Elementary symmetric polynomials, and writing a symmetric polynomial in them #2152

Modenov's Chapter III (#2131) writes the orthocentre, the circumcentre and most of its formulas in the elementary symmetric functions of the vertices: σ₁ = a + b + c, σ₂ = ab + bc + ca, σ₃ = abc. Problem 21 argues "this line equation is symmetric in z₁ … z₄, so it serves both points". Readers counted the σ's in 10–12 rows of each part of Chapter III.

Wanted, on the polynomial layer:

• elemsym(k, a, b, c, …), or another name, since AngouriMath has no rounding functions (floor, ceil, round), nor min, max or gcd #809's research plans divisorsigma and σ would clash;
• rewriting a symmetric polynomial in the σ's, which is the fundamental theorem of symmetric polynomials.

Where these come from today: sum and product exist, and the σ's can be written out by hand.