[Kadir Altintas and Ercole Suppa]:
Let ABC be a triangle and I,O the incenter and circumcenter, resp.
Denote:
DEF = the circumcevian triangle of I.
Ga, Ga' = the centroids of AFE and BDC resp. Similarly Gb, Gb' and Gc, Gc'
Ha, Ha' = the orthocenters of AFE and BDC resp. Similarly Hb, Hb. and Hc, Hc'=
Pa = point of the line GaHa such that GaPa/PaHa = t = real number (or t = ∞ if Pa=Ha). Similarly Pb, Pc
Pa' = point of line Ga'Ha' such that Ga'Pa'/Pa'Ha' = t = real number (or t = ∞ if Pa'=Ha'). Similarly Pb', Pc'
Oa' = the center of the circle PaPb'Pc'; define Ob', Oc' cyclically
(1) Prove that O, Oa', Ob', Oc' lie on same circle ω(t)
(2) The center W(t) of the circle ω(t) has barycentric coordinates:
W(t)= a (-3 a^3 (1+t)+a^2 (b+c) (1+3 t)-(b-c)^2 (b+c) (1+3 t)+a (3 b^2 (1+t)+3 c^2 (1+t)-2 b (c+3 c t))) ::
and the radius ρ is such that
ρ^2(t)= (((R-2 r) R (1+3 t)^2)/(9 (1+t)^2))
where R,r are the circumradius and inradius of ABC resp.
--------------------------------------------------------------------------------------------
[Kadir Altintas and Ercole Suppa]:
Some points:
*** if t=0 then Pa,Pa'=X(2) of AFE and BDC resp., W(0)=X(3576) of ABC
*** if t=-1/3 then Pa,Pa'=X(3) of AFE and BDC resp., W(-1/3)=X(3) of ABC and ρ(-1/3)=0 i.e. O=Oa'=Ob'=Oc'
*** if t=∞ then Pa,Pa'=X(4) of AFE and BDC resp., W(∞)=X(1) of ABC
*** if t=1/3 then Pa,Pa'=X(5) of AFE and BDC resp., W(1/3)=X(1385) of ABC
*** if t=-2/3 then Pa,Pa'=X(20) of AFE and BDC resp., W(-2/3)=X(40) of ABC
*** if t=-1/9 then Pa,Pa'=X(140) of AFE and BDC resp., W(-1/9)=X(13624) of ABC
*** if t=1 then Pa,Pa'=X(381) of AFE and BDC resp., W(1)=X(10246) of ABC
*** if t=-5/3 then Pa,Pa'=X(382) of AFE and BDC resp., W(-5/3)=X(1482) of ABC
*** if t=5/3 then Pa,Pa'=X(546) of AFE and BDC resp., W(5/3)=X(15178) of ABC
*** if t=1/7 then Pa,Pa'=X(547) of AFE and BDC resp. then
W(1/7) = MIDPOINT OF X(1385) AND X(3576)
= a (12 a^3-5 a^2 b-12 a b^2+5 b^3-5 a^2 c+10 a b c-5 b^2 c-12 a c^2-5 b c^2+5 c^3) : : (barys)
= 5*X[1]+7*X[3], 5*X[355]-17*X[3533], X[548]+2*X[3636], 5*X[551]+X[15686], 5*X[1125]-2*X[3850], X[3244]+5*X[15712], 49*X[3523]-X[20054], 2*X[3530]+X[13607], X[3543]-5*X[5886], X[3545]-5*X[3653], X[3626]-4*X[12108], X[3627]-7*X[15808], X[3655]+X[11231], 5*X[3817]-3*X[3845], 7*X[3832]+5*X[18481], 2*X[3853]-5*X[9955], 11*X[5056]-5*X[18480], X[5059]+5*X[22793], 7*X[5657]+X[20049], 5*X[7967]+11*X[15719], 5*X[9812]+3*X[11001], 5*X[9956]-8*X[16239], 5*X[10164]-7*X[19711], 5*X[10165]-3*X[11539], 9*X[15708]-5*X[26446]
= lies on these lines: {1,3}, {104,28180}, {140,28236}, {214,3740}, {355,3533}, {515,547}, {516,15690}, {548,3636}, {551,15686}, {952,4745}, {1125,3850}, {2771,28463}, {3244,15712}, {3523,20054}, {3530,13607}, {3543,5886}, {3545,3653}, {3626,12108}, {3627,15808}, {3655,11231}, {3817,3845}, {3832,18481}, {3853,9955}, {5056,18480}, {5059,22793}, {5603,28202}, {5657,20049}, {5901,28150}, {7967,15719}, {9812,11001}, {9956,16239}, {10164,19711}, {10165,11539}, {10283,28198}, {12100,28234}, {13464,28216}, {15708,26446}, {17536,17614}, {19538,19861}
= midpoint of X(i) and X(j) for these {i,j}: {1385,3576}, {3655,11231}, {10246,17502}
= reflection of X(13624) in X(576)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,10246,16200}, {3,11531,3579}, {1385,13624,15178}, {1385,17502,10246}, {3576,10246,17502}, {3653,5731,11230}
= (6-9-13) search numbers [4.66067356722163204, 4.14301658620041564, -1.37865787802589267]
*** if t=-7/15 then Pa,Pa'=X(548) of AFE and BDC resp. then
W(-7/15) = COMPLEMENT OF X(22793)
= a (4 a^3+a^2 b-4 a b^2-b^3+a^2 c-2 a b c+b^2 c-4 a c^2+b c^2-c^3) : : (barys)
= X[1]-5*X[3], 3*X[2]-X[22793], X[4]-3*X[11231], X[5]-3*X[10164], X[8]+7*X[3528], X[10]+X[550], 3*X[20]+5*X[5818], X[355]+3*X[376], X[382]-5*X[1698], X[546]-2*X[3634], 3*X[547]-2*X[12571], 3*X[549]-X[946], X[551]-3*X[17504], 5*X[631]+3*X[9778], 5*X[632]-3*X[3817], X[944]+3*X[3654], X[962]-9*X[3524], X[1125]-2*X[3530], X[1386]-3*X[17508], X[1657]+3*X[5587], 3*X[1699]-7*X[3526], X[2948]+3*X[15041], X[3241]-9*X[15710], 25*X[3522]+7*X[4678], 7*X[3523]-3*X[5886], 11*X[3525]-3*X[9812], X[3529]+7*X[9780], 17*X[3533]-9*X[9779], 3*X[3534]+X[5691], 5*X[3616]-13*X[10299], 7*X[3624]-11*X[15720], X[3627]-3*X[10175], 2*X[3628]-X[18483], 3*X[3651]+X[3652], 9*X[3653]-5*X[10595], 3*X[3655]+X[12245], X[3656]-5*X[15692], X[3679]+3*X[15688], 3*X[3830]-7*X[7989], 2*X[3850]-3*X[10172], 5*X[4297]+3*X[4669], 9*X[5054]-5*X[8227], 15*X[5071]-7*X[10248], X[5073]-5*X[18492], 13*X[5079]-17*X[19872], X[5493]+3*X[10165], 15*X[5731]+X[20053], 3*X[5790]-7*X[9588], X[5901]-3*X[12100], X[9589]-5*X[18493], 3*X[10171]-4*X[16239], 3*X[10178]-X[13369], 3*X[11204]-X[12262], X[11699]-3*X[15035], X[11826]+3*X[28459], X[11827]+3*X[28458], X[12103]+X[18357], X[12515]+X[22935], X[12619]+X[24466], X[12645]+15*X[14093], X[12778]+3*X[15055], X[13607]-6*X[15759], 3*X[14269]-7*X[19876], 7*X[14869]-5*X[19862], X[15681]+3*X[19875], 7*X[15700]-3*X[25055], X[16117]+X[22936], X[20014]-65*X[21734]
= lies on these lines: {1,3}, {2,22793}, {4,11231}, {5,10164}, {8,3528}, {10,550}, {20,5818}, {21,14496}, {30,3828}, {72,11270}, {100,3916}, {104,8698}, {140,516}, {186,1902}, {355,376}, {382,1698}, {392,4188}, {411,17613}, {515,548}, {518,14810}, {546,3634}, {547,12571}, {549,946}, {551,17504}, {572,16668}, {573,16669}, {582,601}, {631,9778}, {632,3817}, {910,24047}, {944,3654}, {952,4701}, {962,3524}, {971,6796}, {1125,3530}, {1384,9593}, {1386,17508}, {1445,15008}, {1571,3053}, {1572,15815}, {1657,5587}, {1699,3526}, {1702,6398}, {1703,6221}, {1737,15338}, {1766,16675}, {1770,5432}, {1829,3520}, {2771,9943}, {2807,5447}, {2948,15041}, {3241,15710}, {3311,9582}, {3312,9616}, {3474,11374}, {3522,4678}, {3523,5886}, {3525,9812}, {3529,9780}, {3533,9779}, {3534,5691}, {3616,10299}, {3624,15720}, {3626,28224}, {3627,10175}, {3628,18483}, {3647,3740}, {3651,3652}, {3653,10595}, {3655,12245}, {3656,15692}, {3679,15688}, {3683,5506}, {3689,6763}, {3702,4781}, {3753,4189}, {3824,6690}, {3827,15578}, {3830,7989}, {3844,29012}, {3850,10172}, {3853,28158}, {3911,15171}, {4257,4646}, {4294,18527}, {4297,4669}, {4302,24914}, {4309,17728}, {4324,5445}, {4333,10895}, {4512,16408}, {4640,5044}, {4652,5687}, {4857,5442}, {4995,13407}, {5023,9620}, {5054,8227}, {5071,10248}, {5073,18492}, {5079,19872}, {5250,16371}, {5267,5836}, {5399,22053}, {5428,7686}, {5433,7743}, {5439,9352}, {5493,10165}, {5731,20053}, {5790,9588}, {5887,6876}, {5901,12100}, {6451,9615}, {6455,9583}, {6693,29032}, {6706,17729}, {6737,9945}, {6836,18407}, {6865,10525}, {6905,9856}, {6916,10526}, {6918,21153}, {6942,12672}, {6985,11495}, {8583,17573}, {9589,18493}, {9590,13564}, {9947,11499}, {10039,15326}, {10171,16239}, {10178,13369}, {10386,11019}, {10624,15325}, {11111,26062}, {11114,17619}, {11204,12262}, {11699,15035}, {11826,28459}, {11827,28458}, {12085,15592}, {12103,18357}, {12108,28216}, {12515,22935}, {12619,24466}, {12645,14093}, {12778,15055}, {13405,24470}, {13464,28212}, {13587,17614}, {13607,15759}, {14269,19876}, {14869,19862}, {15586,16814}, {15681,19875}, {15700,25055}, {16117,22936}, {17229,29093}, {19513,28257}, {19535,19860}, {19537,19861}, {19543,29229}, {20014,21734}
= complement of X(22793)
= midpoint of X(i) and X(j) for these {i,j}: {3,3579}, {10,550}, {20,18480}, {40,1385}, {2077,10225}, {3651,22937}, {4297,5690}, {5493,22791}, {6684,12512}, {7991,11278}, {9778,11230}, {10222,12702}, {12103,18357}, {12515,22935}, {12619,24466}, {13528,23961}, {16117,22936}
= reflection of X(i) in X(j) for these {i,j}: {546,3634}, {1125,3530}, {6583,9940}, {9955,140}, {9956,6684}, {13624,3}, {15178,13624}, {18483,3628}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3,17502}, {3,40,1385}, {3,165,3579}, {3,1482,7987}, {3,6244,10267}, {3,10679,8273}, {3,11012,23961}, {3,11849,15931}, {3,12702,3576}, {20,26446,18480}, {35,1155,942}, {40,3576,11531}, {40,7987,1482}, {40,11531,12702}, {40,16192,3}, {46,5217,24929}, {165,16192,40}, {631,9778,12699}, {631,12699,11230}, {1385,3579,40}, {1482,7987,1385}, {3057,7280,5126}, {3522,5657,18481}, {3523,6361,5886}, {3576,12702,10222}, {3576,16191,10246}, {4640,25440,5044}, {5119,5204,24928}, {5493,10165,22791}, {5690,8703,4297}, {7991,10246,11278}, {10222,11278,16191}, {15712,22791,10165}
= (6-9-13) search numbers [8.05537649038117363, 6.94611153104999562, -4.88604803514923646]
*** if t=-1/5 then Pa,Pa'=X(549) of AFE and BDC resp. then , W(-1/5)=X(17502) of ABC
*** if t=-5/9 then Pa,Pa'=X(550) of AFE and BDC resp. then , W(-5/9)=X(3579) of ABC
*** if t=-1/6 then Pa,Pa'=X(631) of AFE and BDC resp. then , W(-1/6)=X(7987) of ABC
*** if t=-1/21 then Pa,Pa'=X(632) of AFE and BDC resp. then
W(-1/21) = MIDPOINT OF X(11522) AND X(15696)
= a (10 a^3-3 a^2 b-10 a b^2+3 b^3-3 a^2 c+6 a b c-3 b^2 c-10 a c^2-3 b c^2+3 c^3) : : (barys)
= 3*X[1]+7*X[3], 3*X[10]-8*X[12108], X[20]+9*X[3653], 3*X[355]-13*X[10303], 2*X[546]+3*X[4297], 2*X[548]+3*X[551], 7*X[549]-2*X[4745], 3*X[946]+2*X[12103], 6*X[1125]-X[3627], X[1657]+9*X[25055], 7*X[3090]+3*X[18481], X[3146]-6*X[9955], 7*X[3523]+3*X[3655], 11*X[3525]+9*X[5731], 7*X[3528]+3*X[3656], X[3529]+9*X[5886], 4*X[3530]+X[5882], 3*X[3534]+7*X[9624], 3*X[3616]+X[17538], 21*X[3624]-11*X[5072], 4*X[3628]-9*X[10165], 3*X[3654]-13*X[10299], 9*X[3817]-4*X[12102], 4*X[3850]-9*X[19883], X[4677]-11*X[15718], 2*X[4746]-7*X[6684], X[5076]-3*X[8227], 13*X[5079]-3*X[5691], 21*X[5657]-X[20054], X[5734]+3*X[19708], X[5881]-11*X[15720], 3*X[8703]+2*X[13464], X[9589]+9*X[15688], 3*X[10283]+2*X[12512], 9*X[11231]-14*X[14869], X[11362]-6*X[12100], X[11522]+X[15696], 8*X[12104]-3*X[22936], 2*X[12812]-3*X[19862], 2*X[15704]+3*X[22793], 77*X[15717]+3*X[20049]
= lies on these lines: {1,3}, {10,12108}, {20,3653}, {104,28214}, {355,10303}, {392,17574}, {515,632}, {519,15712}, {546,4297}, {548,551}, {549,4745}, {572,15492}, {631,28204}, {946,12103}, {1125,3627}, {1656,28208}, {1657,25055}, {2771,15034}, {3090,18481}, {3091,28160}, {3146,9955}, {3522,28198}, {3523,3655}, {3525,5731}, {3528,3656}, {3529,5886}, {3530,5882}, {3534,9624}, {3616,17538}, {3624,5072}, {3628,10165}, {3654,10299}, {3817,12102}, {3850,19883}, {4677,15718}, {4746,6684}, {4881,5047}, {5076,8227}, {5079,5691}, {5657,20054}, {5734,19708}, {5881,15720}, {6428,9583}, {8703,13464}, {9589,15688}, {9619,22331}, {10283,12512}, {11231,14869}, {11362,12100}, {11363,14865}, {11522,15696}, {12104,22936}, {12812,19862}, {15704,22793}, {15717,20049}, {16865,17614}, {18493,28154}
= midpoint of X(11522) and X(15696)
= reflection of X(7987) in X(13624)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,1385,10222}, {3,10222,3579}, {3,10246,7991}, {1385,11278,10246}, {1385,13624,17502}, {1385,17502,3579}, {3576,13624,1385}, {10222,17502,3}, {13151,26287,1385}
= (6-9-13) search numbers [5.25474657877455182, 4.63355820154909214, -1.99245115552247783]
Best regards
Ercole Suppa
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου