The center of the sphere-plane intersection is the perpendicular projection of the sphere's center onto the plane:
(1)
The distance from to the plane is , so the intersection is a genuine circle. A right triangle with hypotenuse equal to the sphere radius gives
(2)
Its plane is the given plane; displacements within it are perpendicular to .
Figure 1.
Meridional section of the sphere and its intersection circle
. Original section through the sphere center and the plane normal. The horizontal chord is a diameter of the intersection circle; the right triangle gives . Labels give the centers' original three-dimensional coordinates. Details.

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