[Antreas P. Hatzipolakis]:
Let ABC be a triange, A'B'C' the orthic triangle and L a line,
The parallels to L through A, B, C, intersect the circumcircle again at A", B", C", resp.
The cirumcircles of AA'A", BB'B", CC'C" are coaxial.
Radical trace ?
Special L = Euler line, Brocard axis, OI line.....
GENERALIZATION
Let ABC be a triangle, A'B'C' the orthic triangle and P a point..
PA, PB, PC intersect the circcumcircle again at A", B", C", resp.
The cirumcircles of AA'A", BB'B", CC'C" are coaxial.
(if P lies on the line at infinity we have the original problem)
Radical trace in terms of P?
APH
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[Ercole Suppa]:
Hi Antreas,
Let (u:v:w) be the barycentric coordinates of P and let R=R(P) be the radical trace. We have
R(P) = (b^2-c^2)^2 (v+w) (b^2 (3 u+v-w)+c^2 (3u-v+w))+a^6 (2 u^2+(v-w)^2+u (v+w))+a^4 (b^2 (-4 u^2-v^2+u (v-3 w)+w^2)+c^2 (-4 u^2+v^2-w^2+u (-3 v+w)))+a^2 (2 b^2 c^2 (2 u^2+(v-w)^2+3 u (v+w))+b^4 (2 u^2-(v-w)^2-u (5 v+w))+c^4 (2 u^2-(v-w)^2-u (v+5 w))):: (barys)
*** H, P, R are collinear (H = orthocenter of ABC)
*** Pairs {P=X(i),R=X(j)} for these {i,j}: {1,16869},{2,5159},{3,10257},{5,2072},{6,15341},{11,11},{19,8074},{20,16976},{22,16977},{23,6676},{25,468},{30,3},{33,16870},{53,5523},{107,24930},{113,113},{114,114},{115,115},{116,116},{117,117},{118,118},{119,119},{120,120},{121,121},{122,122},{123,123},{124,124},{125,125},{126,126},{127,127},{128,128},{129,129},{130,130},{131,131},{132,132},{133,133},{134,134},{135,135},{136,136},{137,137},{138,138},{139,139},{186,140},{225,1785},{235,403},{281,5199},{297,11007},{378,15122},{381,10297},{403,5},{427,858},{428,23},{460,1316},{468,2},{497,20130},{523,4},{524,15098},{546,18403},{847,16269},{858,1368},{900,18341},{952,18342},{1312,1312},{1313,1313},{1316,441},{1503,18338},{1519,12608},{1528,6260},{1560,1560},{1566,1566},{1596,11799},{1826,5179},{1839,5011},{1842,242},{1843,5167},{1861,10},{1862,15343},{1877,1},{1884,3109},{1885,2071},{1990,6794},{2039,2039},{2040,2040},{2070,7542},{2071,16196},{2072,11585},{2074,6675},{2489,14700},{2501,6792},{2679,2679},{3153,12362},{3154,1650},{3258,3258},{3259,3259},{3564,18348},{3566,18347},{3575,186},{3845,18323},{4240,12068},{5099,5099},{5101,17615},{5139,5139},{5189,10691},{5190,5190},{5509,5509},{5510,5510},{5511,5511},{5512,5512},{5513,5513},{5514,5514},{5515,5515},{5516,5516},{5517,5517},{5518,5518},{5519,5519},{5520,5520},{5521,5521},{5522,5522},{5895,15427},{5950,5950},{5952,5952},{5993,5993},{6092,6092},{6240,15646},{6748,13509},{6756,2070},{7576,7575},{7649,6788},{8105,8426},{8106,8427},{8754,16278},{8884,16810},{9151,9151},{9152,9152},{9193,9193},{10017,10017},{10151,4},{10257,3548},{10295,549},{10297,18531},{10301,7426},{10989,10300},{11563,10024},{11569,11569},{11744,974},{11792,11792},{11799,15760},{12494,12494},{12624,12624},{13141,13141},{13234,13234},{13249,13249},{13473,20},{13488,18859},{13499,13499},{13517,13517},{13612,13612},{13613,13613},{13619,3530},{13870,13870},{13871,13871},{13994,13994},{13999,13999},{14103,14103},{14119,1375},{14342,4},{14672,14672},{14895,15781},{15169,15169},{15241,15241},{15607,15607},{15608,15608},{15609,15609},{15610,15610},{15611,15611},{15612,15612},{15613,15613},{15614,15614},{16177,16177},{16178,16178},{16188,16188},{16221,16221},{16228,4},{16229,4},{16230,11005},{16240,14847},{16328,13531},{16938,16938},{18402,18402},{18403,12605},{18559,18571},{18809,18809},{20389,20389},{20551,20551},{20619,20619},{20620,20620},{20621,20621},{20622,20622},{20623,20623},{20625,20625},{21284,7499},{21662,21662},{22474,22474},{23047,3153},{23323,18404},{25640,25640},{25641,25641},{25642,25642}
*** Some points:
R(X(7)) = X(4)X(7) n X(3900)X(17069)
= 2 a (a^8 b-10 a^6 b^3+16 a^5 b^4-16 a^3 b^6+10 a^2 b^7-b^9+a^8 c-2 a^7 b c+8 a^6 b^2 c-10 a^5 b^3 c-14 a^4 b^4 c+26 a^3 b^5 c-14 a b^7 c+5 b^8 c+8 a^6 b c^2-8 a^5 b^2 c^2+14 a^4 b^3 c^2-8 a^3 b^4 c^2-36 a^2 b^5 c^2+32 a b^6 c^2-2 b^7 c^2-10 a^6 c^3-10 a^5 b c^3+14 a^4 b^2 c^3-4 a^3 b^3 c^3+26 a^2 b^4 c^3-2 a b^5 c^3-14 b^6 c^3+16 a^5 c^4-14 a^4 b c^4-8 a^3 b^2 c^4+26 a^2 b^3 c^4-32 a b^4 c^4+12 b^5 c^4+26 a^3 b c^5-36 a^2 b^2 c^5-2 a b^3 c^5+12 b^4 c^5-16 a^3 c^6+32 a b^2 c^6-14 b^3 c^6+10 a^2 c^7-14 a b c^7-2 b^2 c^7+5 b c^8-c^9) : : (barys)
= lies on these lines: {4,7}, {3900,17069}
= (6-9-13) search numbers [1.97344638170893145, 2.19616150289697751, 1.20942357295927247]
R(X(8)) = X(4)X(8) n X(36)X(24320)
= 2 a (a+b+c) (a^5 b-3 a^4 b^2-4 a^3 b^3+4 a^2 b^4+3 a b^5-b^6+a^5 c+4 a^3 b^2 c+4 a^2 b^3 c-5 a b^4 c-4 b^5 c-3 a^4 c^2+4 a^3 b c^2-4 a^2 b^2 c^2-2 a b^3 c^2+b^4 c^2-4 a^3 c^3+4 a^2 b c^3-2 a b^2 c^3+8 b^3 c^3+4 a^2 c^4-5 a b c^4+b^2 c^4+3 a c^5-4 b c^5-c^6) : : (barys)
= lies on these lines: {4,8}, {36,24320}, {513,20315}, {3660,28036}, {3814,4138}, {5123,20306}
= (6-9-13) search numbers [4.33735165443347354, 1.73763341618949316, 0.435832891730049897]
R(X(9)) = X(4)X(9) n X(514)X(28984)
= (a-b-c) (a^3-a^2 b-a b^2+b^3-a^2 c-2 a b c-b^2 c-a c^2-b c^2+c^3) (2 a^6-a^5 b-a^4 b^2-2 a^3 b^3+3 a b^5-b^6-a^5 c-2 a^4 b c+4 a^3 b^2 c-3 a b^4 c+2 b^5 c-a^4 c^2+4 a^3 b c^2+b^4 c^2-2 a^3 c^3-4 b^3 c^3-3 a b c^4+b^2 c^4+3 a c^5+2 b c^5-c^6) : : (barys)
= lies on these lines: {4,9}, {514,28984}, {1323,24635}, {4295,27541}, {4512,28118}, {5088,5273}
= (6-9-13) search numbers [3.83062923835495849, 2.47757696098361222, 0.157436168139589800]
R(X(10)) = X(4)X(9) n X(101)X(29219)
= a^5 b-a^3 b^3-a^2 b^4+b^6+a^5 c-2 a^4 b c-a^3 b^2 c+a^2 b^3 c+b^5 c-a^3 b c^2+4 a^2 b^2 c^2-b^4 c^2-a^3 c^3+a^2 b c^3-2 b^3 c^3-a^2 c^4-b^2 c^4+b c^5+c^6 : : (barys)
= 5*X[1698]-X[5018]
= lies on these lines: {4,9}, {101,29219}, {514,20315}, {1146,29016}, {1698,5018}, {3332,27541}, {5044,5074}, {5224,7112}
= (6-9-13) search numbers [3.65568694200709420, 2.31180766893863220, 0.352942122485128719]
Best regards,
Ercole Suppa
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