[Antreas P. Hatzipolakis]:
Let ABC be a triangle, HaHbHc the pedal triangle of H and P a point.
Denote:
H1, H2, H3 = the antipodes of Ha, Hb, Hc in the NPC, resp.
Which is the locus of P such that the reflections of PH1, PH2, PH3 in BC, CA, AB, resp are concurrent?
NPC + Euler line?
And which is the locus of the point of concurrence?
Linf + ?
--------------------------------------------------------------------------------------------
[Ercole Suppa]
*** The locus of point P such that the reflections of PH1, PH2, PH3 in BC, CA, AB, resp are concurrent : {Euler line GH} U {nine point circle NPC}
*** if P ∈ GH the locus of point of concurrence Q=Q(P) is the hyperbola
h: ∑ (b^2-c^2)(a^2-b^2-c^2)(a^6-a^4 b^2-a^2 b^4+b^6-a^4 c^2-2 a^2 b^2 c^2-b^4 c^2-a^2 c^4-b^2 c^4+c^6) x^2 + (b^-c^2)(a^2-b^2-c^2)(a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) y z = 0
-- pairs {P=X(i)∈GH, Q=X(j)} for these {i,j}: {2,4},{3,3},{4,6193},{5,52},{22,8907},{25,9937},{140,18488},{1368,10575},{6676,1209},{7495,23330}
-- Some points :
Z= center Z of hyperbola h = MIDPOINT OF X(110) AND X(19863)
= -a^2 (12 a^6 b^4 c^4+a^12 (b^2+c^2)-4 a^10 (b^4+c^4)-(b^2-c^2)^4 (b^6+2 b^4 c^2+2 b^2 c^4+c^6)+a^8 (5 b^6-4 b^4 c^2-4 b^2 c^4+5 c^6)+4 a^2 (b^2-c^2)^2 (b^8+c^8)-a^4 (5 b^10-9 b^8 c^2+8 b^6 c^4+8 b^4 c^6-9 b^2 c^8+5 c^10)) : : (barys)
= -18 R^4 SB SC+10 R^2 SB SC SW-SB SC SW^2 + (18 R^4+3 R^2 SB+3 R^2 SC-8 R^2 SW-SB SW-SC SW+SW^2)S^2 : : (barys)
= X[4]-3*X[12824], X[30]-X[16104], X[146]+X[17854], X[265]-2*X[11746], X[389]-X[542], X[568]-X[2854], X[1154]-X[3292], X[2777]-X[12605], 3*X[2979]-7*X[15020], 4*X[5462]-3*X[12099], X[5562]-3*X[5642], X[5609]+X[6102], 3*X[5892]-2*X[20397], X[6000]-X[10297], X[6241]+3*X[10706], X[7731]+3*X[15035], 3*X[9140]-7*X[15043], X[9517]-X[16230], 2*X[9729]-X[20417], 5*X[10574]-X[15054], X[10620]-2*X[15151], X[11412]-5*X[15034], 15*X[11451]-11*X[15025], 3*X[11597]-X[12606], X[13201]-5*X[15051], X[13417]+X[16163], 3*X[14644]+X[15102], 5*X[15021]-9*X[20791], 5*X[15059]-X[15100], X[21649]+X[24981]
= lies on these lines: {3,1177},{4,12824},{5,113},{24,110},{26,16165},{30,16104},{52,5095},{74,7503},{146,17854},{186,15136},{265,11746},{389,542},{399,7506},{468,13148},{568,2854},{973,12236},{1092,1511},{1112,3575},{1154,3292},{1539,18377},{1625,2493},{1995,5890},{2777,12605},{2979,15020},{3448,7544},{3548,6293},{5462,12099},{5504,19504},{5562,5642},{5609,6102},{5878,7728},{5892,20397},{5946,18474},{5972,7542},{6000,10297},{6241,10706},{6639,7723},{6644,15132},{7505,7722},{7547,11439},{7731,15035},{9140,15043},{9517,16230},{9729,20417},{10020,10272},{10574,15054},{10620,15151},{11412,15034},{11438,19140},{11451,15025},{11470,13352},{11560,22970},{11597,12606},{12133,23047},{12302,15472},{12359,12827},{12412,13198},{13201,15051},{13417,16163},{14644,15102},{14787,20126},{14982,18917},{15021,20791},{15059,15100},{16776,18440},{21649,24981}
= midpoint of X(i) and X(j) for these {i,j}: {110,1986},{113,11562},{185,15063},{5562,14448},{5609,6102},{12270,12292},{13417,16163},{21649,24981}
= reflection of X(i) in X(j) for these {i,j}: {125,9826},{265,11746},{974,14708},{1112,11557},{12358,5972},{14708,11561},{15738,5},{16003,16270},{20379,12006},{20417,9729}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {125,16223,9826},{265,16222,11746},{5609,20772,10539},{5642,14448,5562},{9730,16003,16270},{9970,15462,15141}
= (6-8-13) search numbers [1.72789371573424176, 1.04110306012467756, 2.12241141763610614]
Q(X(20)) = X(3)X(3620) ∩ X(4)X(9820)
= 15 a^10-35 a^8 b^2+18 a^6 b^4+6 a^4 b^6-a^2 b^8-3 b^10-35 a^8 c^2+52 a^6 b^2 c^2-22 a^4 b^4 c^2-4 a^2 b^6 c^2+9 b^8 c^2+18 a^6 c^4-22 a^4 b^2 c^4+10 a^2 b^4 c^4-6 b^6 c^4+6 a^4 c^6-4 a^2 b^2 c^6-6 b^4 c^6-a^2 c^8+9 b^2 c^8-3 c^10 : : (barys)
= -20 R^2 SB SC + 7 SB SC SW + (20 R^2 - 2 SB - 2 SC - 4 SW)S^2 : : (barys)
= lies on these lines: {3,3620},{4,9820},{110,3529},{186,9937},{376,1216},{631,1209},{3090,3431},{3147,12383},{3524,14516},{3545,12278},{6225,16163},{6353,12118},{7487,21850},{8254,18420},{8889,12038},{8907,21844},{9729,11179},{10299,11442},{11441,17538}
= (6-8-13) search numbers [4.55531390858508167, 2.81636165771102818, -0.411576931239447329]
Q(X(21)) = X(1209)X(6905) ∩ X(1385)X(1798)
= a (a+b) (a+c) (a^10-2 a^9 b-a^8 b^2+4 a^7 b^3-2 a^6 b^4+2 a^4 b^6-4 a^3 b^7+a^2 b^8+2 a b^9-b^10-2 a^9 c+5 a^8 b c-a^7 b^2 c-7 a^6 b^3 c+7 a^5 b^4 c-a^4 b^5 c-3 a^3 b^6 c+3 a^2 b^7 c-a b^8 c-a^8 c^2-a^7 b c^2+4 a^6 b^2 c^2-a^5 b^3 c^2-6 a^4 b^4 c^2+5 a^3 b^5 c^2-3 a b^7 c^2+3 b^8 c^2+4 a^7 c^3-7 a^6 b c^3-a^5 b^2 c^3+2 a^4 b^3 c^3+2 a^3 b^4 c^3-3 a^2 b^5 c^3+3 a b^6 c^3-2 a^6 c^4+7 a^5 b c^4-6 a^4 b^2 c^4+2 a^3 b^3 c^4-2 a^2 b^4 c^4-a b^5 c^4-2 b^6 c^4-a^4 b c^5+5 a^3 b^2 c^5-3 a^2 b^3 c^5-a b^4 c^5+2 a^4 c^6-3 a^3 b c^6+3 a b^3 c^6-2 b^4 c^6-4 a^3 c^7+3 a^2 b c^7-3 a b^2 c^7+a^2 c^8-a b c^8+3 b^2 c^8+2 a c^9-c^10) : : (barys)
= lies on these lines: {1209,6905},{1385,1798},{6906,10575}
= (6-8-13) search numbers [-13.9428717833428584, -15.6098228923991679, 20.8826365381881992]
Q(X(23)) = MIDPOINT OF X(3519) AND X(23236)
= a^2 (a^14-3 a^12 b^2+a^10 b^4+5 a^8 b^6-5 a^6 b^8-a^4 b^10+3 a^2 b^12-b^14-3 a^12 c^2+2 a^10 b^2 c^2+10 a^8 b^4 c^2-16 a^6 b^6 c^2+13 a^4 b^8 c^2-10 a^2 b^10 c^2+4 b^12 c^2+a^10 c^4+10 a^8 b^2 c^4-11 a^6 b^4 c^4+6 a^2 b^8 c^4-6 b^10 c^4+5 a^8 c^6-16 a^6 b^2 c^6+2 a^2 b^6 c^6+3 b^8 c^6-5 a^6 c^8+13 a^4 b^2 c^8+6 a^2 b^4 c^8+3 b^6 c^8-a^4 c^10-10 a^2 b^2 c^10-6 b^4 c^10+3 a^2 c^12+4 b^2 c^12-c^14) : : (barys)
= -9 R^4 SB SC+8 R^2 SB SC SW-2 SB SC SW^2 + (18 R^4+3 R^2 SB+3 R^2 SC-14 R^2 SW+2 SW^2)S^2 : : (barys)
=X[23]-X[1154], X[186]-X[539], X[511]-X[12380], X[542]-X[7512], 2*X[1493]-3*X[11597], 4*X[3628]-3*X[11804], 3*X[5642]-2*X[12242], X[6152]-X[14984], X[7464]-X[18400], X[9934]-X[10628], X[14865]-X[17702]
= lies on these lines : {3,2888},{23,1154},{24,9925},{26,9143},{52,110},{54,575},{186,539},{195,9716},{399,17714},{511,12380},{542,7512},{576,7730},{631,19468},{973,6593},{1092,12291},{1147,12280},{1173,11800},{1209,7550},{1493,11597},{1511,10821},{2917,2930},{2931,6193},{3519,7488},{3628,11804},{5642,12242},{6152,14984},{6288,7527},{7464,18400},{7545,20424},{7691,10575},{9545,13472},{9920,12082},{9934,10628},{10272,18369},{10594,12310},{11803,13621},{12226,15132},{12273,15083},{12899,18951},{14865,17702},{15133,23330},{15246,20379}
= midpoint of X(3519) and X(23236)
= reflection of X(i) in X(j) for these {i,j}: {2914,110},{15089,1511}
= (6-8-13) search numbers [17.4253459340519605, -11.5877142107445421, 3.62046081209123418]
Q(X(24)) = X(3)X(18124) ∩ X(4)X(9932)
= a^2 (a^2-b^2-c^2) (a^18-3 a^16 b^2+8 a^12 b^6-6 a^10 b^8-6 a^8 b^10+8 a^6 b^12-3 a^2 b^16+b^18-3 a^16 c^2+4 a^14 b^2 c^2+6 a^12 b^4 c^2-12 a^10 b^6 c^2+8 a^8 b^8 c^2-4 a^6 b^10 c^2-6 a^4 b^12 c^2+12 a^2 b^14 c^2-5 b^16 c^2+6 a^12 b^2 c^4+8 a^10 b^4 c^4-18 a^8 b^6 c^4+2 a^4 b^10 c^4-8 a^2 b^12 c^4+10 b^14 c^4+8 a^12 c^6-12 a^10 b^2 c^6-18 a^8 b^4 c^6+24 a^6 b^6 c^6+4 a^4 b^8 c^6-28 a^2 b^10 c^6-10 b^12 c^6-6 a^10 c^8+8 a^8 b^2 c^8+4 a^4 b^6 c^8+54 a^2 b^8 c^8+4 b^10 c^8-6 a^8 c^10-4 a^6 b^2 c^10+2 a^4 b^4 c^10-28 a^2 b^6 c^10+4 b^8 c^10+8 a^6 c^12-6 a^4 b^2 c^12-8 a^2 b^4 c^12-10 b^6 c^12+12 a^2 b^2 c^14+10 b^4 c^14-3 a^2 c^16-5 b^2 c^16+c^18) : : (barys)
= -28 R^6 SB SC+24 R^4 SB SC SW-8 R^2 SB SC SW^2+SB SC SW^3 + (28 R^6+2 R^4 SB+2 R^4 SC-30 R^4 SW-R^2 SB SW-R^2 SC SW+10 R^2 SW^2-SW^3)S^2 : : (barys)
= lies on these lines: {3,18124},{4,9932},{22,7689},{52,19141},{578,9813},{1209,7503},{12293,23330}
= (6-8-13) search numbers [-1.75371949609702696, -1.78984721396560362, 5.68919847438996329}]
*** if P ∈ NPC the locus of point of concurrence Q=Q(P) is the infinite line and the Q(P) = isogonal conjugate of point P1 (= image of P under homotety with center H and ratio 2)
Best regards
Ercole Suppa
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου