremos <img src="https://tex.z-dn.net/?f=P%27_1%3D%280%2C0%29%5C%5C%5C%5C%20P_2%27%3D%28x_2%27%2Cy_2%27%29%3D%28x_2-x_1%2Cy_2-y_1%29%5C%5C%5C%5C%20P_3%27%3D%28x_3%27%2Cy_3%27%29%3D%28x_3-x_1%2Cy_3-y_1%29" />

Como se sabe, la ecuación de una circunferencia que pasa por (h, k) y tenga radio R es de la forma : <img src="https://tex.z-dn.net/?f=%28x-h%29%5E2%2B%28y-k%29%5E2%3DR%5E2" />

Según los puntos trasladados tenemos

<img src="https://tex.z-dn.net/?f=h%5E2%2Bk%5E2%3DR%5E2%5C%5C%20%5C%5C%0A%28x_2%27-h%29%5E2%2B%28y_2%27-k%29%5E2%3DR%5E2%5Cto%20x_2%27%5E2%2By_2%27%5E2%3D2%28x_2%27h%2By_2%27k%29%5C%5C%20%5C%5C%0A%28x_3%27-h%29%5E2%2B%28y_3%27-k%29%5E2%3DR%5E2%5Cto%20x_3%27%5E2%2By_3%27%5E2%3D2%28x_3%27h%2By_3%27k%29" />

Nuestro objetivo es hallar h y k de estas ecuaciones

<img src="https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%0Ax_2%27%5E2%2By_2%27%5E2%3D2%28x_2%27h%2By_2%27k%29%5C%5C%0Ax_3%27%5E2%2By_3%27%5E2%3D2%28x_3%27h%2By_3%27k%29%0A%5Cend%7Bcases%7D" />

Ordenamos

<img src="https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%20%0Ax_2%27h%2By_2%27k%3D%5Cdfrac%7B1%7D%7B2%7Dx_2%27%5E2%2B%5Cdfrac%7B1%7D%7B2%7Dy_2%27%5E2%5C%5C%20%5C%5C%0Ax_3%27h%2By_3%27k%3D%5Cdfrac%7B1%7D%7B2%7Dx_3%27%5E2%2B%5Cdfrac%7B1%7D%7B2%7Dy_3%27%5E2%0A%5Cend%7Bcases%7D" />

Por el método de Cramer se tiene

<img src="https://tex.z-dn.net/?f=h%3D%5Cdfrac%7B1%7D%7B2%7D%5Ccdot%5Cdfrac%7B%5Cleft%7C%5Cbegin%7Bmatrix%7Dx_2%27%5E2%2By_2%27%5E2%26y_2%27%5C%5C%20x_3%27%5E2%2By_3%27%5E2%26y_3%27%5Cend%7Bmatrix%7D%5Cright%7C%7D%7B%5Cleft%7C%5Cbegin%7Bmatrix%7Dx_2%27%26y_2%27%5C%5C%20x_3%27%26y_3%27%5Cend%7Bmatrix%7D%5Cright%7C%7D%5C%5C%20%5C%5C%20%5C%5C%0Ak%3D%5Cdfrac%7B1%7D%7B2%7D%5Ccdot%5Cdfrac%7B%5Cleft%7C%5Cbegin%7Bmatrix%7Dx_2%27%26x_2%27%5E2%2By_2%27%5E2%5C%5C%20y_2%27%26x_3%27%5E2%2By_3%27%5E2%5Cend%7Bmatrix%7D%5Cright%7C%7D%7B%5Cleft%7C%5Cbegin%7Bmatrix%7Dx_2%27%26y_2%27%5C%5C%20x_3%27%26y_3%27%5Cend%7Bmatrix%7D%5Cright%7C%7D%5C%5C%20%5C%5C" /> = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Luego para devolver al triángulo a su posición inicial, sumamos a h y k<img src="https://tex.z-dn.net/?f=x_11" /> y<img src="https://tex.z-dn.net/?f=y_1" /> respectivamente :

Centro[img = 10]

[img = 11]

[img = 12]

El radio es[img = 13], es decir

[img = 14].