Characterising Intersections of Spread Polynomials (by area matching)
Moving point P_1 changes the areas of all triangles at once - and since
Spreads are defined as relative areas-squared then matching two triangles with 'equal-areas' shows exactly when their assoicated spreads are also equal.
For the unit circle we see equal areas whenever a triangle passes either 'x=0' or 'y=0' (when their 'heights' match the height at 'P_1' either directly or via reflection)
Thus for s = 0.5, S_3(s) = S_5(s) = s, giving a point on intersection of their respective spread polynomials with the line 's = x' represented by 'P_1'