The cross section of the ball plane

Theorem

Every section of a ball by a plane is a circle. The center of this circle is the base of the perpendicular, which is lowered from the center of the ball to the cutting plane.

Evidence

Let α be a cutting plane and O be the center of the ball. We drop the perpendicular from the center of the ball to the plane α and denote by O` the base of this perpendicular.
Let X be an arbitrary point of the ball belonging to the plane α. According to the Pythagorean theorem

Formula

Since OX is not larger than the radius R of the ball,

Formula 1

those. any point of the ball section of the plane α is from the point O` at a distance not greater than

Formula 2

consequently, it belongs to the circle with the center O `and radius

Formula 2

Back: any point X of this circle belongs to the ball. And this means that the section of the ball with the plane α is a circle with a center at the point O`. The theorem is proved.