Ellipse-inscribed family of cyclic quadrilaterals: vertex centroid is stationary

[@dominiquelaurain6427]

Point P is called the "4th point", the 4th intersection of the ABC circumconic (here the ellliptic billiard with equation in trilinears 1/apha + 1/ beta + 1/gamma =0) with the ABC circumcircle centerd at O. From ETC, P = X(100) lies on the circumcircle and the circumconic with center Mittenpunkt X(9). Fix point C0 is X(3035) = 3*X(2) + X(100).

[@xfs4710]

midpoint F1 - C, A +P(x^2) when quadrilateral perimeter is equal to circle circumference

[@Cursed_math]

The coefficients of x^3 and x^4 in the degree 4 polynomial obtained by computed the intersection of the ellipse and the circle only depends on the center of the circle and the parameters of the ellipse. Thus the sum of its roots P(t), A, B, C is constant when the radius is moving.

[@dreznik]

tks! had u seen this before? apparently four real roots iff O is inside the evolute. Any idea what the locus of C0 is over O on the evolute? Shoot me an email? d r e z n i k --a t-- g m a i l