Jacobi's theorem on the spherical image of the principal normals
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Problem 2310 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"
From the center of a sphere, radii are drawn parallel to the principal normals of a closed curve having no singular points. Prove that the locus of their endpoints divides the surface of the sphere into two parts of equal area. (Jacobi.)
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References
[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.