HYACINTHOS 26695

 [Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the cevian triangle of G.

Denote:

 

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

A1, Aa = the midpoints of MaA, MaA', resp.

B2, Bb = the midpoints of MbB, MbB', resp.

C3, Cc = the midpoints of McC, McC', resp.

P1, P2, P3 = same points on the Euler lines of A1BbCc, B2CcAa, C3AaBb, resp.
Pa, Pb, Pc = same points on the Euler lines of AaB2C3, BbC3A1, CcA1B2, resp.

Conjecture:

The centroids G1, Ga of P1P2P3, PaPbPc, resp. lie on the Euler line of ABC.

[César Lozada]:

 

If P is such that OP=t*OH, then

G1 = (21*t-55)*S^2-21*(3*t-1)*SB*SC : : (barycentrics)

= (21*t-55)*X(3)-(21*t+17)*X(4)

 

Ga = (21*t-23)*S^2-21*(3*t-1)*SB* SC: : (barycentrics)

= (21*t-23)*X(3)-(21*t+1)*X(4)

 

ETC pairs (P,G1): (2, 2), (26, 7544), (3523, 11539), (3830, 3850), (3832, 5055), (8703, 12108), (10020, 7405), (11001, 548), (13371, 7509)

 

ETC pairs (P.Ga): (2, 2), (376, 548), (381, 3850), (549, 12108), (3090, 547), (3138, 8355), (3522, 14093), (3523, 549), (3526, 10124), (3543, 3627), (3832, 381), (3851, 11737), (5071, 12812), (6896, 854)

 

Examples:

 

G1(X(3))  =  MIDPOINT OF X(140) AND X(5054)

= 55*S^2-21*SB*SC : : (barycentrics)

= 19*X(2)-7*X(5) = X(2)-7*X(140) = 13*X(2)-7*X(547) = 5*X(2)+X(548) = 5*X(2)+7*X(549) = 9*X(2)+7*X(3524) = 8*X(2)+7*X(3530) = 23*X(2)-7*X(3545) = 13*X(2)-X(3627) = 10*X(2)-7*X(3628) = 4*X(2)-X(3850) = X(2)+7*X(5054) = 15*X(2)-7*X(5055) = 25*X(2)-7*X(5066) = 16*X(2)-7*X(10109) = 4*X(2)-7*X(10124) = 3*X(2)-7*X(11539) = 5*X(2)-14*X(11540) = 22*X(2)-7*X(11737) = 2*X(2)+7*X(11812)

= Shinagawa coefficients: (55, -21)

= On lines: {2, 3}

= midpoint of X(i) and X(j) for these {i,j}: {140, 5054}, {5066, 10304}

= reflection of X(i) in X(j) for these (i,j): (3861, 3545), (11812, 5054), (14269, 12811)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2, 3627, 547), (4, 549, 12100), (472, 7924, 4201), (1346, 7462, 3144), (3363, 7460, 3145), (3843, 5059, 3627), (3850, 10124, 2), (3850, 12108, 3530), (5071, 6948, 8703), (7450, 7458, 6894), (7563, 8370, 13729), (11305, 13729, 7395), (12812, 14869, 12108)

=  [3.84065873882012, 2.96100706119411, -.181875209143901]

 

Ga(X(3)) =  EULER LINE INTERCEPT OF X(3633)X(3654)

= 23*S^2-21*SB*SC : : (barycentrics)

= X(2)+7*X(3) = 23*X(2)-7*X(4) = 11*X(2)-7*X(5) = 5*X(2)-7*X(140) = 9*X(2)+7*X(376) = 15*X(2)-7*X(381) = 17*X(2)-7*X(546) = 9*X(2)-7*X(547) = 3*X(2)-7*X(549) = 13*X(2)+7*X(550) = 7*X(2)+X(1657) = 2*X(2)-7*X(3530) = 5*X(2)-X(3627) = 8*X(2)-7*X(3628) = 13*X(2)-5*X(3843) = 19*X(2)-7*X(3845) = 31*X(2)-14*X(3856) = 16*X(2)-7*X(3860) = X(3633)+7*X(3654)

= Shinagawa coefficients: (23, -21)

= On lines: {2, 3}, {3633, 3654}, {4114, 5719}, {5355, 8589}, {6053, 13392}, {11694, 12041}

= midpoint of X(i) and X(j) for these {i,j}: {2, 548}, {3, 12100}, {20, 12101}, {140, 8703}, {376, 547}, {546, 3534}, {550, 5066}, {3845, 12103}, {11694, 12041}

= reflection of X(i) in X(j) for these (i,j): (2, 12108), (5, 11540), (3530, 12100), (3628, 11812), (3830, 3856), (3850, 2), (3860, 3628), (3861, 10109), (10109, 140), (10124, 549), (11737, 10124), (11812, 3530), (12101, 12811), (12102, 5066)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (140, 546, 5070), (140, 3861, 3628), (381, 549, 140), (381, 3524, 549), (381, 5070, 5071), (381, 10109, 11737), (3090, 3543, 381), (3530, 3850, 12108), (3627, 3850, 3861), (3843, 5054, 2), (3843, 5072, 3854), (3850, 12102, 3843), (3861, 11737, 381)

= [ 6.263238631482892, 5.37719612693071, -2.972735282036674 ]

 

G1(X(4)) = EULER LINE INTERCEPT OF X(3656)X(4668)

= 17*S^2+21*SB*SC : : (barycentrics)

= 19*X(2)-7*X(3) = X(2)-7*X(5) = 10*X(2)-7*X(140) = 5*X(2)+7*X(381) = 8*X(2)+7*X(546) = 4*X(2)-7*X(547) = 4*X(2)-X(548) = 13*X(2)-7*X(549) = 13*X(2)-X(1657) = 15*X(2)-7*X(3524) = 29*X(2)-14*X(3530) = X(2)+7*X(3545) = 5*X(2)+X(3627) = 11*X(2)-14*X(3628) = 9*X(2)+7*X(3839) = 7*X(2)+5*X(3843) = 11*X(2)+7*X(3845) = X(2)+2*X(3850) = 11*X(2)-7*X(5054) = 3*X(2)-7*X(5055)

= Shinagawa coefficients: (17, 21)

= On lines: {2, 3}, {3630, 11178}, {3656, 4668}, {4691, 9955}, {7687, 11694}, {10113, 11693}

= midpoint of X(i) and X(j) for these {i,j}: {5, 3545}, {549, 14269}, {3839, 11539}, {3845, 5054}, {10113, 11693}

= reflection of X(i) in X(j) for these (i,j): (3545, 11737), (3853, 14269), (5054, 3628), (5066, 3545), (10304, 11812), (12103, 10304), (14269, 3860)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2, 4, 14093), (4, 8226, 7413), (5, 3845, 5071), (140, 3627, 548), (381, 12101, 546), (382, 3851, 13587), (546, 547, 12100), (547, 12101, 140), (1567, 6862, 4234), (3090, 3861, 140), (3627, 5072, 12811), (3839, 5055, 11539), (3845, 5071, 3628)

= [ 0.206788899825936, -0.66327653741078, 4.004414900195258 ]

 

Ga(X(4)) = EULER LINE INTERCEPT OF X(517)X(4532)

= S^2+21*SB*SC : : (barys)

= 11*X(2)-7*X(3) = X(2)+7*X(4) = 5*X(2)-7*X(5) = 23*X(2)-7*X(20) = 8*X(2)-7*X(140) = 15*X(2)-7*X(376) = 3*X(2)-7*X(381) = 13*X(2)+7*X(382) = 2*X(2)-7*X(546) = 6*X(2)-7*X(547) = 9*X(2)-7*X(549) = 17*X(2)-7*X(550) = 5*X(2)-X(1657) = 19*X(2)-14*X(3530) = 19*X(2)-7*X(3534) = 9*X(2)+7*X(3543) = 13*X(2)-14*X(3628) = 5*X(2)+7*X(3830) = X(3630)-7*X(3818) = 5*X(4668)+7*X(12699)

= Shinagawa coefficients: (1, 21)

= On lines: {2, 3}, {265, 14487}, {397, 12816}, {398, 12817}, {517, 4532}, {541, 11801}, {1587, 6499}, {1588, 6498}, {3163, 6748}, {3630, 3818}, {4668, 12699}, {5305, 14537}, {5309, 14075}, {5480, 13687}, {5663, 13451}, {10095, 13474}, {13364, 13570}

= midpoint of X(i) and X(j) for these {i,j}: {2, 3627}, {4, 3845}, {5, 3830}, {382, 8703}, {546, 12101}, {549, 3543}, {3853, 5066}, {3860, 12102}

= reflection of X(i) in X(j) for these (i,j): (2, 3850), (3, 10109), (140, 5066), (376, 10124), (546, 3845), (547, 381), (548, 2), (549, 11737), (550, 11812), (3534, 3530), (3830, 12102), (3845, 3861), (3853, 12101), (5066, 546), (8703, 3628), (10109, 3856), (12100, 5), (12101, 4), (12103, 12100), (13364, 13570)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (5, 3627, 1657), (381, 3543, 549), (382, 3545, 8703), (382, 3858, 3628), (546, 3627, 12812), (1658, 1885, 3529), (3627, 3843, 3850), (3627, 3850, 548), (3830, 3839, 5), (3830, 3845, 3860), (3830, 3860, 12100), (3855, 5073, 632), (5073, 6885, 12108)

= [ -4.638370885499629, -5.49565466888397, 9.586135045980804 ]

 

César Lozada