Theorem
If a plane F cuts a sphere S non-trivially,
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Proof Suppose that the point P lies on C, the curve of intersection. Let Q be the foot of the perpendicular to F from O, the centre of S. Then DOPQ is right-angled at Q. Hence OP2 = OQ2 + QP2, so that QP2 = OP2 - OQ2. But OP is the radius of S, and OQ is the distance from O to F, so are constant. Thus QP is constant, so C is a circle. |
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