[Antreas P. Hatzipolakis]:
Let ABC be a triangle and P a point.
Denote:
Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.
La =: NaP, Lb =: NbP, Lc =: NcP
L1, L2, L3 = the reflections of La, Lb, Lc in AI, BI, CI, resp.
Which is the locus of P such that: L1, L2, L3 are concurrent ?
Is the locus the Euler line of NaNbNc [ = IN line of ABC]?
And which is the locus of the point of concurrence as P moves on the locus (is it the OI line?) ?
--------------------------------------------------------------------------------------------
[Ercole Suppa]
*** locus of point P such that L1, L2, L3 are concurrent:
{trilinear polar of X(655) = IN of ABC} U {c1=circle of center N and radius (1/2)*sqrt(R(R-2r))}
where R,r are circumradius and inradius of ABC.
-- IN pass through ETC points X(i) for these i : {1,5,11,12,80,119,355,495,496,952,1317,1387,1411,1421,1483,1484,1807,1837,2006,2594,2596,2606,3614,4551,5219,5252,5396,5399,5400,5443,5531,5533,5534,5587,5660,5718,5719,5720,5721,5722,5723,5724,5725,5726,5727,5881,5886,5901,6127,6264,6265,6326,7173,7741,7951,7958,7972,7988,7989,7993,8068,8070,8227,9578,9581,9624,9817,9897,10057,10073,10283,10523,10592,10593,10826,10827,10886,10887,10942,10943,10944,10948,10949,10950,10954,10955,10956,10957,10958,10959,11373,11374,11375,11376,11698,11729,12019,12025,12433,12550,12735,12737,12738,12739,12740,12749,12750,12751,13244,14204,14584,14679,15017,15251,15252,15253,15888,15935,15943,15950,16173,17602,17717,17718,17719,17720,17721,17722,17723,17724,17725,17726,17857,18357,19372,19907,20586,22392}
-- c1 pass through ETC points X(i) for these i : {10, 502, 946, 11798, 13604, 15529}
*** The locus of the point of concurrence Q=Q(P):
-- if P moves on the line IN then Q(P) moves on the line IO
-- if P moves on c1 then Q(P) moves on circle c2 of center X(10222) and radius (R-2r)/2
*** pairs (P,Q(P))
-- pairs (P ∈ IN, Q(P)) : {1,1},{5,617},{11,65},{12,3057},{80,11009},{355,1482},{495,9957},{496,942},{952,10222},{1484,6583},{1837,2099},{5219,1697},{5252,2098},{5443,35},{5587,7982},{5881,16200},{5886,3},{5901,1385},{7741,5903},{7951,5697},{7958,7957},{7988,7991},{7989,11531},{8227,40},{9578,7962},{9581,3340},{9624,3576},{10283,15178},{10886,12435},{10944,5048},{10948,5570},{10950,11011},{11373,999},{11374,3295},{11375,55},{11376,56},{15888,5919},{15950,2646},{16173,5563},{17718,3303},{17720,5710},{18357,11278},{19907,11567}
-- pairs (P ∈ c1, Q(P)) : {10,3244}, {946,946}
*** some points Q=Q(P) :
-- P1=Q(X(119)) = MIDPOINT X(4) AND X(3885)
= -2 a^2 b (a+b-3 c) (a-b-c) c (a-b+c) (a+b+c)^2 (a^3-a^2 b-a b^2+b^3-a^2 c+3 a b c-b^2 c-a c^2-b c^2+c^3)^2 (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c-6 a^4 b c+8 a^3 b^2 c+4 a^2 b^3 c-9 a b^4 c+2 b^5 c-a^4 c^2+8 a^3 b c^2-16 a^2 b^2 c^2+8 a b^3 c^2+b^4 c^2-2 a^3 c^3+4 a^2 b c^3+8 a b^2 c^3-4 b^3 c^3+2 a^2 c^4-9 a b c^4+b^2 c^4+a c^5+2 b c^5-c^6) : : (barys)
= X[4]+X[3885], 3*X[392]-2*X[5690], 3*X[944]-X[9961], X[1071]-2*X[1483], X[3625]-2*X[20117], 8*X[3628]-7*X[4002], X[3633]+X[5693], 2*X[3635]-X[5884], 3*X[3655]-2*X[9943], 3*X[3656]-2*X[7686], 5*X[3698]-6*X[11230], 3*X[3753]-4*X[5901], 2*X[3884]-X[11362], 5*X[3890]-3*X[5657], 3*X[3898]-2*X[6684], 5*X[5439]-6*X[10283], 2*X[5836]-3*X[5886], X[7491]-2*X[10624], 3*X[7967]-2*X[13369], 2*X[9856]-X[18525]
= lies on these lines : {1,3},{4,3885},{5,6735},{8,6893},{72,5844},{119,946},{145,912},{355,3880},{392,5690},{496,17622},{519,5887},{944,9961},{952,12672},{962,12115},{971,18526},{1000,5555},{1071,1483},{1210,15558},{1320,12775},{1519,10942},{1699,11929},{1872,1877},{2136,5720},{2800,3244},{2950,12773},{3555,14988},{3560,3872},{3585,12749},{3625,20117},{3628,4002},{3633,5693},{3635,5884},{3655,9943},{3656,7686},{3698,11230},{3753,5901},{3869,6930},{3877,5084},{3878,5795},{3881,15528},{3884,11362},{3890,5657},{3898,6684},{4301,12608},{5053,21853},{5252,10525},{5439,10283},{5552,5603},{5587,11928},{5734,6970},{5761,6848},{5777,12625},{5836,5886},{6256,12699},{6827,9785},{6882,12053},{6923,12700},{6958,11373},{6971,7743},{7330,12629},{7491,10624},{7680,13463},{7967,13369},{7970,13189},{7978,13217},{7983,12189},{7984,12381},{9856,18525},{10526,12701},{10595,17567},{10698,13278},{10705,13118},{10738,12751},{10866,18527},{10912,11496},{12650,12686},{12705,18519},{13099,13313}
= midpoint of X(i) and X(j) for these {i,j}: {4,3885},{3633,5693},{5697,7982}
= reflection of X(i) in X(j) for these {i,j}: {3,9957},{65,10222},{1071,1483},{1482,13600},{3057,10284},{3625,20117},{7491,10624},{10273,10247},{10914,5},{11362,3884},{12645,5777},{18525,9856}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: 1,40,10269},{1,2077,1385},{1,3359,16203},{1,5903,18838},{1,11010,14803},{1,12703,11248},{942,9957,20789},{946,10915,119},{1482,10679,1},{2098,11509,1},{2099,10965,1},{3746,11014,1385},{10596,12245,5554},{10942,22791,1519},{12702,16203,3359}
= (6-8-13) search numbers [-2.24116367933029919, -1.55601084974910435, 5.75228599911697405]
-- P2=Q(X(502)) = X(502)X(8702) ∩ X(7984)X(11009)
= -2 a^2 (a-c) (b-c) (a+b+c)^3 (a^3-a^2 b-a b^2+b^3+3 a^2 c-a b c-b^2 c+3 a c^2-b c^2+c^3) (a^3-a^2 b-a b^2+b^3-a^2 c-a b c+3 b^2 c-a c^2+3 b c^2+c^3) (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8-2 a^6 b c+a^5 b^2 c+4 a^4 b^3 c+3 a^3 b^4 c+3 a^2 b^5 c-4 a b^6 c-5 b^7 c+a^6 c^2+a^5 b c^2+6 a^4 b^2 c^2+3 a^3 b^3 c^2-2 a b^5 c^2-5 b^6 c^2+4 a^4 b c^3+3 a^3 b^2 c^3-4 a^2 b^3 c^3+7 a b^4 c^3+5 b^5 c^3-3 a^4 c^4+3 a^3 b c^4+7 a b^3 c^4+12 b^4 c^4+3 a^2 b c^5-2 a b^2 c^5+5 b^3 c^5+3 a^2 c^6-4 a b c^6-5 b^2 c^6-5 b c^7-c^8) : : (barys)
= lies on these lines : {502,8702},{7984,11009}
= (6-8-13) search numbers [-1.60011240945530656, -2.41419903530916329, 6.05054646456240287]
*** Question: describe geometrically the transformation P -> Q(P)
Best regards
Ercole Suppa
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου