[Antreas P. Hatzipolakis]:
Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.
D = the Poncelet point of ABCP.
A", B", C" = the other than D intersections of PD and (Na), (Nb), (Nc), resp.
The circumcircles of PA'A", PB'B", PC'C" are coaxial.
Which is the other than D intersection in terms of P?
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[Ercole Suppa]
** If P=(x:y:z) (barys) then Q=(2 a^4 x-2 a^2 b^2 x-2 a^2 c^2 x+a^4 y-b^4 y+2 b^2 c^2 y-c^4 y+a^4 z-b^4 z+2 b^2 c^2 z-c^4 z) (a^2 x^2-b^2 x^2-c^2 x^2+a^2 x y-b^2 x y+a^2 x z-c^2 x z+a^2 y z) : : (barys)
** pairs {P=X(i),Q=X(j)} for these {i,j}: {1,11700},{2,5642},{3,1511},{5,10272},{8,1145},{20,16163},{21,16164},{22,16165},{23,1495},{24,20771},{25,20772},{26,20773},{76,5976},{140,13392},{146,1553},{147,6072},{148,6071},{149,6075},{150,14505},{152,6074},{153,6073},{316,325},{382,1539},{549,11694},{671,16092},{858,11064},{962,1537},{1916,16068},{3146,13202},{3153,1568},{3448,6070},{3524,11693},{3557,2029},{3558,2028},{5011,910},{5057,908},{5080,17757},{5134,17747},{5176,6735},{5196,18653},{5523,16318},{6785,6784},{6787,6786},{6788,3756},{6792,6791},{6794,6793},{7464,10564},{7471,3233},{10152,1552},{10296,1531},{10989,13857},{11050,15354},{12084,25487},{12699,12611},{12833,15631},{13137,15630},{13509,8779},{14262,10354},{14360,6077},{14807,14499},{14808,14500},{15342,14999},{17511,3258},{18328,1146},{18339,2968},{18343,4904},{20344,14506},{21290,14507}
** some points:
Q(X(6)) = MIDPOINT OF X(6) AND X(112)
= a^2 (a^4-b^4+b^2 c^2-c^4) (2 a^6-a^4 b^2-b^6-a^4 c^2+b^4 c^2+b^2 c^4-c^6) :: (barys)
= X[127]-2*X[3589], X[141]-2*X[6720], X[1297]-3*X[5085], 5*X[3618]-X[13219], 3*X[5050]+X[13310], X[10749]-3*X[14561], 5*X[12017]-X[13115], X[12384]+3*X[25406], X[13200]+3*X[14853], X[13221]+3*X[16475], 3*X[16225]-X[19161], 2*X[19130]-X[19163]
= lies on these lines: {6,74}, {127,3589}, {132,1503}, {141,6720}, {518,11722}, {611,13312}, {613,13311}, {1297,5085}, {1384,14649}, {1428,3320}, {1691,13195}, {1974,13166}, {2330,6020}, {2492,6593}, {2794,5480}, {2799,5026}, {3618,13219}, {5039,14676}, {5050,13310}, {8744,18374}, {9019,10317}, {9142,21309}, {9157,17810}, {10749,14561}, {11610,14495}, {12017,13115}, {12145,19124}, {12384,25406}, {13200,14853}, {13221,16475}, {16225,19161}, {19130,19163}
= midpoint of X(6) and X(112)
= reflection of X(i) in X(j) for these {i,j}: {127,3589}, {141,6720}, {19163,19130}
= (6-8-13) search numbers [0.396222886788812052, 0.916360699355107193, 2.82338881922062241]
Q(X(7)) = MIDPOINT OF X(7) AND X(934)
= (a+b-c) (a-b+c) (2 a^2-a b-b^2-a c+2 b c-c^2) (a^4 b-2 a^3 b^2+2 a b^4-b^5+a^4 c+2 a^3 b c-2 a b^3 c-b^4 c-2 a^3 c^2+2 b^3 c^2-2 a b c^3+2 b^2 c^3+2 a c^4-b c^4-c^5) :: (barys)
= 2*X[142]-X[5514], X[972]-3*X[21151]
= lies on these lines: {7,104}, {142,5514}, {658,13257}, {971,1543}, {972,21151}, {1360,3323}, {3321,12831}, {4617,15252}, {6366,10427}
= midpoint of X(7) and X(934)
= reflection of X(5514) in X(142)
= (6-8-13) search numbers [0.393551197518337608, 0.365477276887734759, 3.20600273751517614]
Q(X(9)) = MIDPOINT OF X(9) AND X(101)
= a (a^2-2 a b+b^2-2 a c+b c+c^2) (2 a^3-a^2 b-b^3-a^2 c+b^2 c+b c^2-c^3) :: (barys)
= X[103]-3*X[21153], X[116]-2*X[6666], X[142]-2*X[6710], X[150]-5*X[18230]
= lies on these lines: {2,14154}, {9,48}, {103,21153}, {116,6666}, {118,516}, {142,6710}, {150,18230}, {518,11712}, {528,21090}, {954,11028}, {1001,2809}, {3022,15837}, {3887,6594}, {5375,16586}, {5526,15730}
= midpoint of X(9) and X(101)
= reflection of X(i) in X(j) for these {i,j}: {116,6666}, {142,6710}
= (6-8-13) search numbers [2.44623971779084346, 0.550310010072905776, 2.13064691287659351]
Q(X(10)) = MIDPOINT OF X(10) AND X(101)
= (a^2+a b-b^2+a c-b c-c^2) (2 a^3-a^2 b-b^3-a^2 c+b^2 c+b c^2-c^3) :: (barys)
= 3*X[2]+X[1282], X[103]-3*X[10164], X[116]-2*X[3634], X[150]-5*X[1698], X[152]+3*X[165], 3*X[551]-X[10695], 7*X[9780]+X[20096], 3*X[10175]-X[10739]
= lies on these lines: {2,1282}, {10,98}, {103,10164}, {116,3634}, {118,516}, {120,24685}, {150,1698}, {152,165}, {519,11712}, {544,3828}, {551,10695}, {1125,2809}, {1362,3911}, {2786,9508}, {2801,3035}, {2808,6684}, {2810,6686}, {3033,6685}, {3842,6690}, {4712,24582}, {6541,17927}, {9780,20096}, {10175,10739}, {11028,13405}, {13411,18413}, {14543,21914}
= midpoint of X(10) and X(101)}
= reflection of X(i) in X(j) for these {i,j}: {116,3634}, {1125,6710}
= (6-8-13) search numbers [2.20970283517814542, 0.326175810443645090, 2.39498761228732692]
Q(X(11)) = MIDPOINT OF X(11) AND (2720)
= (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c+5 a^3 b c-2 a^2 b^2 c-3 a b^3 c+b^4 c-2 a^3 c^2-2 a^2 b c^2+4 a b^2 c^2+2 a^2 c^3-3 a b c^3+a c^4+b c^4-c^5) (2 a^7-2 a^6 b-3 a^5 b^2+3 a^4 b^3+a b^6-b^7-2 a^6 c+8 a^5 b c-3 a^4 b^2 c-4 a^3 b^3 c+4 a^2 b^4 c-4 a b^5 c+b^6 c-3 a^5 c^2-3 a^4 b c^2+8 a^3 b^2 c^2-4 a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2+3 a^4 c^3-4 a^3 b c^3-4 a^2 b^2 c^3+8 a b^3 c^3-3 b^4 c^3+4 a^2 b c^4-a b^2 c^4-3 b^3 c^4-4 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7) :: (barys)
= X[11]+X[2720], X[1737]+X[15524], X[2745]-3*X[21154]
= lies on these lines: {11,2720}, {521,3035}, {522,10271}, {1737,15524}, {2745,21154}, {3660,6001}
= midpoint of X(i) and X(j) for these {i,j}: {11,2720}, {1737,15524}
= (6-8-13) search numbers [1.24261794046012197, 1.81750078681728600, 1.80887873389850942]
Best regards
Ercole Suppa
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