Si ℓt es la recta OT, con T=(3 a + (a + b + c) t : 3 b + (a + b + c) t : 3 c + (a + b + c) t) (en coordenadas baricéntricas), la ecuación de la parábola 𝒫t es:

Σ abc xyz (3 a b c (-b + c) + a^2 (-b^2 + c^2) t + (b^2 - c^2) (3 b c + b^2 t + c^2 t))^2 (9 a^10 (b - c)^2 + 6 a^9 (b - c)^2 (b + c) t + 6 a (b - c)^4 (b + c)^3 (b^4 + 6 b^2 c^2 + c^4) t - 24 a^7 (b^5 - b^4 c - 2 b^3 c^2 - 2 b^2 c^3 - b c^4 + c^5) t + (b^2 - c^2)^4 (b^4 + 6 b^2 c^2 + c^4) t^2 - 24 a^3 (b - c)^2 (b + c) (2 b^3 c^3 (-3 + t) + b^6 t + b^4 c^2 t + b^2 c^4 t + c^6 t) + 12 a^5 (b + c) (-12 b^3 c^3 + 3 b^6 t - 6 b^5 c t + b^4 c^2 t + b^2 c^4 t - 6 b c^5 t + 3 c^6 t) + 2 a^4 (36 b^7 c + 36 b c^7 + 12 b^5 c^3 (3 + 2 t) + 12 b^3 c^5 (3 + 2 t) + 3 b^8 (-6 + t^2) + 3 c^8 (-6 + t^2) + 18 b^4 c^4 (6 + t^2) - 8 b^6 c^2 (9 + t^2) - 8 b^2 c^6 (9 + t^2)) - 2 a^6 (54 b^5 c + 54 b c^5 + 12 b^3 c^3 (3 + 4 t) + b^6 (-27 + 2 t^2) + 3 b^4 c^2 (-27 + 2 t^2) + 3 b^2 c^4 (-27 + 2 t^2) + c^6 (-27 + 2 t^2)) - a^2 (b^2 - c^2)^2 (18 b^5 c + 18 b c^5 - 12 b^3 c^3 (-9 + 4 t) + b^6 (-9 + 4 t^2) + c^6 (-9 + 4 t^2) - b^4 c^2 (63 + 4 t^2) - b^2 c^4 (63 + 4 t^2)) + a^8 (72 b^3 c + 72 b c^3 + b^4 (-36 + t^2) + c^4 (-36 + t^2) + 2 b^2 c^2 (-36 + 7 t^2)))x^2 -2 (a - b) (a - c) (a^18 t^4 + a^17 (b + c) t^3 (6 + t) - b^2 c^2 (b^2 - c^2)^6 t^2 (3 b^2 t + 3 c^2 t + b c (9 + t^2)) + a^16 t^2 (b^2 (9 + 6 t - t^2) + c^2 (9 + 6 t - t^2) + b c (36 + t^2)) - a^15 t (b^3 t (-9 + 21 t + t^2) + c^3 t (-9 + 21 t + t^2) + b^2 c (-54 + 9 t - 9 t^2 + t^3) + b c^2 (-54 + 9 t - 9 t^2 + t^3)) - a b^2 (b - c)^6 c^2 (b + c)^5 t (b^2 t (18 + 3 t + t^2) + c^2 t (18 + 3 t + t^2) + 2 b c (27 + 6 t^2 + t^3)) + a^12 (3 b^6 t^2 (45 + 4 t + 5 t^2) + 3 c^6 t^2 (45 + 4 t + 5 t^2) - b^5 c t^2 (108 + 7 t^2) - b c^5 t^2 (108 + 7 t^2) + 3 b^3 c^3 (297 - 180 t + 222 t^2 + 5 t^4) - 3 b^4 c^2 (135 - 108 t + 111 t^2 + 2 t^3 + 5 t^4) - 3 b^2 c^4 (135 - 108 t + 111 t^2 + 2 t^3 + 5 t^4)) - a^14 (b^3 c t^2 (90 + t^2) + b c^3 t^2 (90 + t^2) + b^4 t^2 (54 + 21 t + 7 t^2) + c^4 t^2 (54 + 21 t + 7 t^2) - 3 b^2 c^2 (27 - 36 t + 60 t^2 + 6 t^3 + 5 t^4)) - a^13 (b + c) (108 b^3 c t (3 - t + t^2) + 108 b c^3 t (3 - t + t^2) + b^4 t^2 (54 - 12 t + 7 t^2) + c^4 t^2 (54 - 12 t + 7 t^2) - 3 b^2 c^2 (-27 + 198 t - 36 t^2 + 66 t^3 + 5 t^4)) - a^7 (b - c)^2 (b + c) (b^8 t^2 (-135 - 69 t + 11 t^2) + c^8 t^2 (-135 - 69 t + 11 t^2) + 2 b^7 c t (-405 + 66 t^2 + 11 t^3) + 2 b c^7 t (-405 + 66 t^2 + 11 t^3) - 54 b^5 c^3 (-24 + 57 t + 6 t^3 + t^4) - 54 b^3 c^5 (-24 + 57 t + 6 t^3 + t^4) - 2 b^6 c^2 (405 - 702 t + 36 t^2 + 33 t^3 + 8 t^4) - 2 b^2 c^6 (405 - 702 t + 36 t^2 + 33 t^3 + 8 t^4) - 6 b^4 c^4 (378 - 612 t + 99 t^2 - 5 t^3 + 9 t^4)) + 3 a^11 (b - c)^2 (b + c) (5 b^4 t^2 (9 + 3 t + t^2) + 5 c^4 t^2 (9 + 3 t + t^2) + 10 b^3 c t (27 + 6 t^2 + t^3) + 10 b c^3 t (27 + 6 t^2 + t^3) + b^2 c^2 (135 - 270 t + 108 t^2 - 7 t^3 + 10 t^4)) + a^2 (b^2 - c^2)^4 (3 b^8 t^3 + 3 c^8 t^3 - 3 b^7 c t^2 (-6 + t^2) - 3 b c^7 t^2 (-6 + t^2) - b^6 c^2 t^2 (18 + 15 t + t^2) - b^2 c^6 t^2 (18 + 15 t + t^2) + 2 b^4 c^4 (81 - 54 t + 27 t^2 - 12 t^3 + t^4) - b^5 c^3 (81 - 54 t + 27 t^2 + 13 t^4) - b^3 c^5 (81 - 54 t + 27 t^2 + 13 t^4)) - a^8 (b - c)^2 (9 b^7 c t^2 (70 + 20 t + 3 t^2) + 9 b c^7 t^2 (70 + 20 t + 3 t^2) + b^8 t^2 (-135 + 90 t + 11 t^2) + c^8 t^2 (-135 + 90 t + 11 t^2) + b^5 c^3 (-1134 + 648 t - 81 t^2 + 90 t^3 - 8 t^4) + b^3 c^5 (-1134 + 648 t - 81 t^2 + 90 t^3 - 8 t^4) + 3 b^6 c^2 (270 + 180 t + 60 t^2 + 45 t^3 - 2 t^4) + 3 b^2 c^6 (270 + 180 t + 60 t^2 + 45 t^3 - 2 t^4) + 2 b^4 c^4 (1296 - 324 t + 576 t^2 + 45 t^3 + 14 t^4)) - a^4 (b - c)^4 (b + c)^2 (b^7 c t^2 (162 + 48 t - 5 t^2) + b c^7 t^2 (162 + 48 t - 5 t^2) + 3 b^8 t^2 (-3 + 8 t + t^2) + 3 c^8 t^2 (-3 + 8 t + t^2) + b^5 c^3 (-81 - 108 t + 486 t^2 + 72 t^3 - 14 t^4) + b^3 c^5 (-81 - 108 t + 486 t^2 + 72 t^3 - 14 t^4) + 3 b^6 c^2 (27 + 72 t + 18 t^2 + 20 t^3 - 2 t^4) + 3 b^2 c^6 (27 + 72 t + 18 t^2 + 20 t^3 - 2 t^4) - 2 b^4 c^4 (162 - 324 t + 45 t^2 - 36 t^3 + 16 t^4)) - a^3 (b - c)^4 (b + c)^3 (3 b^8 t^2 (-3 - t + t^2) + 3 c^8 t^2 (-3 - t + t^2) + 6 b^7 c t (-9 + 6 t^2 + t^3) + 6 b c^7 t (-9 + 6 t^2 + t^3) + 2 b^5 c^3 t (27 + 78 t^2 + 13 t^3) + 2 b^3 c^5 t (27 + 78 t^2 + 13 t^3) + 2 b^4 c^4 (81 - 54 t + 135 t^2 + 12 t^3 + 13 t^4) + b^6 c^2 (-81 + 54 t + 90 t^2 + 15 t^3 + 16 t^4) + b^2 c^6 (-81 + 54 t + 90 t^2 + 15 t^3 + 16 t^4)) + a^6 (b - c)^2 (b^10 t^2 (-54 + 69 t + 11 t^2) + c^10 t^2 (-54 + 69 t + 11 t^2) + b^9 c t^2 (486 + 138 t + 11 t^2) + b c^9 t^2 (486 + 138 t + 11 t^2) + b^8 c^2 (405 + 648 t + 90 t^2 + 135 t^3 - 4 t^4) + b^2 c^8 (405 + 648 t + 90 t^2 + 135 t^3 - 4 t^4) - 6 b^5 c^5 (324 - 180 t + 144 t^2 + 10 t^3 + 3 t^4) + 6 b^7 c^3 (-54 - 18 t + 87 t^2 + 22 t^3 + 5 t^4) + 6 b^3 c^7 (-54 - 18 t + 87 t^2 + 22 t^3 + 5 t^4) + b^6 c^4 (891 - 216 t + 540 t^2 + 36 t^3 + 25 t^4) + b^4 c^6 (891 - 216 t + 540 t^2 + 36 t^3 + 25 t^4)) - a^9 (b - c)^2 (b + c) (5 b^6 t^2 (36 + 18 t + t^2) + 5 c^6 t^2 (36 + 18 t + t^2) + 10 b^5 c t (108 + 6 t^2 + t^3) + 10 b c^5 t (108 + 6 t^2 + t^3) + 2 b^3 c^3 (-486 + 1539 t + 246 t^3 + 41 t^4) + b^4 c^2 (810 - 1728 t + 486 t^2 + 45 t^3 + 46 t^4) + b^2 c^4 (810 - 1728 t + 486 t^2 + 45 t^3 + 46 t^4)) - a^10 (b - c)^2 (5 b^6 t^2 (36 - 9 t + t^2) + 5 c^6 t^2 (36 - 9 t + t^2) - 5 b^5 c t^2 (54 + 18 t + t^2) - 5 b c^5 t^2 (54 + 18 t + t^2) + 2 b^4 c^2 (-405 + 54 t - 171 t^2 - 12 t^3 + 8 t^4) + 2 b^2 c^4 (-405 + 54 t - 171 t^2 - 12 t^3 + 8 t^4) + b^3 c^3 (1053 - 702 t + 513 t^2 + 42 t^3 + 52 t^4)) + a^5 (b - c)^4 (b + c) (b^8 t^2 (-54 - 24 t + 11 t^2) + c^8 t^2 (-54 - 24 t + 11 t^2) + 4 b^7 c t (-81 - 27 t + 21 t^2 + 11 t^3) + 4 b c^7 t (-81 - 27 t + 21 t^2 + 11 t^3) + 4 b^5 c^3 (-81 - 162 t + 81 t^2 + 102 t^3 + 40 t^4) + 4 b^3 c^5 (-81 - 162 t + 81 t^2 + 102 t^3 + 40 t^4) + b^6 c^2 (-405 - 162 t + 108 t^2 + 204 t^3 + 95 t^4) + b^2 c^6 (-405 - 162 t + 108 t^2 + 204 t^3 + 95 t^4) + 2 b^4 c^4 (-567 + 270 t + 162 t^2 + 312 t^3 + 98 t^4)))y z = 0.