[Antreas P. Hatzipolakis]:
GENERALIZATION
Let ABC be a triangle, A'B'C' the pedal triangle of I and P a point.
Denote:
Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.
D = the Poncelet point of ABCI = Feuerbach point X(11).
P1, P2, P3 = the reflections of P in NbNc, NcNa, NaNb, resp.
La =: DP1, Lb =: DP2, Lc =: DP3
L1, L2, L3 = the reflections of La, Lb, Lc in BC, CA, AB, resp.
Which is the locus of P such that:
1. L1, L2, L3 are concurrent ?
2. The parallels to L1, L2, L3 through A', B', C' are concurrent ?
Are both loci the Euler line of NaNbNc [= IN line of ABC] ?
And which are the loci of the points of concurrence as P moves on the loci?
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[Ercole Suppa]
(1)
*** The locus of point P such that L1, L2, L3 are concurrent: {polar trilineal of X(655) = IN of ABC} U {c1=circle centered at X(1385) with radius R/2} , where R=circumradius 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 : {214, 11700, 11709, 11710, 11711, 11712, 11713, 11714, 11715, 11716, 11717, 11718, 11719, 11720, 11721, 11722, 11796, 11797, 12265, 13497, 15746, 22477}
*** The locus of the point of concurrence as P moves on the IN line is the hyperbola h1
-- equation of h1
∑ [(a-b-c) (b-c) (a^7-5 a^5 b^2+2 a^4 b^3+7 a^3 b^4-4 a^2 b^5-3 a b^6+2 b^7+4 a^4 b^2 c-7 a^3 b^3 c-a^2 b^4 c+7 a b^5 c-3 b^6 c-5 a^5 c^2+4 a^4 b c^2+9 a^3 b^2 c^2-5 a^2 b^3 c^2-6 a b^4 c^2+3 b^5 c^2+2 a^4 c^3-7 a^3 b c^3-5 a^2 b^2 c^3+12 a b^3 c^3-2 b^4 c^3+7 a^3 c^4-a^2 b c^4-6 a b^2 c^4-2 b^3 c^4-4 a^2 c^5+7 a b c^5+3 b^2 c^5-3 a c^6-3 b c^6+2 c^7) x^2+2 (b-c) (a^8-a^7 b-a^6 b^2+3 a^3 b^5-a^2 b^6-2 a b^7+b^8-a^7 c+4 a^5 b^2 c+a^4 b^3 c-7 a^3 b^4 c+4 a b^6 c-b^7 c-a^6 c^2+4 a^5 b c^2-9 a^4 b^2 c^2+5 a^3 b^3 c^2+2 a^2 b^4 c^2+a b^5 c^2-2 b^6 c^2+a^4 b c^3+5 a^3 b^2 c^3-2 a^2 b^3 c^3-3 a b^4 c^3+b^5 c^3-7 a^3 b c^4+2 a^2 b^2 c^4-3 a b^3 c^4+2 b^4 c^4+3 a^3 c^5+a b^2 c^5+b^3 c^5-a^2 c^6+4 a b c^6-2 b^2 c^6-2 a c^7-b c^7+c^8) y z] = 0
-- center of h1
W1 = -(a^4 (b^2+c^2)-2 a^3 b c (b+c)-2 a^2 (b^4-b^3 c-b^2 c^2-b c^3+c^4)+2 a b c (b-c)^2 (b+c)+(b-c)^4 (b+c)^2) (2 a^9-4 a^8 (b+c)+12 a^7 b c+a^6 (5 b^3-9 b^2 c-9 b c^2+5 c^3)-a^5 (5 b^4+2 b^3 c-16 b^2 c^2+2 b c^3+5 c^4)+a^4 (b-c)^2 (2 b^3+11 b^2 c+11 b c^2+2 c^3)+2 a^3 (b-c)^2 (b^4-3 b^3 c-4 b^2 c^2-3 b c^3+c^4)-a^2 (b-c)^2 (3 b^5-b^4 c-4 b^3 c^2-4 b^2 c^3-b c^4+3 c^5)+a (b^2-c^2)^4-b c (b-c)^4 (b+c)^3) : : (barys)
= lies on this line: {1830,7649}
= (6-8-13) search numbers [-1.62990407826707286, -0.820833345202799041, 4.96119714163265822]
-- h1 pass through ETC points X(i) for these i : {11, 65, 1155}
*** Let Q1=Q1(P) denote the point of concurrence. pairs (P,Q1(P)) :
-- pairs (P ∈ IN) : {11,1155},{952, 11},{6265,65}
-- pairs (P ∈ c1, Q1(P)) : no pairs
*** some points Q1=Q1(P) :
-- P1=Q1(X(1)) = REFLECTION OF X(3057) IN X(13756)
= -a (a^7 (b+c)-(b-c)^6 (b+c)^2-a^6 (b^2+6 b c+c^2)-a^5 (b^3-7 b^2 c-7 b c^2+c^3)+a^4 (b^4-10 b^2 c^2+c^4)+a^2 (b-c)^2 (b^4+4 b^3 c-b^2 c^2+4 b c^3+c^4)+a (b-c)^2 (b^5-5 b^4 c+2 b^3 c^2+2 b^2 c^3-5 b c^4+c^5)-a^3 (b^5+b^4 c-3 b^3 c^2-3 b^2 c^3+b c^4+c^5)) : :
= 3*X[354]-2*X[3025]
= lies on these lines: {80,517},{88,105},{354,3025},{513,17660},{953,2646},{1319,4351},{3057, 3326},{3259,17605},{5048,10702}
= reflection of X(3057) in X(13756)
= (6-8-13) search numbers [-2.76686390500768185, -0.198758879236307174, 5.05528089292075262]
-- P2=Q1(X(5)) = -a (2 a^11-5 a^10 (b+c)-(b-c)^8 (b+c)^3-a^9 (b^2-20 b c+c^2)+a^8 (13 b^3-19 b^2 c-19 b c^2+13 c^3)+a (b-c)^4 (b+c)^2 (b^4-10 b^3 c+15 b^2 c^2-10 b c^3+c^4)-a^7 (10 b^4+23 b^3 c-64 b^2 c^2+23 b c^3+10 c^4)-2 a^4 (b-c)^2 (2 b^5+20 b^4 c-17 b^3 c^2-17 b^2 c^3+20 b c^4+2 c^5)-a^6 (8 b^5-57 b^4 c+46 b^3 c^2+46 b^2 c^3-57 b c^4+8 c^5)-a^3 (b-c)^2 (8 b^6-25 b^5 c-20 b^4 c^2+59 b^3 c^3-20 b^2 c^4-25 b c^5+8 c^6)+a^5 (16 b^6-26 b^5 c-59 b^4 c^2+136 b^3 c^3-59 b^2 c^4-26 b c^5+16 c^6)+a^2 (b-c)^2 (5 b^7+4 b^6 c-35 b^5 c^2+25 b^4 c^3+25 b^3 c^4-35 b^2 c^5+4 b c^6+5 c^7)) : : (barys)
= lies on this line : {36,1411}
= (6-8-13) search numbers [-0.622057667801356489, -2.80670445887121149, 5.87087110780353621]
----------------------------------------
(2)
*** The locus of point P such that the parallels to L1, L2, L3 through A', B', C' concurr: same as locus (1)
*** The locus of the point of concurrence as P moves on the IN line is the hyperbola h2
-- equation of h2:
∑ [-b (a-b-c)^2 (2 a-b-c) (b-c) c x^2-a (b-c) (a+b-c) (a-b+c) (a^2-b^2+b c-c^2) y z] = 0
-- center of h2:
W2 = -a (a+b-c) (a-b+c) (a^7 (b-c)^2+2 a^6 b c (b+c)+b (b-c)^2 c (b+c)^3 (b^2-3 b c+c^2)-a^5 (b+c)^2 (3 b^2-2 b c+3 c^2)+3 a^4 b c (3 b^3+b^2 c+b c^2+3 c^3)-a (b^2-c^2)^2 (b^4-4 b^3 c-3 b^2 c^2-4 b c^3+c^4)+a^2 b c (-12 b^5+9 b^4 c+5 b^3 c^2+5 b^2 c^3+9 b c^4-12 c^5)+a^3 (3 b^6+2 b^5 c-16 b^4 c^2+6 b^3 c^3-16 b^2 c^4+2 b c^5+3 c^6)) : : (barys)
= lies on this line : {8758,22465}
= (6-8-13) search numbers [0.457897650963244096, 0.589376933786446766, 3.02129691961072637]
-- h2 pass through ETC points X(i) for these i : {1, 1319, 4017, 13756, 17705, 22464}
*** Let Q2=Q2(P) denote the point of concurrence. Pairs (P,Q2(P))
-- pairs (P ∈ IN) : {1, 13756}, {11, 1319}, {952, 1}
-- pairs (P ∈ c1, Q2(P)) : no pairs
*** some points Q2=Q2(P) :
-- P3=Q2(X(5)) = -a (2 a^6-3 a^5 (b+c)+a^2 b c (-9 b^2+16 b c-9 c^2)+a^4 (-3 b^2+10 b c-3 c^2)+(b^2-c^2)^2 (b^2-b c+c^2)-a (b-c)^2 (3 b^3-2 b^2 c-2 b c^2+3 c^3)+a^3 (6 b^3-5 b^2 c-5 b c^2+6 c^3)) (a^6-a^5 (b+c)+(b-c)^4 (b+c)^2-a^4 (b^2-4 b c+c^2)-a (b-c)^2 (b^3-2 b^2 c-2 b c^2+c^3)+a^3 (2 b^3-3 b^2 c-3 b c^2+2 c^3)-a^2 (b^4+2 b^3 c-7 b^2 c^2+2 b c^3+c^4)) : : (barys)
= lies on this line : {3025,17705}
= (6-8-13) search numbers [0.899161391181984672, -0.740376732228415518, 3.73823542367390461]
-- P4=Q2(X(12)) = -a (a+b-c) (a-b+c) (a^7+6 a^5 b c-2 a^6 (b+c)-(b-c)^4 (b+c)^3+a (b^2-c^2)^2 (2 b^2-5 b c+2 c^2)+a^4 (3 b^3-7 b^2 c-7 b c^2+3 c^3)+2 a^2 b c (4 b^3-5 b^2 c-5 b c^2+4 c^3)-a^3 (3 b^4+b^3 c-17 b^2 c^2+b c^3+3 c^4)) (2 a^6-3 a^5 (b+c)+6 a^3 (b-c)^2 (b+c)+a^2 b c (-9 b^2+20 b c-9 c^2)+a^4 (-3 b^2+10 b c-3 c^2)+(b^2-c^2)^2 (b^2-b c+c^2)-a (3 b^5-9 b^4 c+7 b^3 c^2+7 b^2 c^3-9 b c^4+3 c^5)) : : (barys)
= (6-8-13) search numbers [-0.0961998509199611735, 2.77610392379448820, 1.76314631201279243]
Best regards
Ercole Suppa
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