[Antreas P. Hatzipolakis]:
Let ABC be a triangle, A'B'C' the pedal triangle of O and P a point.
Denote:
(N), (Na), (Nb), (Nc) = the NPCs of ABC, PBC, PCA, PAB, resp.
The line A'N intersects again (Na) at A"
The line A'Na intersects again (N) at A*
The line B'N intersects again (Nb) at B"
The line B'Nb intersects again (N) at B*
The line C'N intersects again (Nc) at C"
The line C'Nc intersects again (N) at C*
Ma, Mb, Mc = the midpoints of A"A*, B"B*, C"C*, resp.
For P = I:
A'B'C', MaMbMc are perspective.
Perspector?
Locus?
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[Ercole Suppa]
*** For P=I the perspector of A'B'C' and MaMbMc is the point Q=Q(I) such that
Q1=Q(I) = COMPLEMENT OF X(35)
= a^2 b^2-b^4+a b^2 c+a^2 c^2+a b c^2+2 b^2 c^2-c^4 : : (barys)
= 3*X[2]-X[35], X[12]-X[519], X[30]-X[4999], X[141]-X[9047], X[515]-X[6842], X[516]-X[6831], X[528]-X[6668], X[535]-X[2975], X[758]-X[6734], X[2802]-X[8068], 3*X[3584]-X[3871], 3*X[3679]+X[11280], X[4324]-3*X[17549], X[5288]+X[20060], X[5850]-X[6067], X[9397]-X[15280], X[15310]-X[24251]
= isotomic conjugate of isogonal conjugate of X(20961)
= complement of X(35)
= complementary conjugate of X(3647)
= lies on these lines: {1,2476},{2,35},{4,993},{5,10},{8,4867},{9,6990},{11,214},{12,519},{21,3583},{30,4999},{36,2475},{40,6830},{56,17532},{58,14009},{72,17605},{75,7752},{79,3218},{100,7504},{113,124},{115,1107},{116,17050},{119,15863},{141,9047},{149,3746},{165,6943},{191,5057},{226,3874},{238,24880},{325,20888},{355,6980},{377,499},{381,958},{386,17717},{404,6681},{405,10896},{427,1900},{443,10200},{495,3244},{496,551},{497,6856},{498,3434},{515,6842},{516,6831},{528,6668},{535,2975},{567,9701},{626,21264},{758,6734},{908,3678},{936,6991},{942,3838},{956,10895},{997,6829},{1001,9669},{1086,24167},{1089,3006},{1100,5949},{1203,24883},{1210,5883},{1376,1656},{1478,6871},{1484,15178},{1506,1575},{1532,19925},{1574,7603},{1594,1861},{1621,4857},{1698,4193},{1699,5705},{1706,6975},{1724,24892},{1737,3754},{2077,6952},{2140,17046},{2478,19854},{2550,3090},{2551,3545},{2599,6358},{2802,8068},{3035,3628},{3086,5177},{3091,19843},{3120,3670},{3136,3454},{3295,10197},{3336,20292},{3419,11375},{3576,6937},{3582,5253},{3584,3871},{3614,3626},{3624,4197},{3625,10592},{3634,3925},{3635,15888},{3647,5745},{3671,15844},{3679,11280},{3698,17619},{3703,4066},{3704,4717},{3706,21081},{3742,3824},{3743,24210},{3753,17606},{3811,5219},{3816,8728},{3826,3847},{3828,17533},{3831,21241},{3833,12446},{3851,9708},{3868,10129},{3869,18393},{3872,10827},{3881,13407},{3884,24987},{3892,21620},{3898,12053},{3918,6702},{3934,20541},{4015,25006},{4119,21067},{4202,19864},{4292,4973},{4297,6907},{4302,6910},{4316,5303},{4324,17549},{4386,7746},{4426,5475},{4511,5443},{4640,22793},{4691,21031},{4847,21077},{4868,5530},{4880,14450},{5025,17030},{5046,5251},{5051,19863},{5055,9709},{5080,5258},{5082,10588},{5154,9780},{5192,19846},{5225,6857},{5231,9612},{5249,6701},{5252,22837},{5255,17734},{5280,17737},{5288,20060},{5289,18493},{5432,20104},{5433,11112},{5450,6923},{5499,13624},{5587,6941},{5603,6874},{5691,6932},{5697,17057},{5714,24477},{5726,12629},{5777,21635},{5791,24703},{5794,5886},{5832,18232},{5850,6067},{5882,10943},{6224,24926},{6256,6982},{6284,7483},{6684,6882},{6690,15171},{6700,10171},{6796,6863},{6832,10598},{6836,12511},{6853,10902},{6862,10525},{6903,7688},{6913,10893},{6919,19855},{6922,10164},{6945,7989},{7280,17579},{7514,9712},{7828,20179},{7887,20172},{7958,12447},{8087,11013},{8226,12571},{9397,15280},{9654,12513},{9655,11194},{9678,23261},{9702,18350},{9713,13861},{9957,21630},{10056,10585},{10106,10957},{10172,23513},{10199,17528},{10572,24541},{10826,19860},{10894,22770},{11019,16193},{11113,24953},{11114,18514},{11281,12433},{12953,16370},{13747,20107},{15310,24251},{15792,19642},{16601,21090},{16819,17669},{17062,17761},{17173,20961},{17443,21018},{17529,19878},{17674,19847},{17792,24206},{17866,21207},{17889,24046},{19861,23708},{23537,24239}
= midpoint of X(i) and X(j) for these {i,j}: {1,5086},{8,11009},{12,24390},{2975,3585},{5288,20060},{6734,12047}
= reflection of X(i) in X(j) for these {i,j}: {2646,1125},{5267,4999},{14526,6701}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,2476,3822},{1,11680,24387},{2,1479,5248},{2,7741,3825},{5,10,3814},{5,2886,10},{5,7681,3817},{8,5141,7951},{10,946,3878},{10,3817,21616},{10,11813,960},{10,21616,10176},{11,442,1125},{226,10916,3874},{443,10589,10200},{495,3813,3244},{497,6856,10198},{498,3434,8715},{942,3838,11263},{960,9955,11813},{1125,17647,214},{1210,12609,5883},{1478,10527,8666},{1699,5705,12514},{1699,6828,12558},{2476,11680,1},{2975,17577,3585},{3419,11375,22836},{3434,6933,498},{3816,8728,19862},{3820,9710,10},{3822,24387,1},{3825,3841,2},{3826,3847,17527},{3918,6702,24982},{3925,4187,3634},{3925,7173,4187},{5836,9956,10},{6871,10527,1478},{8728,10593,3816},{17530,24390,12}
= (6-8-13) search numbers [2.30170842367355298, 1.12595349395990891, 1.79883125208587925]
-------------------
**** The locus of points P such that A'B'C' and MaMbMc are perspective = {c1 = circumcircle} U {K003 = McCay cubic} U {q4 = circumquartic}
*** c1: c^2 x y + b^2 x z + a^2 y z = 0
-- if P∈c1 then Q(P) is the midpoint of segment PH, where H is the orthocenter of ABC
-- locus of point Q when P moves on the circumcircle is the NPC of ABC = image of circumcircle under an homotety with center H and radius R/2 (R=circumradius)
-- pairs {P=X(i) ∈ c1, Q=X(j)} for these {i,j}: {74,125},{98,115},{99,114},{100,119},{101,118},{102,124},{103,116},{104,11},{105,5511},{106,5510},{107,133},{109,117},{110,113},{111,5512},{112,132},{477,3258},{691,16188},{842,5099},{915,5521},{917,5190},{925,131},{930,128},{933,18402},{953,3259},{972,5514},{1113,1312},{1114,1313},{1141,137},{1292,120},{1293,121},{1294,122},{1295,123},{1296,126},{1297,127},{1298,130},{1299,135},{1300,136},{1303,129},{1304,18809},{1379,2040},{1380,2039},{2373,14672},{2687,5520},{2693,16177},{2698,2679},{2723,15612},{2724,1566},{2734,10017},{3563,5139},{5606,5950},{5951,5952},{6082,6092},{6233,13234},{6323,12494},{10121,15169},{11568,13994},{12507,13249},{13238,12624},{13597,11792},{14720,21662},{15323,5518},{15324,13613},{18401,20625},{22751,14103},{23232,138},{23233,139}
*** K003 : ∑ a^2 c^2 (a^2 + b^2 - c^2) y^2 z - a^2 b^2 (a^2 - b^2 + c^2) y z^2 = 0
-- points X(i) ∈ K003 for these i: {1, 3, 4, 1075, 1745, 3362, 13855}
-- pairs {P=X(i) ∈ K003, Q=X(j)} for these {i,j}: {3, 140}, {4, 5}
*** q4 : ∑ (a^4-2 a^2 b^2+b^4-2 a^2 c^2-4 b^2 c^2+c^4) x^2 y z-2 a^2 c^2 y^3 z+a^2 (a^2-2 b^2-2 c^2) y^2 z^2-2 a^2 b^2 y z^3 = 0
-- q4 does not contain ETC points
*** Some points
Q2=Q(X(108)) = ISOGONAL CONJUGATE OF X(15405)
= (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2 b-b^3+a^2 c-2 a b c+b^2 c+b c^2-c^3) (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+2 a^3 b^2 c-3 a b^4 c-a^4 c^2+2 a^3 b c^2-4 a^2 b^2 c^2+2 a b^3 c^2+b^4 c^2-2 a^3 c^3+2 a b^2 c^3+2 a^2 c^4-3 a b c^4+b^2 c^4+a c^5-c^6) : : (barys)
= 3*X[2]-X[1295], X[3]-2*X[6717], 2*X[5]-X[123], X[113]-X[2850], X[114]-X[2798], X[116]-X[2823], X[117]-X[2849], X[118]-X[2812], X[119]-X[2804], X[120]-X[9521], X[121]-X[9525], X[126]-X[9531], X[133]-X[2845], 3*X[381]-X[10746], X[515]-X[11719], X[1528]-X[6001], X[1596]-X[2834], X[2840]-X[5510], X[2851]-X[5512], 3*X[3545]-X[10715], 3*X[5603]-X[10702], 3*X[5886]-2*X[11733], X[10763]-3*X[14853]
= lies on the nine point circle and these lines : {2,1295},{3,6717},{4,11},{5,123},{12,3318},{113,2850},{114,2798},{115,1865},{116,2823},{117,2849},{118,2812},{119,2804},{120,9521},{121,9525},{122,442},{124,946},{125,429},{126,9531},{133,2845},{136,431},{225,20620},{235,5521},{281,5514},{381,10746},{403,5520},{406,11496},{515,11719},{1465,1532},{1519,1875},{1528,6001},{1537,1845},{1596,2834},{1878,3259},{1881,5190},{2840,5510},{2851,5512},{3545,10715},{4242,24466},{5603,10702},{5886,11733},{6907,17073},{7952,10271},{10763,14853},{15908,17555}
= isogonal conjugate of X(15405)
= midpoint of X(4) and X(108)
= complement of X(1295)
= complementary conjugate of X(6001)
= perspector of circumconic centerd at X(14571)
= reflection of X(i) in X(j) for these {i,j}: {3,6717},{123,5}
= (6-8-13) search numbers [-2.87919007190569828, -3.20107781410495185, 7.18565223255197575]
Q3=Q(X(476)) = ISOGONAL CONJUGATE OF X(15396)
=-54 R^4 S^2+S^4-9 R^2 S^2 SB-9 R^2 S^2 SC-162 R^4 SB SC-3 S^2 SB SC+33 R^2 S^2 SW+2 S^2 SB SW+2 S^2 SC SW+63 R^2 SB SC SW-5 S^2 SW^2-5 SB SC SW^2 : : (barys)
= -(a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2+2 a^4 b^2 c^2-2 a^2 b^4 c^2-b^6 c^2-3 a^4 c^4-2 a^2 b^2 c^4+4 b^4 c^4+3 a^2 c^6-b^2 c^6-c^8) (2 a^8-2 a^6 b^2-a^4 b^4+b^8-2 a^6 c^2+4 a^4 b^2 c^2-4 b^6 c^2-a^4 c^4+6 b^4 c^4-4 b^2 c^6+c^8) : : (barys)
= 3*X[2]-X[477], X[4]+X[476], 2*X[5]-X[3258], X[20]+X[14989], X[30]-X[125], X[113]-X[523], 5*X[3091]-X[14731], 2*X[3154]-3*X[23515], X[3448]-3*X[5627], 2*X[5446]-X[16978], 2*X[5972]-X[14934], X[7728]+3*X[14993], X[9179]-X[23699], 2*X[12079]-X[16003], X[14611]-2*X[16534], 3*X[14644]-X[17511], X[14915]-X[16280], X[16340]-2*X[20304]
= lies on these lines : {2,477},{3,16177},{4,476},{5,3258},{20,14989},{30,125},{113,523},{115,3003},{122,2072},{127,10297},{135,10151},{136,403},{137,11563},{235,16178},{381,2453},{1522,1523},{3091,14731},{3154,23515},{3448,5627},{5446,16978},{5520,6841},{5972,14934},{7471,15468},{7728,14993},{9179,23699},{10024,12091},{11251,18809},{12079,16003},{14611,16534},{14644,17511},{14915,16280},{16188,18556},{16340,20304},{18402,23290},{18403,20625}
= isogonal conjugate of X(15396)
= complement of X(477)
= complementary conjugate of X(5663)
= perspector of circumconic centerd at X(3018)
= midpoint of X(i) and X(j) for these {i,j}: {4,476},{5,18319},{20,14989},{1553,6070}
= reflection of X(i) in X(j) for these {i,j}: {3,22104},{3258,5},{10113,21316},{14611,16534},{14934,5972},{16003,12079},{16340,20304},{16978,5446}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {16340,21315,20304}
= (6-8-13) search numbers [-2.85156919150931209, -3.24537256812643147, 7.20356973284543724]
Q4=Q(X(675)) = MIDPOINT OF X(4) AND X(675)
= (b-c)^2 (-a^4+2 a^3 b-2 a^2 b^2+b^4+2 a^3 c-2 a^2 b c-2 a^2 c^2-2 b^2 c^2+c^4) (a^3 b-a^2 b^2-a b^3+b^4+a^3 c-a^2 b c-a b^2 c+b^3 c-a^2 c^2-a b c^2-2 b^2 c^2-a c^3+b c^3+c^4) : : (barys)
= X[4]+X[675], 2*X[5]-X[5513]
= lies on the nine point circle and these lines: {4,675},{5,5513},{12,6025},{117,5805},{118,381},{120,6881},{132,15762},{1596,20622}
= midpoint of X(4) and X(675)
= reflection of X(5513) in X(5)
= (6-8-13) search numbers [0.972852319167607773, 3.11319374266298307, 1.03636774352495647]
Best regards
Ercole Suppa
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