Sphere through four points

Method. The general sphere is x² + y² + z² + 2ux + 2vy + 2wz + d = 0, with four unknowns. Four points give four equations. Write D = 2u, E = 2v, F = 2w, G = d:

x² + y² + z² + Dx + Ey + Fz + G = 0.

Substitute the points

(0, 0, 0):   G = 0
(0, 1, −1):  0 + 1 + 1 + E − F + G = 0     →  E − F = −2        ... (1)
(−1, 2, 0):  1 + 4 + 0 − D + 2E + G = 0    →  −D + 2E = −5      ... (2)
(1, 2, 3):   1 + 4 + 9 + D + 2E + 3F + G = 0 → D + 2E + 3F = −14 ... (3)

Solve

Add (2) and (3): 4E + 3F = −19. From (1), F = E + 2, so 4E + 3E + 6 = −19, 7E = −25, E = −25/7. Then F = −25/7 + 2 = −11/7. From (2): D = 2E + 5 = −50/7 + 35/7 = −15/7.

Equation

x² + y² + z² − (15/7)x − (25/7)y − (11/7)z = 0

Multiplying by 7:

7(x² + y² + z²) − 15x − 25y − 11z = 0

Check with (1, 2, 3): 7(14) − 15 − 50 − 33 = 98 − 98 = 0 ✓. With (−1, 2, 0): 35 + 15 − 50 = 0 ✓.

The centre is (15/14, 25/14, 11/14).