Answer to Question #348286 in Analytic Geometry for Kalai

Question

Show that the centre of all sections of the sphere x² + y ² +z² = r² by the planes through a point (a,b,c) lie on the sphere x(x-a)+y(y-b)+z(z-c)=0

Answer 1

All sections of the sphere x² + y ² +z² = r² by the planes through a point (a,b,c) is poin (0, 0, 0), which lies on the sphere x(x-a)+y(y-b)+z(z-c)=0.

Check:

0*(0-a) + 0*(0-b) + 0*(0-c) = 0 + 0 + 0 = 0