## Πέμπτη, 20 Οκτωβρίου 2016

### TRIANGLE CENTERS FROM HYACINTHOS

H001 = HATZIPOLAKIS - MONTESDEOCA

Barycentrics (a (2 a^3 - 3 a^2 (b + c) + 3 (b - c)^2 (b + c) - 2 a (b^2 - 3 b c + c^2)) :

Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

A1, B1, C1 = the orthogonal projections of Na, Nb, Nc on IA, IB, IC, resp.

A2, B2, C2 = the reflections of Na, Nb, Nc, in IA, IB, IC, resp.

The Euler line of A2B2C2 is the OI line of ABC.

The point is the O of ABC wrt the triangle A2B2C2

The orthocenter of A1B1C1 is the point X942

(Antreas Hatzipolakis and Angel Montesdeoca, Sept. 13, 2016. See: Hyacinthos #24380)

The point lies on these lines: {1, 3}, {4, 1392}, {5, 519}, {8, 3090}, {10, 3628}, {20, 3655}, {30, 4301}, {72, 1173}, {140, 551}, {145, 355}, {381, 5881}, {392, 5047}, {515, 1483}, {518, 576}, {546, 946}, {547, 4669}, {548, 5493}, {573, 3723}, {575, 1386}, {631, 3654}, {632, 1125}, {944, 3146}, {956, 3951}, {962, 3529}, {1000, 5703}, {1056, 4323}, {1058, 4345}, {1210, 1387}, {1320, 1389}, {1339, 6048}, {1457, 5399}, {1656, 3679}, {1657, 9589}, {1837, 7743}, {1870, 1872}, {2771, 7984}, {2800, 3881}, {3058, 7491}, {3419, 6984}, {3485, 6982}, {3488, 5812}, {3523, 3653}, {3525, 3616}, {3555, 5887}, {3584, 5559}, {3585, 7972}, {3621, 5818}, {3622, 5657}, {3632, 5079}, {3633, 5072}, {3636, 6684}, {3680, 6918}, {3872, 3984}, {3892, 5884}, {3913, 6911}, {3915, 5398}, {3940, 4853}, {3962, 5288}, {3991, 4919}, {4004, 5253}, {4511, 6946}, {4677, 5055}, {4870, 6980}, {4902, 5059}, {4930, 6913}, {5044, 5289}, {5054, 9588}, {5076, 5691}, {5258, 7489}, {5722, 5761}, {5727, 9669}, {6419, 7969}, {6420, 7968}, {6447, 9583}, {6519, 9616}, {6863, 10056}, {6914, 8666}, {6924, 8715}, {6958, 10072}, {6988, 7320}

= Midpoint of X(i) and X(j) for these {i,j}: {1, 1482}, {3, 7982}, {40, 8148}, {145, 355}, {381, 2487}, {946, 3244}, {1320, 6265}, {1657, 9589}, {3241, 3656}, {3555, 5887}, {4301, 5882}.

= Reflection of X(i) in X(j) for these {i,j}: {8, 9956}, {10, 5901}, {65, 6583}, {355, 9955}, {1385, 1}, {1483, 3635}, {3579, 1385}, {4669, 547}, {5493, 548}, {5690, 1125}, {6684, 3636}.

H002 = HATZIPOLAKIS - MONTESDEOCA

Barycentrics( a^2 (2 a^8-5 a^6 (b^2+c^2)+a^4 (3 b^4+4 b^2 c^2+3 c^4)+a^2 (b^2-c^2)^2 (b^2+c^2)-(b^2-c^2)^2 (b^4+c^4)):

Let ABC be a triangle.

Denote:

Oa, Ob, Oc = the circumcenters of OBC, OCA, OAB, resp.

N1, N2, N3 = the NPC centers of OObOc, OOcOa, OOaOb, resp.

ABC, N1N2N3 are orthologic. The orthologic center (ABC, N1N2N3) is X74

The point is the orthologic center (N1N2N3, ABC)

(Antreas Hatzipolakis and Angel Montesdeoca, Oct. 20, 2016. See: Hyacinthos #24665)

The point lies on these lines: {2, 9833}, {3, 64}, {4, 1495}, {5, 5944}, {6, 3517}, {22, 1092}, {24, 184}, {25, 578}, {26, 206}, {30, 5448}, {39, 1971}, {49, 52}, {51, 54}, {110, 5562}, {125, 10018}, {140, 1503}, {143, 5097}, {156, 1658}, {159, 182}, {161, 569}, {185, 186}, {216, 3463}, {376, 5878}, {394, 9715}, {436, 8884}, {468, 6146}, {549, 6247}, {550, 1511}, {567, 9920}, {568, 9704}, {575, 2393}, {1181, 3515}, {1216, 7502}, {1660, 6644}, {1853, 3526}, {1899, 3147}, {1970, 3199}, {1994, 9706}, {2781, 7555}, {3060, 9545}, {3270, 9638}, {3292, 7556}, {3522, 5656}, {3528, 6225}, {3530, 6696}, {3534, 5895}, {3574, 7576}, {3917, 7512}, {5010, 6285}, {5050, 9924}, {5447, 7525}, {5449, 10020}, {5480, 7715}, {5651, 7509}, {5889, 9544}, {5894, 8703}, {6001, 7508}, {6102, 7575}, {6243, 9703}, {7280, 7355}, {8681, 9937}, {8718, 9934}.

H003 = HATZIPOLAKIS - MOSES

Barycentrics 4 a^4-4 a^3 b-5 a^2 b^2+4 a b^3+b^4-4 a^3 c+8 a^2 b c-4 a b^2 c-5 a^2 c^2-4 a b c^2-2 b^2 c^2+4 a c^3+c^4::

Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

Oib, Oib, Oic = the circumcenters of INbNc, INcNa, INaNb, resp.

The point is the centroid of OiaOibOic lying on the IN line.

(Antreas Hatzipolakis and Peter Moses, Oct. 20, 2016. See: Hyacinthos #24667)

The point lies on these lines: {1,5},{2,5844},{3,3622},{8, 3628},{30,5603},{140,1482},{ 145,1656},{165,3653},...

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5,1483),(1,5901,5),(1,7951, 1317),(1,9624,355),(355,5886, 7988),(1482,3616,140),(7988, 9624,5886).

= Reflection of X(i) and X(j) for these {i,j}: {{5,5886},{5657,140},{5790, 547},{5886,5901},{8703,3576}}.

= Midpoint of X(i) and X(j) for these {i,j}: {{1,5886},{2,10247},{381,7967} ,{1482,5657},{1699,3655},{ 3241,5790},{3576,3656},{5603, 10246}}.

= 2 X[1] + X[5], 5 X[5] - 2 X[355], 5 X[1] + X[355], 2 X[140] + X[1482], 4 X[1] - X[1483], 2 X[5] + X[1483], 4 X[355] + 5 X[1483], 4 X[1387] - X[1484], X[145] + 5 X[1656], 2 X[140] - 5 X[3616], X[1482] + 5 X[3616], X[3] - 7 X[3622], X[8] - 4 X[3628], X[165] - 3 X[3653], 3 X[355] - 5 X[5587], 3 X[5] - 2 X[5587], 3 X[1] + X[5587], 3 X[1483] + 4 X[5587], 5 X[3616] - X[5657], 3 X[5603] + X[5731], 11 X[355] - 5 X[5881], 11 X[5587] - 3 X[5881], 11 X[5] - 2 X[5881], 11 X[1] + X[5881], 11 X[1483] + 4 X[5881], X[5881] - 11 X[5886], X[355] - 5 X[5886], X[5587] - 3 X[5886], X[1483] + 4 X[5886], X[355] - 10 X[5901], X[5587] - 6 X[5901], X[5] - 4 X[5901], X[1] + 2 X[5901], X[1483] + 8 X[5901], 5 X[5587] - 9 X[7988], 5 X[5] - 6 X[7988], X[355] - 3 X[7988], 5 X[5886] - 3 X[7988], 10 X[5901] - 3 X[7988], 5 X[1] + 3 X[7988], 5 X[1483] + 12 X[7988], 17 X[5] - 14 X[7989], 17 X[5886] - 7 X[7989], 17 X[1] + 7 X[7989], 7 X[5587] - 15 X[8227], 7 X[5] - 10 X[8227], 7 X[5886] - 5 X[8227], 14 X[5901] - 5 X[8227], 7 X[1] + 5 X[8227], 5 X[7989] - 17 X[9624], 5 X[5] - 14 X[9624], X[355] - 7 X[9624], 5 X[5886] - 7 X[9624], 10 X[5901] - 7 X[9624], 3 X[7988] - 7 X[9624], 5 X[1] + 7 X[9624], 5 X[5603] - X[9812], 5 X[5731] + 3 X[9812], X[5731] - 3 X[10246], X[9812] + 5 X[10246],...

H004 = HUNG - MONTESDEOCA

Barycentrics a (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+7 a^3 b^2 c+4 a^2 b^3 c-8 a b^4 c+2 b^5 c-a^4 c^2+7 a^3 b c^2-14 a^2 b^2 c^2+7 a b^3 c^2+b^4 c^2-2 a^3 c^3+4 a^2 b c^3+7 a b^2 c^3-4 b^3 c^3+2 a^2 c^4-8 a b c^4+b^2 c^4+a c^5+2 b c^5-c^6)::

= (2 r – 3 R) X[1] - (2 r - R) X[3] =

Let ABC be a triangle.

A1B1C1 is pedal triangle of incenter I.

A2,B2,C2 are reflections of A1,B1,C1 through I.

A3,B3,C3 are reflections of A,B,C through A2,B2,C2, reps.

The point is the NPC center of A3B3C3 lying on the OI line of ABC.

(Tran Quang Hung and Angel Montesdeoca, Sept. 20, 2016. See: Hyacinthos #24438)

The point lies on these lines: {1,3},{5,2802},{8,6965},....

H005 = HUNG - MOSES - EULER 1

Barycentrics 2 a^16-9 a^14 b^2+15 a^12 b^4-9 a^10 b^6-5 a^8 b^8+13 a^6 b^10-11 a^4 b^12+5 a^2 b^14-b^16-9 a^14 c^2+22 a^12 b^2 c^2-13 a^10 b^4 c^2-15 a^6 b^8 c^2+34 a^4 b^10 c^2-27 a^2 b^12 c^2+8 b^14 c^2+15 a^12 c^4-13 a^10 b^2 c^4+4 a^8 b^4 c^4-7 a^6 b^6 c^4-22 a^4 b^8 c^4+51 a^2 b^10 c^4-28 b^12 c^4-9 a^10 c^6-7 a^6 b^4 c^6-2 a^4 b^6 c^6-29 a^2 b^8 c^6+56 b^10 c^6-5 a^8 c^8-15 a^6 b^2 c^8-22 a^4 b^4 c^8-29 a^2 b^6 c^8-70 b^8 c^8+13 a^6 c^10+34 a^4 b^2 c^10+51 a^2 b^4 c^10+56 b^6 c^10-11 a^4 c^12-27 a^2 b^2 c^12-28 b^4 c^12+5 a^2 c^14+8 b^2 c^14-c^16::

Let ABC be a triangle.

Denote: Ha, Hb, Hc = the orthocenters of NBC, NCA, NAB, resp.

Oa, Ob, Oc = the circumcenters of NBC, NCA, NAB, resp.

Nha,Nhb,Nhc = the NPC centers of NHbHc,NHcHa,NHaHb, resp.

The point is the NPC center of NhaNhbNhc lying on the Euler line of ABC.

(Tran Quang Hung and Peter Moses, Oct. 20, 2016. See: Hyacinthos #24664)

The point lies on these lines:{2,3},{54,1263}.

= Anticomplement X[10126].

= Reflection of X(i) in X(j) for these {i,j}: {{5, 5501}, {10205, 140}}.

X(54) {Nha,Nhb,Nhc} = N ABC.

X(1141) {Nha,Nhb,Nhc} = H ABC.

X(1157) {Nha,Nhb,Nhc} = X(1263) ABC.

X(8254) {Nha,Nhb,Nhc} = X(5501) ABC.

H006 = HUNG - MOSES - EULER 2

Barycentrics 2 a^22-15 a^20 b^2+50 a^18 b^4-93 a^16 b^6+92 a^14 b^8-14 a^12 b^10-84 a^10 b^12+110 a^8 b^14-62 a^6 b^16+13 a^4 b^18+2 a^2 b^20-b^22-15 a^20 c^2+82 a^18 b^2 c^2-172 a^16 b^4 c^2+139 a^14 b^6 c^2+41 a^12 b^8 c^2-125 a^10 b^10 c^2+3 a^8 b^12 c^2+97 a^6 b^14 c^2-54 a^4 b^16 c^2-a^2 b^18 c^2+5 b^20 c^2+50 a^18 c^4-172 a^16 b^2 c^4+160 a^14 b^4 c^4+52 a^12 b^6 c^4-94 a^10 b^8 c^4-65 a^8 b^10 c^4+58 a^6 b^12 c^4+32 a^4 b^14 c^4-14 a^2 b^16 c^4-7 b^18 c^4-93 a^16 c^6+139 a^14 b^2 c^6+52 a^12 b^4 c^6-72 a^10 b^6 c^6-39 a^8 b^8 c^6-84 a^6 b^10 c^6+98 a^4 b^12 c^6+4 a^2 b^14 c^6-5 b^16 c^6+92 a^14 c^8+41 a^12 b^2 c^8-94 a^10 b^4 c^8-39 a^8 b^6 c^8-18 a^6 b^8 c^8-89 a^4 b^10 c^8+76 a^2 b^12 c^8+22 b^14 c^8-14 a^12 c^10-125 a^10 b^2 c^10-65 a^8 b^4 c^10-84 a^6 b^6 c^10-89 a^4 b^8 c^10-134 a^2 b^10 c^10-14 b^12 c^10-84 a^10 c^12+3 a^8 b^2 c^12+58 a^6 b^4 c^12+98 a^4 b^6 c^12+76 a^2 b^8 c^12-14 b^10 c^12+110 a^8 c^14+97 a^6 b^2 c^14+32 a^4 b^4 c^14+4 a^2 b^6 c^14+22 b^8 c^14-62 a^6 c^16-54 a^4 b^2 c^16-14 a^2 b^4 c^16-5 b^6 c^16+13 a^4 c^18-a^2 b^2 c^18-7 b^4 c^18+2 a^2 c^20+5 b^2 c^20-c^22::

Let ABC be a triangle.

Denote:

Oa, Ob, Oc = the circumcenters of NBC, NCA, NAB, resp.

Noa,Nob,Noc = the NPC centers of NObOc,NOcOa,NOaOb, resp.

The point is the NPC center of NoaNobNoc lying on the Euler line of ABC.

(Tran Quang Hung and Peter Moses, Oct. 20, 2016. See: Hyacinthos #24664)

The point lies on these lines:{2,3} = Midpoint of X[5] and X[5500].

H007 = KIRIKAMI - MONTESDEOCA - EULER 1

Barycentrics (b^2+c^2-a^2) (R F1 + a^2 G1 |OH|)::

where F1 = a^30 (b^2+c^2)-6 a^28 (2 b^4+b^2 c^2+2 c^4)+9 a^26 (6 b^6+5 b^4 c^2+5 b^2 c^4+6 c^6)-a^24 (111 b^8+245 b^6 c^2+42 b^4 c^4+245 b^2 c^6+111 c^8)+3 a^22 (21 b^10+242 b^8 c^2+41 b^6 c^4+41 b^4 c^6+242 b^2 c^8+21 c^10)+a^20 (174 b^12-1145 b^10 c^2-856 b^8 c^4+750 b^6 c^6-856 b^4 c^8-1145 b^2 c^10+174 c^12)-8 a^18 (47 b^14-81 b^12 c^2-306 b^10 c^4+150 b^8 c^6+150 b^6 c^8-306 b^4 c^10-81 b^2 c^12+47 c^14)+a^16 (207 b^16+982 b^14 c^2-4004 b^12 c^4+410 b^10 c^6+2794 b^8 c^8+410 b^6 c^10-4004 b^4 c^12+982 b^2 c^14+207 c^16)+a^14 (207 b^18-2387 b^16 c^2+4142 b^14 c^4+498 b^12 c^6-2076 b^10 c^8-2076 b^8 c^10+498 b^6 c^12+4142 b^4 c^14-2387 b^2 c^16+207 c^18)-4 a^12 (94 b^20-498 b^18 c^2+441 b^16 c^4+582 b^14 c^6-615 b^12 c^8+24 b^10 c^10-615 b^8 c^12+582 b^6 c^14+441 b^4 c^16-498 b^2 c^18+94 c^20)+a^10 (b^2-c^2)^4 (174 b^14+493 b^12 c^2-1435 b^10 c^4-1232 b^8 c^6-1232 b^6 c^8-1435 b^4 c^10+493 b^2 c^12+174 c^14)+a^8 (b^2-c^2)^4 (63 b^16-845 b^14 c^2+1176 b^12 c^4-123 b^10 c^6+322 b^8 c^8-123 b^6 c^10+1176 b^4 c^12-845 b^2 c^14+63 c^16)-a^6 (b^2-c^2)^6 (111 b^14-402 b^12 c^2-36 b^10 c^4+7 b^8 c^6+7 b^6 c^8-36 b^4 c^10-402 b^2 c^12+111 c^14)+a^4 (b^2-c^2)^8 (b^2+c^2)^2 (54 b^8-149 b^6 c^2+124 b^4 c^4-149 b^2 c^6+54 c^8)-6 a^2 (b^2-c^2)^10 (b^2+c^2)^3 (2 b^4-3 b^2 c^2+2 c^4)+(b^2-c^2)^12 (b^2+c^2)^4,

and G1 = a^26 (b^4+c^4)-7 a^24 (b^6+b^4 c^2+b^2 c^4+c^6)+a^22 (18 b^8+59 b^6 c^2+16 b^4 c^4+59 b^2 c^6+18 c^8)-a^20 (14 b^10+205 b^8 c^2+73 b^6 c^4+73 b^4 c^6+205 b^2 c^8+14 c^10)+a^18 (-25 b^12+349 b^10 c^2+363 b^8 c^4-134 b^6 c^6+363 b^4 c^8+349 b^2 c^10-25 c^12)+a^16 (63 b^14-194 b^12 c^2-992 b^10 c^4+291 b^8 c^6+291 b^6 c^8-992 b^4 c^10-194 b^2 c^12+63 c^14)-2 a^14 (18 b^16+169 b^14 c^2-808 b^12 c^4+31 b^10 c^6+492 b^8 c^8+31 b^6 c^10-808 b^4 c^12+169 b^2 c^14+18 c^16)-2 a^12 (18 b^18-371 b^16 c^2+765 b^14 c^4+191 b^12 c^6-443 b^10 c^8-443 b^8 c^10+191 b^6 c^12+765 b^4 c^14-371 b^2 c^16+18 c^18)+a^10 (63 b^20-518 b^18 c^2+319 b^16 c^4+1488 b^14 c^6-1406 b^12 c^8+236 b^10 c^10-1406 b^8 c^12+1488 b^6 c^14+319 b^4 c^16-518 b^2 c^18+63 c^20)-a^8 (b^2-c^2)^2 (25 b^18+109 b^16 c^2-1104 b^14 c^4+1160 b^12 c^6-126 b^10 c^8-126 b^8 c^10+1160 b^6 c^12-1104 b^4 c^14+109 b^2 c^16+25 c^18)-a^6 (b^2-c^2)^4 (14 b^16-303 b^14 c^2+608 b^12 c^4-57 b^10 c^6+148 b^8 c^8-57 b^6 c^10+608 b^4 c^12-303 b^2 c^14+14 c^16)+a^4 (b^2-c^2)^6 (18 b^14-157 b^12 c^2+39 b^10 c^4+52 b^8 c^6+52 b^6 c^8+39 b^4 c^10-157 b^2 c^12+18 c^14)-a^2 (b^2-c^2)^8 (b^2+c^2)^2 (7 b^8-47 b^6 c^2+38 b^4 c^4-47 b^2 c^6+7 c^8)+(b^2-c^2)^10 (b^2+c^2)^3 (b^4-5 b^2 c^2+c^4)

The point is the point of concurrence of the Euler lines of AHX(1113), BHX(1113), CHX(1113)

(Seiichi Kirikami and Angel Montesdeoca, Oct 6, 2016. See: Hyacinthos #24541) and #24545)

= X(3)X(2575) /\ X(5)X(523)

H008 = KIRIKAMI - MONTESDEOCA - EULER 2

Barycentrics (b^2+c^2-a^2) (R F1 - a^2 G1 |OH|)::

where F1 = a^30 (b^2+c^2)-6 a^28 (2 b^4+b^2 c^2+2 c^4)+9 a^26 (6 b^6+5 b^4 c^2+5 b^2 c^4+6 c^6)-a^24 (111 b^8+245 b^6 c^2+42 b^4 c^4+245 b^2 c^6+111 c^8)+3 a^22 (21 b^10+242 b^8 c^2+41 b^6 c^4+41 b^4 c^6+242 b^2 c^8+21 c^10)+a^20 (174 b^12-1145 b^10 c^2-856 b^8 c^4+750 b^6 c^6-856 b^4 c^8-1145 b^2 c^10+174 c^12)-8 a^18 (47 b^14-81 b^12 c^2-306 b^10 c^4+150 b^8 c^6+150 b^6 c^8-306 b^4 c^10-81 b^2 c^12+47 c^14)+a^16 (207 b^16+982 b^14 c^2-4004 b^12 c^4+410 b^10 c^6+2794 b^8 c^8+410 b^6 c^10-4004 b^4 c^12+982 b^2 c^14+207 c^16)+a^14 (207 b^18-2387 b^16 c^2+4142 b^14 c^4+498 b^12 c^6-2076 b^10 c^8-2076 b^8 c^10+498 b^6 c^12+4142 b^4 c^14-2387 b^2 c^16+207 c^18)-4 a^12 (94 b^20-498 b^18 c^2+441 b^16 c^4+582 b^14 c^6-615 b^12 c^8+24 b^10 c^10-615 b^8 c^12+582 b^6 c^14+441 b^4 c^16-498 b^2 c^18+94 c^20)+a^10 (b^2-c^2)^4 (174 b^14+493 b^12 c^2-1435 b^10 c^4-1232 b^8 c^6-1232 b^6 c^8-1435 b^4 c^10+493 b^2 c^12+174 c^14)+a^8 (b^2-c^2)^4 (63 b^16-845 b^14 c^2+1176 b^12 c^4-123 b^10 c^6+322 b^8 c^8-123 b^6 c^10+1176 b^4 c^12-845 b^2 c^14+63 c^16)-a^6 (b^2-c^2)^6 (111 b^14-402 b^12 c^2-36 b^10 c^4+7 b^8 c^6+7 b^6 c^8-36 b^4 c^10-402 b^2 c^12+111 c^14)+a^4 (b^2-c^2)^8 (b^2+c^2)^2 (54 b^8-149 b^6 c^2+124 b^4 c^4-149 b^2 c^6+54 c^8)-6 a^2 (b^2-c^2)^10 (b^2+c^2)^3 (2 b^4-3 b^2 c^2+2 c^4)+(b^2-c^2)^12 (b^2+c^2)^4,

and G1 = a^26 (b^4+c^4)-7 a^24 (b^6+b^4 c^2+b^2 c^4+c^6)+a^22 (18 b^8+59 b^6 c^2+16 b^4 c^4+59 b^2 c^6+18 c^8)-a^20 (14 b^10+205 b^8 c^2+73 b^6 c^4+73 b^4 c^6+205 b^2 c^8+14 c^10)+a^18 (-25 b^12+349 b^10 c^2+363 b^8 c^4-134 b^6 c^6+363 b^4 c^8+349 b^2 c^10-25 c^12)+a^16 (63 b^14-194 b^12 c^2-992 b^10 c^4+291 b^8 c^6+291 b^6 c^8-992 b^4 c^10-194 b^2 c^12+63 c^14)-2 a^14 (18 b^16+169 b^14 c^2-808 b^12 c^4+31 b^10 c^6+492 b^8 c^8+31 b^6 c^10-808 b^4 c^12+169 b^2 c^14+18 c^16)-2 a^12 (18 b^18-371 b^16 c^2+765 b^14 c^4+191 b^12 c^6-443 b^10 c^8-443 b^8 c^10+191 b^6 c^12+765 b^4 c^14-371 b^2 c^16+18 c^18)+a^10 (63 b^20-518 b^18 c^2+319 b^16 c^4+1488 b^14 c^6-1406 b^12 c^8+236 b^10 c^10-1406 b^8 c^12+1488 b^6 c^14+319 b^4 c^16-518 b^2 c^18+63 c^20)-a^8 (b^2-c^2)^2 (25 b^18+109 b^16 c^2-1104 b^14 c^4+1160 b^12 c^6-126 b^10 c^8-126 b^8 c^10+1160 b^6 c^12-1104 b^4 c^14+109 b^2 c^16+25 c^18)-a^6 (b^2-c^2)^4 (14 b^16-303 b^14 c^2+608 b^12 c^4-57 b^10 c^6+148 b^8 c^8-57 b^6 c^10+608 b^4 c^12-303 b^2 c^14+14 c^16)+a^4 (b^2-c^2)^6 (18 b^14-157 b^12 c^2+39 b^10 c^4+52 b^8 c^6+52 b^6 c^8+39 b^4 c^10-157 b^2 c^12+18 c^14)-a^2 (b^2-c^2)^8 (b^2+c^2)^2 (7 b^8-47 b^6 c^2+38 b^4 c^4-47 b^2 c^6+7 c^8)+(b^2-c^2)^10 (b^2+c^2)^3 (b^4-5 b^2 c^2+c^4)

The point is the point of concurrence of the Euler lines of AHX(1114), BHX(1114), CHX(1114)

(Seiichi Kirikami and Angel Montesdeoca, Oct 6, 2016. See: Hyacinthos #24541) and #24545)

= X(3)X(2574) /\ X(5)X(523)

H009 = HATZIPOLAKIS - MOSES

Barycentrics 2 a^16-13 a^14 b^2+43 a^12 b^4-89 a^10 b^6+115 a^8 b^8-87 a^6 b^10+33 a^4 b^12-3 a^2 b^14-b^16-13 a^14 c^2+62 a^12 b^2 c^2-113 a^10 b^4 c^2+64 a^8 b^6 c^2+61 a^6 b^8 c^2-94 a^4 b^10 c^2+33 a^2 b^12 c^2+43 a^12 c^4-113 a^10 b^2 c^4+68 a^8 b^4 c^4+17 a^6 b^6 c^4+46 a^4 b^8 c^4-81 a^2 b^10 c^4+20 b^12 c^4-89 a^10 c^6+64 a^8 b^2 c^6+17 a^6 b^4 c^6+30 a^4 b^6 c^6+51 a^2 b^8 c^6-64 b^10 c^6+115 a^8 c^8+61 a^6 b^2 c^8+46 a^4 b^4 c^8+51 a^2 b^6 c^8+90 b^8 c^8-87 a^6 c^10-94 a^4 b^2 c^10-81 a^2 b^4 c^10-64 b^6 c^10+33 a^4 c^12+33 a^2 b^2 c^12+20 b^4 c^12-3 a^2 c^14-c^16::

Let ABC be a triangle and A'B'C' the pedal triangle of N.

Denote:

Oa, Ob, Oc = the circumcenters of NB'C', NC'A', NA'B', resp.

Ooa, Oob, Ooc = the circumcenters of NObOc,NOcOa,NOaOb, resp.

The point is the NPC center of OoaOobOoc lying on the Euler line of ABC.

(Antreas Hatzipolakis and Peter Moses, Oct 21, 2016. See: Hyacinthos #24670)

The point lies on these lines:{2,3}

## Κυριακή, 16 Οκτωβρίου 2016

### TRIANGLE CENTERS FROM HYACINTHOS H011 - H020

H011 = HATZIPOLAKIS - MOSES

Barycentrics 2 a^9-2 a^8 b-3 a^7 b^2+a^6 b^3+a^5 b^4+5 a^4 b^5-a^3 b^6-5 a^2 b^7+a b^8+b^9-2 a^8 c+8 a^7 b c-a^6 b^2 c-2 a^5 b^3 c-3 a^4 b^4 c-12 a^3 b^5 c+9 a^2 b^6 c+6 a b^7 c-3 b^8 c-3 a^7 c^2-a^6 b c^2+2 a^5 b^2 c^2-2 a^4 b^3 c^2+a^3 b^4 c^2+19 a^2 b^5 c^2-16 a b^6 c^2+a^6 c^3-2 a^5 b c^3-2 a^4 b^2 c^3+24 a^3 b^3 c^3-23 a^2 b^4 c^3-6 a b^5 c^3+8 b^6 c^3+a^5 c^4-3 a^4 b c^4+a^3 b^2 c^4-23 a^2 b^3 c^4+30 a b^4 c^4-6 b^5 c^4+5 a^4 c^5-12 a^3 b c^5+19 a^2 b^2 c^5-6 a b^3 c^5-6 b^4 c^5-a^3 c^6+9 a^2 b c^6-16 a b^2 c^6+8 b^3 c^6-5 a^2 c^7+6 a b c^7+a c^8-3 b c^8+c^9::

Let ABC be a triangle and A'B'C' the pedal triangle of I.

Denote:

Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.

M1, M2, M3 = the midpoints of IA', IB', IC', resp.

MaMbMc, M1M2M3 are cyclologic. The point is the cyclologic center (MaMbMc, M1M2M3). The other cyclologic center (M1M2M3, MaMbMc) is the point X(1387)

(Antreas Hatzipolakis and Peter Moses, Sept. 20, 2016. See: Hyacinthos #24436)

The point lies on these lines: {1,1537},{55,108},{123,3816}, ...

H012 = HATZIPOLAKIS - MOSES

Barycentrics(2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6)::

3 X[3] + X[146], 3 X[5] - X[265], 3 X[110] + X[265], 3 X[2] + X[399], X[74] - 3 X[549], 3 X[113] - X[1539], 3 X[1511] + X[1539], 5 X[1656] - X[3448], 2 X[3628] + X[5609], X[1511] - 3 X[5642], X[113] + 3 X[5642], X[1539] + 9 X[5642], X[2931] + 3 X[5654], X[74] + 3 X[5655], X[2948] + 3 X[5886], 3 X[5972] + X[6053], 3 X[140] + 2 X[6053], 3 X[140] - 2 X[6699], 3 X[5972] - X[6699], 3 X[5066] - 2 X[7687], 3 X[5055] + X[9143], 3 X[597] - X[9976].

Let ABC be a triangle, NaNbNc the pedal triangle of N and OaObOc the pedal triangle of O.

Denote:

N1, N2, N3 = the reflections of N in BC, CA, AB, resp.

O1,O2, O3 = the reflections of O in BC, CA, AB, resp.

The point is the intersection of the parallels to O1N1, O2N2, O3N3 through Na,Nb,Nc, resp. The parallels to O1N1, O2N2, O3N3 through Oa,Ob,Oc, resp. concur at X(1511). The lines O1N1, O2N2, O3N3 concur at X(110)

(Antreas Hatzipolakis and Peter Moses, Sept. 21, 2016. See: Hyacinthos #24449)

The point lies on these lines:{2,399},{3,146},{4,7666},{5,4 9},{30,113},{69,10201},{74,549 },{125,3628},{140,5663},{403,3 043},{468,1986},{495,10091},{4 96,10088},{542,547},{546,9820} ,{548,2777},{550,7728},{597, 9976},{1125,2771},{1154,10096} ,{1656,3448},{2931,5654},{ 2948,5886},{3564,6593},{3582, 6126},{3584,7343},{3850,10113} ,{5055,9143},{5066,7687},{ 5432,7727},{5876,10125},{5898, 7693},{6140,6592},{6153,10095} ,{6677,9826},{7525,10117},{754 2,7723},{7722,10018} = Complement of the complement of X(399).

= Midpoint of X(i) and X(j) for these {i,j}: {5,110},{113,1511},{125,5609} ,{549,5655},{550,7728},{6053,6 699}}. Reflection of X(i) in X(j) for these {i,j}: {{125,3628},{140,5972},{10113, 3850}.

= Crossdifference of every pair of points on line {2081,2433}.

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (113,5642,1511), (5972,6053,6699).

H013 = HATZIPOLAKIS - MOSES

Barycentrics a (3 a^5 b-3 a^4 b^2-6 a^3 b^3+6 a^2 b^4+3 a b^5-3 b^6+3 a^5 c-6 a^4 b c+8 a^3 b^2 c-11 a b^4 c+6 b^5 c-3 a^4 c^2+8 a^3 b c^2-16 a^2 b^2 c^2+8 a b^3 c^2+3 b^4 c^2-6 a^3 c^3+8 a b^2 c^3-12 b^3 c^3+6 a^2 c^4-11 a b c^4+3 b^2 c^4+3 a c^5+6 b c^5-3 c^6)::

(3 r + 2 R) X[1] - (3 r - R) X[3]. 2 X[942]-X[10247],X[3057]-4 X[5885],X[3576]+X[5903],3 X[10202]-2 X[10246],2 X[5]-5 X[4004],X[355]-4 X[10107],X[3817]-3 X[3919],5 X[3698]-2 X[5694],4 X[3754]-X[5887],2 X[3754]-X[10175],X[5887]-2 X[10175],7 X[3922]-4 X[9956],X[4018]+2 X[5690].

Let ABC be a triangle.

Denote:

A', B', C' = the reflections of I in BC, CA, AB, resp.

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

N1, N2, N3 = the reflections of Na, Nb, Nc in B'C', C'A', A'B', resp.

The point is the centroid of N1N2N3 lying on the OI line of ABC.

(Antreas Hatzipolakis and Peter Moses, Oct. 13, 2016. See: Hyacinthos #24611)

The point lies on these lines: {1,3},{5,4004},{355,10107},{1 864,6797},{2800,3817},{3698,56 94},{3754,5887},{3922,9956},{ 4018,5690},{4323,6961},{4848,6 842},{5927,9952}

= Midpoint of X(3576) and X(5903).

= Reflection of X(i) in X(j) for these {i,j}: {{5887, 10175}, {10175, 3754}, {10247, 942}}.

H014 = HATZIPOLAKIS - LOZADA - MOSES

Barycentrics a^4 (a^12-4 a^10 b^2+5 a^8 b^4-5 a^4 b^8+4 a^2 b^10-b^12-4 a^10 c^2+9 a^8 b^2 c^2-5 a^6 b^4 c^2+a^4 b^6 c^2-3 a^2 b^8 c^2+2 b^10 c^2+5 a^8 c^4-5 a^6 b^2 c^4+2 a^4 b^4 c^4-a^2 b^6 c^4-b^8 c^4+a^4 b^2 c^6-a^2 b^4 c^6-5 a^4 c^8-3 a^2 b^2 c^8-b^4 c^8+4 a^2 c^10+2 b^2 c^10-c^12)::

Trilinears cos(2*A)*cos(3*A)-cos(4*A)* cos(B-C) ::

R^2*X(4)+(7*R^2-2*SW)*X(54)

2 X[54] + X[6759], 3 X[154] - X[9920].

X[4] + (J^2 - 2) X[54], (J = OH/R).

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P. Denote: O', Oa, Ob, Oc = the circumcenters of A'B'C', PBC, PCA, PAB, resp. The reflections of O'Oa, O'Ob, O'Oc in BC, CA, AB, resp. are concurrent for P = O. The point is the point of concurrence for P = O

(Antreas P. Hatzipolakis, Peter Moses, Cesar Lozada, Oct. 8, 2016. See: Hyacinthos #24566 and #24574)

The point lies on these lines:{3,8157},{4,54},{49,52},{110, 2888},{154,9704},{156,9927},{ 182,6689},{206,576},{539, 10201},{569,6145},{1092,7691}, {1147,1154},{1209,6639},{1971, 9697},{2904,9707},{3518,7730}, {6288,10254},{9813,9827},{ 10182,10203}

= Midpoint of X(195) and X(2917)

= X(324)-Ceva conjugate of X(571).

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54,3574,578).

H015 = HATZIPOLAKIS - MOSES

Barycentrics 14 a^16-103 a^14 b^2+335 a^12 b^4-633 a^10 b^6+765 a^8 b^8-609 a^6 b^10+313 a^4 b^12-95 a^2 b^14+13 b^16-103 a^14 c^2+454 a^12 b^2 c^2-729 a^10 b^4 c^2+342 a^8 b^6 c^2+459 a^6 b^8 c^2-786 a^4 b^10 c^2+469 a^2 b^12 c^2-106 b^14 c^2+335 a^12 c^4-729 a^10 b^2 c^4+360 a^8 b^4 c^4+15 a^6 b^6 c^4+480 a^4 b^8 c^4-837 a^2 b^10 c^4+376 b^12 c^4-633 a^10 c^6+342 a^8 b^2 c^6+15 a^6 b^4 c^6-14 a^4 b^6 c^6+463 a^2 b^8 c^6-758 b^10 c^6+765 a^8 c^8+459 a^6 b^2 c^8+480 a^4 b^4 c^8+463 a^2 b^6 c^8+950 b^8 c^8-609 a^6 c^10-786 a^4 b^2 c^10-837 a^2 b^4 c^10-758 b^6 c^10+313 a^4 c^12+469 a^2 b^2 c^12+376 b^4 c^12-95 a^2 c^14-106 b^2 c^14+13 c^16::

Let A be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of NBC, NCA, NAB, resp.

Denote:

Nab, Nac = the orthogonal projections of Na on BNb, CNc, resp.

Nbc, Nba = the orthogonal projections of Nb on CNc, ANa, resp.

Nca, Ncb = the orthogonal projections of Nc on ANa, BNb, resp.

Let Oa, Ob, Oc be the circumcenters of NaNabNac, NbNbcNba, NcNcaNc, resp.

ABC, OaObOc are orthologic.

ABC, OaObOc are orthologic at X(1263)

(Antreas Hatzipolakis and Peter Moses, Sept. 30, 2016. See: Hyacinthos #24515)

The point is the orthologic center (OaObOc,ABC)

The point lies on these lines: {140, 930}.

= Midpoint of X(140), X(1487).