[PASS] a''^2 = a^3(a+2b)/(a+b)^2 and b''^2 = b^3(2a+b)/(a+b)^2
[PASS] member I: chosen square root squares to the discriminant
[PASS] member I: all vertices on E
[PASS] member I: closes after six reflections (P7 = P1, direction returns)
[PASS] member I: every chord tangent to C_lam6
   contact parameters along the six chords: [b/(a + b), 1/2, a/(a + b), b/(a + b), 1/2, a/(a + b)] (all y-parts zero: True)
[PASS] member II: chosen square root squares to the discriminant
[PASS] member II: all vertices on E
[PASS] member II: closes after six reflections (P7 = P1, direction returns)
[PASS] member II: every chord tangent to C_lam6
   contact parameters along the six chords: [a/(a + b), 1/2, b/(a + b), a/(a + b), 1/2, b/(a + b)] (all y-parts zero: True)
[PASS] b''^2 = (b s/(a+b))^2 and a''^2 = (a t/(a+b))^2
[PASS] member I: vertices as printed in Prop. 3.1, in the printed order
[PASS] member II: vertices as printed in Prop. 3.1, in the printed order
[PASS] outer polygon of I: (+-a, +-b^2/s), (0, +-b(a+b)/s)
[PASS] outer polygon of II: (+-a^2/t, +-b), (+-a(a+b)/t, 0)
[PASS] Ant(I;O): (+-a, +-(b^2+ab-a^2)/s), (0, +-(a^3-a^2b+ab^2+b^3)/(bs))
[PASS] Ant(II;O): (+-(a^2+ab-b^2)/t, +-b), (+-(a^3+a^2b-ab^2+b^3)/(at), 0)
[PASS] A(I) = 2abs(2a+b)/(a+b)^2
[PASS] A'(I) = 2ab(a+2b)/s
[PASS] A*_O(I) = 2a(a^3-2a^2b+2ab^2+2b^3)/(bs)
[PASS] A(II) = 2abt(a+2b)/(a+b)^2
[PASS] A'(II) = 2ab(2a+b)/t
[PASS] A*_O(II) = 2b(2a^3+2a^2b-2ab^2+b^3)/(at)
[PASS] eq. (3.1): A'A*_O(I) as printed
[PASS] eq. (3.1): A'A*_O(II) as printed
[PASS] eq. (3.1): A'A(I) = A'A(II) = 4a^2b^2(2a+b)(a+2b)/(a+b)^2
[PASS] eq. (3.1): difference = 4(a-b)^3(a+b)^5/(ab(2a+b)(a+2b))
   factored difference: 4*(a - b)**3*(a + b)**5/(a*b*(a + 2*b)*(2*a + b))
[PASS] proof of Prop. 3.1: first product = a^8+2a^7b-2a^6b^2+2a^5b^3+16a^4b^4+8a^3b^5
[PASS] proof of Prop. 3.1: second product = first with a, b exchanged
[PASS] proof of Prop. 3.1: difference expansion = (a-b)^3(a+b)^5
[PASS] proof of Prop. 3.1: b^3+2ab^2-a^3 = (a+b)(b^2+ab-a^2) and a^4+2ab^3+b^4 = (a+b)(a^3-a^2b+ab^2+b^3)
[PASS] kappa(lam6) = (a+b)^2(2b-a)/(b^2(2a+b))
[PASS] member I: A*_f1 = A*_f2 = kappa6*A for every ellipse
[PASS] member II: A*_f1 = A*_f2 = kappa6*A for every ellipse
[PASS] Corollary 3, N=6: kappa6 * A'A = 4a^2(4b^2-a^2)
[PASS] Corollary 3, N=4: kappa(lam4) = 2
[PASS] proof of Cor. 3: b^2-lam4 = b^4/(a^2+b^2), numerator 4a^2b^6/(a^2+b^2)
[PASS] proof of Cor. 3: b^2-lam6 = b^3(2a+b)/(a+b)^2, numerator 2a^2b^3(2b-a)
[PASS] rhombus: side x/a + y/b = 1 tangent to C_lam4 (a''^2/a^2 + b''^2/b^2 = 1)
[PASS] rhombus: A = 2ab, outer rectangle area 4ab, A'A = 8a^2b^2, A'A*_f = 16a^2b^2
[PASS] rhombus: A*_f1 = 2A = 4ab directly
[PASS] common points of E and C: x0^2 = a^2(a^2-lam)/c^2, y0^2 = -b^2(b^2-lam)/c^2
[PASS] gradient determinant = 4 x0 y0 lam c^2/(a^2 b^2 (a^2-lam)(b^2-lam))
[PASS] system at infinity: determinant = lam c^2/(a^2 b^2 (a^2-lam)(b^2-lam))
[PASS] kappa = 0 iff lam = a^2 b^2 (a^2-3b^2)/(a^2-b^2)^2 (unique root of the numerator, linear in lam)
[PASS] kappa6 = 0 iff a = 2b (numerator factor 2b - a)
[PASS] circle limit a = b: kappa = b^2/b''^2

46 of 46 checks pass; ALL PASS: True  (elapsed 3.4 s)
