HYACINTHOS 28829

• [buratinogigle = Tran Quang Hung]:

  Let ABC be a triangle inscribed in circle (O) and A’B’C’ its medial triangle. Let P be a point on the Euler line of ABC , A*B*C* the circumcevian triangle of P and A”, B”, C” the midpoints of AA*, BB*, CC*, resp. Then the lines A’A”, B’B”, C’C” are concurrent. (See AOPS Concurrent lines with a point on Euler line (buratinogigle)).

   

  [César Lozada]:

   

  First than all, the locus of P such that A’A”, B’B”, C’C” are concurrent is {sidelines of ABC} ∪ {circumcircle} ∪ {Euler line of ABC}

   

  ·         If P lies on the circumcircle of ABC then the given lines concur at Q(P) = complement(complement(P)), ie, Q(P) lies on the circle (X(140), R/4) through ETCs {620, 3035, 5972, 6036, 6699, 6710, 6711, 6712, 6713, 6714, 6715, 6716, 6717, 6718, 6719, 6720, 10120, 13372, 15240, 16760, 22102, 22103, 22104} and vertices of 6^th mixtilinear triangle.

   

  ETC pairs (P,Q(P)) = (74,6699), (98,6036), (99,620), (100,3035), (101,6710), (102,6711), (103,6712), (104,6713), (105,6714), (106,6715), (107,6716), (108,6717), (109,6718), (110,5972), (111,6719), (112,6720), (476,22104), (805,22103), (842,16760), (901,22102), (930,13372)

   

  ·         If P lies on the Euler line and OP/OH=t, then

  Q(P) = ((t+1)*S^2-(3*t-1)*SA^2)*((t-1)*S^2-(3*t-1)*SB*SC)*(4*(t-1)*t*S^2-(3*t-1)*(SB+SC)*(2*(3*t-1)*R^2-(SB+SC)*t)) : :  (barys)

   

  ETC pairs (P,Q(P)): (2,2), (3,3), (4,1147), (21,4999)

   

  Some others:

   

  Q(X(5)) = COMPLEMENT OF X(252)

  = (S^2+SB*SC)*(3*S^2-SA^2)*(4*S^2+(SB+SC)*(2*R^2-SB-SC)) : : (barys)

  = lies on these lines: {2, 252}, {3, 24573}, {5, 128}, {140, 6150}, {233, 5421}, {1209, 1493}, {2072, 10600}, {5501, 6592}, {7575, 15848}, {13372, 21975}

  = midpoint of X(3) and X(24573)

  = reflection of X(i) in X(j) for these (i,j): (5, 23281), (23280, 3628)

  = complement of X(252)

  = {X(5), X(15345)}-harmonic conjugate of X(137)

  = [ 2.5600921329564780, 0.8715528465604969, 1.8557007575395070 ]

   

  Q(X(20)) = COMPLEMENT OF X(6526)

  = (3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(a^8-4*(b^2-c^2)^2*a^4+4*(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)^4)*(-a^2+b^2+c^2)^2 : : (barys)

  = SA^2*(S^2-2*SB*SC)*(2*S^2-(SB+SC)*(8*R^2-SB-SC)) : : (barys)

  = lies on these lines: {2, 1105}, {3, 1661}, {4, 12096}, {20, 122}, {131, 3548}, {140, 20207}, {216, 631}, {1073, 6696}, {3546, 10600}, {5895, 27089}, {6225, 11589}, {6389, 16196}, {6523, 6716}, {6760, 14216}, {11457, 14385}

  = complement of X(6526)

  = [ 4.6873637620998600, 3.6627407163306680, -1.0584777503675560 ]

   

  Q(X(30)) = COMPLEMENT OF X(5627)

  = (S^2-3*SB*SC)*(3*SA^2-S^2)*(4*S^2-3*(SB+SC)*(6*R^2-SB-SC)) : : (barys)

  = 4*X(140)-X(6070), 3*X(549)+X(18285), 5*X(631)+X(14480), 2*X(1511)+X(3258), X(1553)-4*X(10272), 2*X(3154)+X(30714), 2*X(5972)+X(14934), 4*X(5972)-X(25641), 2*X(6699)+X(14611), X(10564)+2*X(16319), X(14508)+5*X(20125), 2*X(14934)+X(25641), 5*X(15034)+X(17511), 5*X(15040)+X(20957)

  = lies on the cubics K515, K900 and these lines: {2, 5627}, {30, 113}, {128, 6760}, {140, 6070}, {186, 14920}, {476, 1138}, {541, 15468}, {549, 18285}, {631, 14480}, {3003, 3163}, {3154, 30714}, {5972, 14934}, {6699, 14611}, {14508, 20125}, {14993, 22104}, {15034, 17511}, {15040, 20957}

  = midpoint of X(476) and X(1138)

  = reflection of X(14993) in X(22104)

  = complement of X(5627)

  = complementary conjugate of X(20304)

  = {X(5972), X(14934)}-harmonic conjugate of X(25641)

  = [ 3.8824128476923860, 2.5344569869207810, 0.0941575612581225 ]

   

  Q(X(477)) = COMPLEMENT OF X(25641)

  = S^4-(3*R^2*(90*R^2+3*SA-40*SW)-2*SA^2-SB*SC+13*SW^2)*S^2+(18*R^2-5*SW)*(9*R^2-SW)*SB*SC : : (barys)

  = 3*X(2)+X(477), 3*X(3)+X(20957), X(476)-5*X(631), 5*X(632)-X(18319), X(1553)-3*X(14643), 5*X(3091)-X(14989), 3*X(3258)-X(20957), 7*X(3523)+X(14731), X(6070)-3*X(15061), X(11749)+7*X(14869), 3*X(15035)+X(17511)

  = lies on these lines: {2, 477}, {3, 3258}, {30, 5972}, {125, 14934}, {140, 16168}, {476, 631}, {523, 6699}, {620, 15122}, {632, 18319}, {1511, 16340}, {1553, 14643}, {3091, 14989}, {3154, 17702}, {3523, 14731}, {5446, 12052}, {6070, 15061}, {10625, 16978}, {11749, 14869}, {12079, 20397}, {14611, 16003}, {14915, 16319}, {15035, 17511}, {15088, 21316}

  = midpoint of X(i) and X(j) for these {i,j}: {3, 3258}, {125, 14934}, {477, 25641}, {1511, 16340}, {10625, 16978}, {14611, 16003}

  = reflection of X(i) in X(j) for these (i,j): (5446, 12052), (12079, 20397), (21316, 15088), (22104, 140)

  = complement of X(25641)

  = {X(2), X(477)}-harmonic conjugate of X(25641)

  = [ 5.3698377844877330, 4.2520550765519330, -1.7814526254689910 ]

   

  Q(X(675)) = COMPLEMENT OF X(5513)

  = 2*a^6-2*(b+c)*a^5-2*(b^2-b*c+c^2)*a^4+(b+c)*(5*b^2-6*b*c+5*c^2)*a^3-(3*b^4+3*c^4-(b^2+c^2)*b*c)*a^2-(b^4-c^4)*(b-c)*a+(b^4+c^4+(b-c)^2*b*c)*(b-c)^2 : : (barys)

  = 3*X(2)+X(675)

  = lies on these lines: {2, 101}, {3, 25642}, {142, 6718}, {1054, 4859}, {3035, 3739}, {3315, 8458}, {5432, 6025}, {5972, 6707}, {6678, 6720}, {10165, 15746}, {16056, 25468}

  = midpoint of X(i) and X(j) for these {i,j}: {3, 25642}, {675, 5513}

  = complement of X(5513)

  = orthoptic circle of Steiner inellipse-inverse-of X(150)

  = {X(2), X(675)}-harmonic conjugate of X(5513)

  = [ 3.8776076066819770, 4.5040723346971930, -1.2672044913515130 ]

   

  César Lozada