Antipodal $n$-angles inscribed into the regular $(2n-1)$-angle and half-circulant Hadamard matrices of order $4n$

It's discovered a necessary and sufficient conditions for existence the half-circulant Hadamard matrix of order $4n$ and besides in two forms — geometric and analitic ones. The geometrical necessary and sufficient conditions are being reduced to a question of existence antipodal $n$-angles inscribed into the regular $(2n-1)$-angle while the analitical one — to solvability in the field of real numbers a nonhomogeneous system square $5n-3$ equations with $4n-4$ unknown quantities, which closely connect with a some cubic nonreducible smoth hypersurface in $(2n-1)$-dimensional projective space.