Lecture
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.
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