Let ABC be a triangle and P a point on the MacBeath Inconic ( = the inconic with foci O,H)
Denote:
DEF = the medial triangle
A'B'C' = the circumcevian triangle of P
A", B", C" = the reflections of A', B', C' in D, E, F, resp.
Then:
1. A", B", C" and H are concyclic (Isogonal P - Hagge circle)
2. The circles (A"B"C") and (O) are tangent
[Antreas P. Hatzipolakis]:
Which are the centers of the circles and the touchpoints for some points on the inconic P = X(339, 1312, 1313, 2968, 2969, 2970, 2971, 2972, 2973, 2974, ......) ?
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[Ercole Suppa]
Hi Antreas,
Let Q = Q(P), T = T(P) the center of the circles and the touchpoint respectively.
We have: Q(P) = reflection of P in X(5)
***Some points:
Q(X(339)) = X(2967)
T(X(339)) = X(112)
--------------------------------
Q(X(2968)) = X(21664)
T(X(2968)) = X(108)
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Q(X(2969)) = REFLECTION OF X(2969) IN X(5)
= (a^2-b^2-c^2) (a^3 b-a^2 b^2-a b^3+b^4+a^3 c+a b^2 c-a^2 c^2+a b c^2-2 b^2 c^2-a c^3+c^4)^2 : : (barys)
= 2*X[5]-X[2969]
= lies on the MacBeath inconic and these lines: {3,8}, {5,2969}, {117,131}, {121,20315}, {339,1234}, {442,2970}, {912,914}, {1060,1411}, {1441,2973}, {1782,12134}, {2834,10743}, {2971,30444}, {2972,21530}, {3549,15253}, {6842,21666}
= reflection of X(2969) in X(5)
= ETC search numbers: [-0.8809957035549931882, -7.6365315921050680324, 9.3341074473132692124]
T(X(2969)) = X(100)
-------------------------------
Q(X(2970)) = REFLECTION OF X(2970) IN X(5)
= a^2 (a^2-b^2-c^2) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6)^2 : : (barys)
= S^4 + (12 R^4-8 R^2 SB-8 R^2 SC-SB SC+2 SB SW+2 SC SW-SW^2)S^2 + 36 R^4 SB SC-12 R^2 SB SC SW+SB SC SW^2 : : (barys)
= 2*X[5]-X[2970]
= lies on the MacBeath inconic and these lines: {3,74}, {4,13556}, {5,2970}, {25,1299}, {30,16933}, {113,131}, {157,18451}, {216,3016}, {265,20975}, {311,339}, {381,2971}, {454,12164}, {1147,13496}, {1624,25711}, {1986,15329}, {2790,6033}, {2969,6841}, {5158,11060}, {5502,20771}, {5562,23217}, {7687,18114}, {8154,13557}, {10257,31945}, {12091,18403}, {12134,31976}, {14575,18445}, {16163,16186}
= reflection of X(2970) in X(5)
= ETC search numbers: [-2.0209629287137045005, -4.2883191630464812227, 7.5422529464998066422]
T(X(2970)) = X(110)
-------------------------------
Q(X(2971)) = X(2974)
T(X(2971)) = X(99)
-------------------------------
Q(X(2972)) = REFLECTION OF X(2972) IN X(5)
= b^2 c^2 (-a^2+b^2-c^2) (a^2+b^2-c^2) (-2 a^4+a^2 b^2+b^4+a^2 c^2-2 b^2 c^2+c^4)^2 : : (barys)
= (12 R^4+4 R^2 SB+4 R^2 SC+SB SC-7 R^2 SW-SB SW-SC SW+SW^2)S^2 36 R^4 SB SC-9 R^2 SB SC SW : : (barys)
= 2*X[5]-X[2972]
=lies on the MacBeath inconic and these lines: {2,18317}, {3,107}, {4,94}, {5,2972}, {25,1300}, {113,133}, {131,132}, {264,339}, {290,3531}, {324,3845}, {338,7687}, {382,1093}, {399,648}, {546,14978}, {1511,4240}, {1596,2971}, {1941,18350}, {2052,3830}, {2968,6841}, {2969,15763}, {2973,15762}, {3534,15466}, {3548,6523}, {3627,13450}, {5667,20127}, {6525,18850}, {6530,11799}, {7480,16168}, {10272,14920}, {10575,14363}, {11563,24977}, {12188,18535}, {12918,18531}, {13202,18507}, {14934,31510}, {15454,20771}, {16163,16240}, {18279,32743}, {18402,23290}
= reflection of X(2972) in X(5)
= isotomic conjugate of isogonal conjugate of X(16240)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {113,133,11251}
= ETC search numbers: [-2.8736078556880893112, -3.2887733464683600757, 7.2437881167031311453]
T(X(2972)) = X(107)
-------------------------------
Q(X(2973)) = REFLECTION OF X(2973) IN X(5)
= a^2 (a^2-b^2-c^2) (a^3 b^2-a^2 b^3-a b^4+b^5+a^3 c^2+2 a b^2 c^2-b^3 c^2-a^2 c^3-b^2 c^3-a c^4+c^5)^2 : : (barys)
= 2*X[5]-X[2973]
=lies on the MacBeath inconic and these lines: {3,101}, {5,2973}, {321,2968}, {440,2972}, {916,2253}, {2969,8226}, {2970,3136}
= reflection of X(2973) in X(5)
= ETC search numbers: [-1.6562964690544747640, -4.9615943065198629318, 7.8400589106001956393]
T(X(2973)) = X(101)
-------------------------------
Q(X(2974)) = X(2971)
T(X(2974)) = X(3563)
Best regards
Ercole Suppa
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