a=8/5 b=1 lambda=64/169; foci (+-sqrt(39)/5, 0)

=== ORBIT I (vertex (a,0)): P1 = (8/5, 0), d1 = (-8*sqrt(105)/105, 1)
  P1 = (8/5, 0)
  P2 = (64/65, sqrt(105)/13)
  P3 = (-64/65, sqrt(105)/13)
  P4 = (-8/5, 0)
  P5 = (-64/65, -sqrt(105)/13)
  P6 = (64/65, -sqrt(105)/13)
  contact ratios along sides: [5/13, 1/2, 8/13, 5/13, 1/2, 8/13]
  check all_on_ellipse: True
  check closes_after_6: True
  check direction_returns: True
  check no_earlier_return: True
  check all_sides_tangent: True
  check contacts_interior: True
  check convex_ccw: True
  areas:
    A = 336*sqrt(105)/845  ~ 4.07452716848
    A' = 96*sqrt(105)/175  ~ 5.62118442018
    A'' = 41472*sqrt(105)/142805  ~ 2.97581696835
    A*_O = 2784*sqrt(105)/4375  ~ 6.52057392741
    Aped_O = 43344*sqrt(105)/142805  ~ 3.11014203984
    A'*_O = 11323*sqrt(105)/14700  ~ 7.89294037571
    A'ped_O = 21*sqrt(105)/50  ~ 4.30371932170
    A*_f1 = 32*sqrt(105)/125  ~ 2.62321939609
    Aped_f1 = 56448*sqrt(105)/142805  ~ 4.05041754026
    A'*_f1 = 9152*sqrt(105)/18375  ~ 5.10367855293
    A'ped_f1 = 168*sqrt(105)/325  ~ 5.29688531902
    A*_f2 = 32*sqrt(105)/125  ~ 2.62321939609
    Aped_f2 = 56448*sqrt(105)/142805  ~ 4.05041754026
    A'*_f2 = 9152*sqrt(105)/18375  ~ 5.10367855293
    A'ped_f2 = 168*sqrt(105)/325  ~ 5.29688531902
  products/ratios:
    A'*A*_O [k401] = 801792/21875  ~ 36.6533485714
    A'/A*_O [k402] = 25/29  ~ 0.862068965517
    Aped*A*_O [k403] = 362009088/17850625  ~ 20.2799110955
    A*/Aped_O [k404] = 1656538/790125  ~ 2.09655181142
    A/Aped_O [k204] = 169/129  ~ 1.31007751938
    A'*A'ped_O [k303a] = 3024/125  ~ 24.1920000000
    A'*A'*_O [antipedal of outer] = 271752/6125  ~ 44.3676734694
    A*_O/A = 9802/6125  ~ 1.60032653061
    A'*A*_f1 [k401] = 9216/625  ~ 14.7456000000
    A'/A*_f1 [k402] = 15/7  ~ 2.14285714286
    Aped*A*_f1 [k403] = 37933056/3570125  ~ 10.6251338539
    A*/Aped_f1 [k404] = 28561/44100  ~ 0.647641723356
    A/Aped_f1 [k204] = 169/168  ~ 1.00595238095
    A'*A'ped_f1 [k303a] = 48384/1625  ~ 29.7747692308
    A'*A'*_f1 [antipedal of outer] = 878592/30625  ~ 28.6887183673
    A*_f1/A = 338/525  ~ 0.643809523810
    A'*A*_f2 [k401] = 9216/625  ~ 14.7456000000
    A'/A*_f2 [k402] = 15/7  ~ 2.14285714286
    Aped*A*_f2 [k403] = 37933056/3570125  ~ 10.6251338539
    A*/Aped_f2 [k404] = 28561/44100  ~ 0.647641723356
    A/Aped_f2 [k204] = 169/168  ~ 1.00595238095
    A'*A'ped_f2 [k303a] = 48384/1625  ~ 29.7747692308
    A'*A'*_f2 [antipedal of outer] = 878592/30625  ~ 28.6887183673
    A*_f2/A = 338/525  ~ 0.643809523810
    A'*A [k106] = 96768/4225  ~ 22.9036686391
    A'/A'' [k113] = 28561/15120  ~ 1.88895502646

=== ORBIT II (vertex (0,b)): P1 = (0, 1), d1 = (-12/5, -1)
  P1 = (0, 1)
  P2 = (-96/65, 5/13)
  P3 = (-96/65, -5/13)
  P4 = (0, -1)
  P5 = (96/65, -5/13)
  P6 = (96/65, 5/13)
  contact ratios along sides: [8/13, 1/2, 5/13, 8/13, 1/2, 5/13]
  check all_on_ellipse: True
  check closes_after_6: True
  check direction_returns: True
  check no_earlier_return: True
  check all_sides_tangent: True
  check contacts_interior: True
  check convex_ccw: True
  areas:
    A = 3456/845  ~ 4.08994082840
    A' = 28/5  ~ 5.60000000000
    A'' = 84672/28561  ~ 2.96460207976
    A*_O = 463/80  ~ 5.78750000000
    Aped_O = 445824/142805  ~ 3.12190749624
    A'*_O = 9464/1125  ~ 8.41244444444
    A'ped_O = 108/25  ~ 4.32000000000
    A*_f1 = 2304/875  ~ 2.63314285714
    Aped_f1 = 580608/142805  ~ 4.06573999510
    A'*_f1 = 1144/225  ~ 5.08444444444
    A'ped_f1 = 1728/325  ~ 5.31692307692
    A*_f2 = 2304/875  ~ 2.63314285714
    Aped_f2 = 580608/142805  ~ 4.06573999510
    A'*_f2 = 1144/225  ~ 5.08444444444
    A'ped_f2 = 1728/325  ~ 5.31692307692
  products/ratios:
    A'*A*_O [k401] = 3241/100  ~ 32.4100000000
    A'/A*_O [k402] = 448/463  ~ 0.967602591793
    Aped*A*_O [k403] = 12901032/714025  ~ 18.0680396345
    A*/Aped_O [k404] = 13223743/7133184  ~ 1.85383455691
    A/Aped_O [k204] = 169/129  ~ 1.31007751938
    A'*A'ped_O [k303a] = 3024/125  ~ 24.1920000000
    A'*A'*_O [antipedal of outer] = 264992/5625  ~ 47.1096888889
    A*_O/A = 78247/55296  ~ 1.41505714699
    A'*A*_f1 [k401] = 9216/625  ~ 14.7456000000
    A'/A*_f1 [k402] = 1225/576  ~ 2.12673611111
    Aped*A*_f1 [k403] = 191102976/17850625  ~ 10.7056742271
    A*/Aped_f1 [k404] = 28561/44100  ~ 0.647641723356
    A/Aped_f1 [k204] = 169/168  ~ 1.00595238095
    A'*A'ped_f1 [k303a] = 48384/1625  ~ 29.7747692308
    A'*A'*_f1 [antipedal of outer] = 32032/1125  ~ 28.4728888889
    A*_f1/A = 338/525  ~ 0.643809523810
    A'*A*_f2 [k401] = 9216/625  ~ 14.7456000000
    A'/A*_f2 [k402] = 1225/576  ~ 2.12673611111
    Aped*A*_f2 [k403] = 191102976/17850625  ~ 10.7056742271
    A*/Aped_f2 [k404] = 28561/44100  ~ 0.647641723356
    A/Aped_f2 [k204] = 169/168  ~ 1.00595238095
    A'*A'ped_f2 [k303a] = 48384/1625  ~ 29.7747692308
    A'*A'*_f2 [antipedal of outer] = 32032/1125  ~ 28.4728888889
    A*_f2/A = 338/525  ~ 0.643809523810
    A'*A [k106] = 96768/4225  ~ 22.9036686391
    A'/A'' [k113] = 28561/15120  ~ 1.88895502646

=== comparison of the two members ===
  A'*A*_O [k401]: I = 801792/21875, II = 3241/100 -> DIFFERENT
  A'/A*_O [k402]: I = 25/29, II = 448/463 -> DIFFERENT
  Aped*A*_O [k403]: I = 362009088/17850625, II = 12901032/714025 -> DIFFERENT
  A*/Aped_O [k404]: I = 1656538/790125, II = 13223743/7133184 -> DIFFERENT
  A/Aped_O [k204]: I = 169/129, II = 169/129 -> EQUAL
  A'*A'ped_O [k303a]: I = 3024/125, II = 3024/125 -> EQUAL
  A'*A'*_O [antipedal of outer]: I = 271752/6125, II = 264992/5625 -> DIFFERENT
  A*_O/A: I = 9802/6125, II = 78247/55296 -> DIFFERENT
  A'*A*_f1 [k401]: I = 9216/625, II = 9216/625 -> EQUAL
  A'/A*_f1 [k402]: I = 15/7, II = 1225/576 -> DIFFERENT
  Aped*A*_f1 [k403]: I = 37933056/3570125, II = 191102976/17850625 -> DIFFERENT
  A*/Aped_f1 [k404]: I = 28561/44100, II = 28561/44100 -> EQUAL
  A/Aped_f1 [k204]: I = 169/168, II = 169/168 -> EQUAL
  A'*A'ped_f1 [k303a]: I = 48384/1625, II = 48384/1625 -> EQUAL
  A'*A'*_f1 [antipedal of outer]: I = 878592/30625, II = 32032/1125 -> DIFFERENT
  A*_f1/A: I = 338/525, II = 338/525 -> EQUAL
  A'*A*_f2 [k401]: I = 9216/625, II = 9216/625 -> EQUAL
  A'/A*_f2 [k402]: I = 15/7, II = 1225/576 -> DIFFERENT
  Aped*A*_f2 [k403]: I = 37933056/3570125, II = 191102976/17850625 -> DIFFERENT
  A*/Aped_f2 [k404]: I = 28561/44100, II = 28561/44100 -> EQUAL
  A/Aped_f2 [k204]: I = 169/168, II = 169/168 -> EQUAL
  A'*A'ped_f2 [k303a]: I = 48384/1625, II = 48384/1625 -> EQUAL
  A'*A'*_f2 [antipedal of outer]: I = 878592/30625, II = 32032/1125 -> DIFFERENT
  A*_f2/A: I = 338/525, II = 338/525 -> EQUAL
  A'*A [k106]: I = 96768/4225, II = 96768/4225 -> EQUAL
  A'/A'' [k113]: I = 28561/15120, II = 28561/15120 -> EQUAL

k401 at O: I - II = 371293/87500   (claim 371293/87500: True)
I = 801792/21875: True;  II = 3241/100: True
k401 at f1: I = 9216/625, II = 9216/625; both 9216/625: True
k401 at f2: I = 9216/625, II = 9216/625; both 9216/625: True
kappa_6 = 338/525; A*_f1/A: I = 338/525, II = 338/525 -> equal kappa_6: True
A*_f1 = A*_f2 (central symmetry): I True, II True
4a^2(4b^2-a^2) = 9216/625
