Paper25_1

The first pdf is (almost) the submission to CMI (Oct 2025).
The second pdf contains that submitted to The Mathematical Gazette (Jan 2026), with much additional material.

In both Lemma psiMax paraphrased is that, for `circum-3-gons' with given perimeter and and total angle sum, the slender isosceles one has the greatest PSP3/second moment.
The following Mathematica uses the same notation as the paper and has Mathematica Reduce (Cylindrical Algebraic Decomposion) establishing this.

tauFn[T1_, T2_, T3_] := (T1*T2*T3 - (T1 + T2 + T3))/(-1 + T1*T2 + T2*T3 + T3*T1);
p1T = T1 + T2 + T3;
consT = {T1 > 0, T2 > 0, T3 > 0};

p3T = T1^3 + T2^3 + T3^3;  p3isos = 2*X^3 + (p1T - 2*X)^3;
consX = {X > 0, p1T/3 - X >= 0};  (*slender isos*)
ans = Reduce[p3isos < p3T && tauFn[X, X, p1T - 2*X] == tauFn[T1, T2, T3] &&
Apply[And, Join[consT, consX]], {T1, T2, T3, X}]
(* False , i.e. the slender isos has larger PSP3 *)

Above CMI draft Oct  2025

At given area and perimeter which tangential n-gons have the largest second moment

At given area and perimeter which tangential $n$-gons

have the largest second momeAt given area and perimeter which tangential $n$-gons

have the largest second momeAt given area and perimeter which tangential $n$-gonhave the largest second momenAt

A 20page version was submitted,  in Jan 2026,  to The Mathematical Gazette.
Attached below is a very first draft towards what might be put up on arXiv.
Separately, I have recorded  Mathematica code for the proofs of Lemma phi_isos and Lemma psi_max.
(Email me if you want the sympy equivalents, just for the routine algebra, not the Reduce (Cylindrical Algebraic Decomposition).