Last geometric statement of Jacobi
In differential geometry the last geometric statement of Jacobi is a conjecture named after Carl Gustav Jacob Jacobi, which states:

Every caustic from any point  on an ellipsoid other than umbilical points has exactly four cusps. Numerical experiments had indicated the statement is true before it was proven rigorously in 2004 by Itoh and Kiyohara. It has since been extended to a wider class of surfaces beyond the ellipsoid. See also 

 Four-vertex theorem

References

Differential geometry
Algebraic geometry
Conjectures that have been proved