HYACINTHOS 29426

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

A', B', C' = the midpoints of AO, BO, CO, resp.

Na, Nb, Nc = the NPC centers of OBC, OCA, OAB, resp.

Ra = the radical axis of (Nb, NbB'), (Nc, NcC')
Rb = the radical axis of (Nc, NcC'), (Na, NaA')
Rc =  the radical axis of (Na, NaA'), (Nb, NbB')

1. The reflections of Ra, Rb, Rc in BC, CA, AB, resp. concur at X(30)
(parallels to Euler line)

2. The reflections of Ra, Rb, Rc in AO, BO, CO, resp. concur on the Euler line.
Point?

3. Which is the point of concurrence of Ra, Rb, Rc (radical center of the circles)?  


[Peter Moses]:


Hi Antreas,

2)
= a^2*(2*a^8 - 4*a^6*b^2 + 4*a^2*b^6 - 2*b^8 - 4*a^6*c^2 + 12*a^4*b^2*c^2 - 7*a^2*b^4*c^2 - b^6*c^2 - 7*a^2*b^2*c^4 + 6*b^4*c^4 + 4*a^2*c^6 - b^2*c^6 - 2*c^8) : :

= 7 X[3] - X[23], 3 X[3] - X[186], 5 X[3] - X[2070], 9 X[3] - X[5899], 5 X[3] + X[7464], 4 X[3] - X[7575], 11 X[3] - 2 X[12105], 3 X[5] - 2 X[10151], 3 X[23] - 7 X[186], 5 X[23] - 7 X[2070], X[23] + 7 X[2071], 9 X[23] - 7 X[5899], 5 X[23] + 7 X[7464], 4 X[23] - 7 X[7575], 11 X[23] - 14 X[12105], 2 X[23] - 7 X[15646], 3 X[140] - X[11558], 3 X[140] - 2 X[15350], 5 X[186] - 3 X[2070], X[186] + 3 X[2071], 3 X[186] - X[5899], 5 X[186] + 3 X[7464], 4 X[186] - 3 X[7575], 11 X[186] - 6 X[12105], 2 X[186] - 3 X[15646], 3 X[376] + X[3153], 3 X[403] - 2 X[11558], 3 X[403] - 4 X[15350], 2 X[548] + X[858], 3 X[549] - X[11563], X[550] + 2 X[15122], X[2070] + 5 X[2071], 9 X[2070] - 5 X[5899], 4 X[2070] - 5 X[7575], 11 X[2070] - 10 X[12105], 2 X[2070] - 5 X[15646], 9 X[2071] + X[5899], 5 X[2071] - X[7464], 4 X[2071] + X[7575], 11 X[2071] + 2 X[12105], 2 X[2071] + X[15646], 5 X[3522] + X[7574], 5 X[3522] - X[13619], 4 X[3530] - X[11799], X[3627] - 4 X[5159], 5 X[5899] + 9 X[7464], 4 X[5899] - 9 X[7575], 11 X[5899] - 18 X[12105], 2 X[5899] - 9 X[15646], X[7426] - 4 X[14891], 4 X[7464] + 5 X[7575], 11 X[7464] + 10 X[12105], 2 X[7464] + 5 X[15646], 11 X[7575] - 8 X[12105], X[10096] - 3 X[12100], X[10151] - 3 X[10257], X[10296] + 5 X[15696], 2 X[10297] + X[15704], X[10540] - 3 X[15035], X[10989] + 5 X[14093], 2 X[11064] + X[14677], 4 X[12105] - 11 X[15646], X[13445] + 3 X[15035], X[14157] - 5 X[15051]

= lies on these lines: {2, 3}, {36, 10149}, {74, 22115}, {539, 20417}, {974, 1154}, {1092, 32138}, {1493, 13382}, {1511, 6000}, {1568, 16111}, {2693, 6760}, {2777, 14156}, {3098, 10250}, {3564, 5621}, {5562, 32210}, {9590, 28190}, {9625, 28182}, {10263, 32411}, {10264, 12901}, {10540, 13445}, {10575, 32171}, {10606, 15068}, {10979, 16328}, {11064, 14677}, {11440, 31834}, {11454, 23039}, {11468, 18436}, {11649, 14810}, {11695, 13446}, {11793, 22966}, {11809, 14794}, {12038, 13491}, {12041, 13754}, {12121, 25739}, {13293, 15311}, {13366, 13630}, {13391, 32110}, {13399, 30714}, {13568, 20424}, {14157, 15051}, {14805, 20791}, {15515, 16308}, {16163, 30522}, {21230, 22978}, {22549, 30507}

= midpoint of X(i) and X(j) for these {i,j}: {3, 2071}, {74, 22115}, {1568, 16111}, {2070, 7464}, {2693, 6760}, {7574, 13619}, {10540, 13445}, {10564, 21663}, {12121, 25739}, {13399, 30714}

= reflection of X(i) in X(j) for these {i,j}: {5, 10257}, {403, 140}, {7575, 15646}, {10263, 32411}, {11558, 15350}, {13446, 11695}, {15646, 3}

= circumcircle inverse of X(1657)

= orthoptic circle of the Steiner inellipse inverse of X(31101)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 20, 15331}, {3, 1657, 21844}, {3, 2937, 17506}, {3, 3520, 140}, {3, 3534, 10298}, {3, 7464, 18571}, {3, 11250, 5}, {3, 11410, 7514}, {3, 11413, 1658}, {3, 14118, 3530}, {3, 15246, 14891}, {3, 18570, 549}, {3, 18859, 186}, {3, 21312, 18324}, {140, 1885, 5}, {140, 11558, 15350}, {186, 378, 10151}, {186, 2071, 18859}, {550, 15122, 18572}, {1113, 1114, 1657}, {1657, 21844, 12107}, {1658, 11413, 15704}, {3522, 23040, 3}, {3534, 10298, 7555}, {7514, 11410, 18570}, {11558, 15350, 403}, {13445, 15035, 10540}


3)  COMPLEMENT OF X(12902)  =

= 4*a^10 - 9*a^8*b^2 + 4*a^6*b^4 + 2*a^4*b^6 - b^10 - 9*a^8*c^2 + 16*a^6*b^2*c^2 - 7*a^4*b^4*c^2 - 3*a^2*b^6*c^2 + 3*b^8*c^2 + 4*a^6*c^4 - 7*a^4*b^2*c^4 + 6*a^2*b^4*c^4 - 2*b^6*c^4 + 2*a^4*c^6 - 3*a^2*b^2*c^6 - 2*b^4*c^6 + 3*b^2*c^8 - c^10 : :

= 3 X[2] - 5 X[15040], 3 X[3] - X[3448], X[4] - 3 X[32609], 3 X[5] - 4 X[5972], 5 X[5] - 4 X[7687], 3 X[5] - 2 X[10113], 7 X[5] - 8 X[12900], 3 X[5] - 5 X[22251], 5 X[110] - 3 X[5655], 3 X[110] - X[7728], 7 X[110] - 3 X[10706], 5 X[110] - X[10721], 2 X[125] - 3 X[549], 2 X[140] - 3 X[15035], 6 X[140] - 5 X[15059], 2 X[143] - 3 X[16223], X[265] - 3 X[15035], 3 X[265] - 5 X[15059], 3 X[376] - X[10620], 3 X[376] + X[14683], 2 X[546] - 3 X[14643], 10 X[546] - 13 X[15029], 2 X[546] - 5 X[15034], 3 X[546] - 7 X[22250], 2 X[548] + X[23236], 5 X[550] - 2 X[10990], 3 X[550] - 2 X[16111], 3 X[550] + 2 X[24981], X[550] + 2 X[30714], 5 X[632] - 4 X[20304], 3 X[1350] + X[25336], 3 X[1511] - 2 X[5972], 5 X[1511] - 2 X[7687], 3 X[1511] - X[10113], 7 X[1511] - 4 X[12900], 6 X[1511] - 5 X[22251], 3 X[2979] + X[15102], X[3146] - 7 X[15039], X[3146] - 5 X[20125], 2 X[3448] - 3 X[10264], X[3448] + 3 X[12383], 5 X[3522] - X[12317], 5 X[3522] - 3 X[15041], 7 X[3526] - 5 X[15081], 4 X[3530] - 5 X[15051], 4 X[3530] - 3 X[15061], 3 X[3534] - X[12244], 3 X[3534] + X[12308], 4 X[3628] - 3 X[14644], 4 X[3628] - 7 X[15020], 7 X[3832] - 9 X[15046], 3 X[3845] - 2 X[12295], 2 X[5609] + X[15704], 3 X[5642] - X[12295], 9 X[5655] - 5 X[7728], 7 X[5655] - 5 X[10706], 3 X[5655] - X[10721], 3 X[5655] + 5 X[12121], 3 X[5946] - 2 X[11800], 5 X[5972] - 3 X[7687], 7 X[5972] - 6 X[12900], 4 X[5972] - 5 X[22251], 4 X[6699] - 5 X[15712], 8 X[6723] - 9 X[11539], 3 X[7575] - 2 X[32269], 6 X[7687] - 5 X[10113], 7 X[7687] - 10 X[12900], 12 X[7687] - 25 X[22251], 7 X[7728] - 9 X[10706], 5 X[7728] - 3 X[10721], X[7728] + 3 X[12121], 3 X[8703] - 2 X[12041], 3 X[9140] - 7 X[15036], 3 X[9143] + X[12244], 3 X[9143] - X[12308], 3 X[9730] - 2 X[13358], 7 X[10113] - 12 X[12900], 2 X[10113] - 5 X[22251], X[10264] + 2 X[12383], 2 X[10272] - 3 X[32609], 3 X[10519] - X[32306], 15 X[10706] - 7 X[10721], 3 X[10706] + 7 X[12121], X[10721] + 5 X[12121], X[10733] - 4 X[13392], X[10733] - 3 X[14643], 5 X[10733] - 13 X[15029], X[10733] - 5 X[15034], 3 X[10733] - 14 X[22250], 4 X[10990] - 5 X[14677], 3 X[10990] - 5 X[16111], X[10990] - 5 X[16163], 3 X[10990] + 5 X[24981], X[10990] + 5 X[30714], 2 X[11801] - 5 X[15040], 5 X[12017] - 3 X[25320], 4 X[12068] - 3 X[21315], 6 X[12100] - 7 X[15036], 2 X[12103] + X[14094], 8 X[12108] - 5 X[15027], X[12281] - 3 X[23039], X[12317] - 3 X[15041], X[12407] - 3 X[26446], 10 X[12812] - 7 X[15044], 24 X[12900] - 35 X[22251], X[12902] - 5 X[15040], X[13201] - 3 X[13340], 4 X[13392] - 3 X[14643], 20 X[13392] - 13 X[15029], 4 X[13392] - 5 X[15034], 6 X[13392] - 7 X[22250], 15 X[14643] - 13 X[15029], 3 X[14643] - 5 X[15034], 9 X[14643] - 14 X[22250], 3 X[14644] - 7 X[15020], 3 X[14677] - 4 X[16111], X[14677] - 4 X[16163], 3 X[14677] + 4 X[24981], X[14677] + 4 X[30714], 7 X[14869] - 6 X[34128], 4 X[14934] - 3 X[33855], 13 X[15029] - 25 X[15034], 39 X[15029] - 70 X[22250], 15 X[15034] - 14 X[22250], 9 X[15035] - 5 X[15059], 7 X[15039] - 5 X[20125], 13 X[15042] - 11 X[15717], 5 X[15051] - 3 X[15061], 3 X[15055] - 4 X[33923], 8 X[15088] - 9 X[15699], 3 X[15462] - 2 X[18583], X[15545] - 3 X[21166], X[16111] - 3 X[16163], X[16111] + 3 X[30714], 3 X[16163] + X[24981], X[24981] - 3 X[30714]

= lies on these lines: {2, 11801}, {3, 2888}, {4, 7666}, {5, 1511}, {20, 399}, {30, 110}, {52, 11561}, {74, 548}, {113, 3627}, {125, 549}, {140, 265}, {143, 16223}, {146, 1657}, {376, 10620}, {381, 11694}, {495, 18968}, {496, 12896}, {511, 25329}, {516, 11699}, {541, 15686}, {542, 8703}, {546, 10733}, {550, 5562}, {567, 27866}, {632, 20304}, {952, 12778}, {974, 31804}, {1154, 11562}, {1350, 25336}, {1352, 12302}, {1353, 14708}, {1503, 12584}, {1539, 16534}, {1595, 12140}, {2771, 4297}, {2777, 5609}, {2929, 2931}, {2935, 9833}, {2948, 18481}, {2979, 15102}, {3043, 6240}, {3146, 15039}, {3521, 9705}, {3522, 12317}, {3526, 15081}, {3530, 15051}, {3534, 9143}, {3564, 32233}, {3575, 15463}, {3580, 18571}, {3589, 32273}, {3628, 14644}, {3832, 15046}, {3845, 5642}, {4325, 6126}, {4330, 7343}, {5012, 15089}, {5092, 25328}, {5159, 32227}, {5844, 12898}, {5946, 11800}, {6101, 10628}, {6247, 25564}, {6593, 21850}, {6644, 12310}, {6699, 15712}, {6723, 11539}, {6756, 15472}, {7471, 18319}, {7487, 11566}, {7575, 32269}, {7583, 10819}, {7584, 10820}, {7722, 10295}, {7727, 15338}, {7978, 28212}, {9140, 12100}, {9730, 13358}, {9820, 19479}, {10088, 18990}, {10091, 15171}, {10224, 12278}, {10226, 14516}, {10263, 11557}, {10519, 32306}, {11061, 33878}, {11064, 18572}, {11449, 13406}, {11591, 21650}, {11597, 20424}, {11720, 22791}, {12017, 25320}, {12068, 21315}, {12084, 12168}, {12103, 14094}, {12108, 15027}, {12227, 13568}, {12228, 31833}, {12270, 15332}, {12281, 23039}, {12368, 28186}, {12407, 26446}, {12812, 15044}, {12893, 15646}, {12904, 15325}, {13201, 13340}, {13391, 13417}, {13605, 13624}, {13630, 21649}, {14869, 34128}, {15042, 15717}, {15055, 33923}, {15088, 15699}, {15326, 19470}, {15462, 18583}, {15545, 21166}, {15806, 34007}, {16160, 16164}, {17701, 21659}, {18400, 23315}, {18533, 19504}, {19140, 29181}, {22584, 31834}, {23335, 25487}

 

= complement of X(12902)  

= anticomplement of X(11801)

= midpoint of X(i) and X(j) for these {i,j}: {3, 12383}, {20, 399}, {74, 23236}, {110, 12121}, {146, 1657}, {2931, 12118}, {2935, 9833}, {2948, 18481}, {3534, 9143}, {5898, 12254}, {10620, 14683}, {11061, 33878}, {12244, 12308}, {12270, 18436}, {14094, 20127}, {16111, 24981}, {16163, 30714}

= reflection of X(i) in X(j) for these {i,j}: {4, 10272}, {5, 1511}, {52, 11561}, {74, 548}, {265, 140}, {381, 11694}, {546, 13392}, {550, 16163}, {1539, 16534}, {3580, 18571}, {3627, 113}, {3845, 5642}, {6247, 25564}, {9140, 12100}, {10113, 5972}, {10263, 11557}, {10264, 3}, {10733, 546}, {12902, 11801}, {13605, 13624}, {14677, 550}, {16160, 16164}, {18319, 7471}, {18572, 11064}, {19479, 9820}, {20127, 12103}, {20424, 11597}, {21649, 13630}, {21650, 11591}, {21850, 6593}, {22584, 31834}, {22791, 11720}, {23306, 12038}, {23335, 25487}, {25328, 5092}, {32273, 3589}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 12902, 11801}, {4, 32609, 10272}, {5, 22251, 5972}, {110, 10721, 5655}, {265, 15035, 140}, {376, 14683, 10620}, {546, 13392, 14643}, {1511, 5972, 22251}, {1511, 10113, 5972}, {3522, 12317, 15041}, {3534, 12308, 12244}, {5972, 10113, 5}, {9143, 12244, 12308}, {10733, 14643, 546}, {10733, 15034, 14643}, {12902, 15040, 2}, {14643, 15034, 13392}, {15051, 15061, 3530}, {16111, 30714, 24981}, {16163, 24981, 16111}

Best regards,
Peter Moses.