HYACINTHOS 25590

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and P a point.

Denote:

A'B'C' = the midway triangle of P

(ie A', B', C' = the midpoints of AP, BP, CP, resp.)

A", B", C" = the orthogonal projections of A', B', C' on BC, CA, AB, resp.

 

Ma, Mb, Mc = the midpoints of A'A", B'B", C'C" resp.

The circumcenter of MaMbMc lies on the line OH', where H' is the orthocenter of A'B'C' (the point the A'A", B'B", C'C" concur at).

If P lies on the Euler line, then H' lies on the Euler line and the circumcenter of  MaMbMc lies on the Euler line.

Which point is it?

 

[César Lozada]:

For P such that OP/OH=t

 

H’(t) = (3*t-1)*SB*SC-(t+1)*S^2 : :  (barycentrics) = midpoint(P,H)

Om(t) = (9*t+1)*SB*SC+(5-3*t)*S^2 : : (barycentrics)

= Shinagawa coefficients: (5-3*t, 9*t+1)

 

ETC pairs (P,H’): (2,381), (3,5), (4,4), (5,546), (20,3), (21,6841), (23,11799), (24,235), (25,1596), (140,3850), (186,403), (376,2), (378,427), (381,3845), (382,3627), (403,10151), (411,6842), (443,6849), (546,3861), (547,3860), (548,3628), (549,5066), (550,140), (631,3091), (632,3859), (858,10297), (1006,8226), (1012,8727), (1113,1312), (1114,1313), (1593,1595), (1656,3858), (1657,550), (2070,11563), (2071,2072), (3079,6624), (3090,3832), (3091,3843), (3146,382), (3520,1594), (3522,1656), (3523,3851), (3524,3545), (3525,3855), (3526,3857), (3528,3090), (3529,20), (3533,3854), (3534,549), (3537,7392), (3541,7507), (3543,3830), (3545,3839), (3575,6756), (3627,3853), (3628,3856), (3651,442), (3853,12102), (4240,11251), (5059,1657), (5189,7574), (6240,3575), (6353,6623), (6815,7528), (6827,6929), (6836,6928), (6850,6917), (6853,7548), (6857,6866), (6865,6893), (6868,3560), (6869,6985), (6875,6828), (6876,2476), (6880,6968), (6897,6835), (6899,2478), (6901,6894), (6903,5046), (6905,1532), (6906,6831), (6909,6882), (6916,6826), (6925,6923), (6934,3149), (6935,6844), (6938,1012), (6942,6941), (6947,6957), (6948,6911), (6950,6830), (6951,6839), (6987,6913), (6988,6867), (7411,6881), (7414,429), (7418,3143), (7421,3142), (7422,868), (7425,3140), (7429,3139), (7430,3136), (7440,3138), (7444,3141), (7454,3137), (7464,858), (7470,6656), (7487,1598), (7488,10024), (7503,7403), (7576,428), (7580,6907), (8703,547), (10018,10019), (10295,468), (10298,10254), (10299,5068), (10304,5055), (10323,7399), (10594,1906), (10996,7401), (11001,376), (11250,10224), (11413,11585), (11414,6823), (11676,1513), (11845,11897), (12100,11737), (12103,3530), (20004 (*),402)

 
(*) = the orthologic center Gossard -> ABC

 

ETC pairs (P,Om): (2,5), (3,3628), (4,546), (20,3), (376,140), (381,3850), (382,12102), (550,12108), (1658,12010), (2071,5159), (3091,3091), (3146,3627), (3153,10297), (3522,632), (3523,3090), (3524,547), (3529,12103), (3533,4235), (3534,3530), (3543,4), (3545,5066), (3830,3861), (3832,3857), (3839,381), (3845,3856), (5054,10109), (5055,11737), (5056,5072), (6644,4235), (7486,3544), (7500,7530), (10303,5079), (10304,2), (11001,548). Fixed point: X(3091).

 

Om(N) = 11*SB*SC+7*S^2 : : (barycentrics)

=   27*X(2)-11*X(3), 3*X(2)-11*X(5), 15*X(2)-11*X(140), 5*X(2)+11*X(381), 9*X(2)+11*X(546), 7*X(2)-11*X(547), 19*X(2)-11*X(549), 21*X(2)-11*X(3530), X(4701)+11*X(9955), X(5609)+3*X(11801)

= Shinagawa coefficients: (7, 11)

= midpoint of X(i) and X(j) for these {i,j}: {3,12102}, {4,3530}, {5,3850}, {140,3861}, {381,10109}, {546,3628}, {547,3860}, {3845,10124}, {5066,11737}

= reflection of X(i) in X(j) for these (i,j): (3856,3850), (11540,547), (11695,12046), (12108,3628)

= On lines: {2,3}, {517,4540}, {3303,10592}, {3304,10593}, {3614,3746}, {4701,5844}, {5418,10147}, {5420,10148}, {5563,7173}, {5609,11801}, {6488,8253}, {6489,8252}, {11695,12046}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (4,5,547), (4,547,3530), (4,632,12103), (4,3859,3860), (4,5070,8703), (4,5079,632), (5,3627,3090), (5,3845,1656), (140,12101,20), (381,3627,546), (546,3627,3861), (632,3627,8703), (1656,3845,548), (3090,3627,140), (3091,3146,3855), (3525,5076,550), (3628,12102,3), (3843,5056,549), (3850,3861,381), (3861,10109,140), (5055,5076,3525)

= [ -0.225814652435276, -1.09473887057803, 4.502782770354682 ]

 

Om(X(631)) = 7*SB*SC+11*S^2 : : (barycentrics)

= 27*X(2)-7*X(3), 3*X(2)+7*X(5), 12*X(2)-7*X(140), 13*X(2)+7*X(381), 2*X(2)-7*X(547), 6*X(2)-X(548), 17*X(2)-7*X(549), 15*X(2)-7*X(631), 9*X(373)+X(5876), 7*X(576)+3*X(3630)

= Shinagawa coefficients: ( 11, 7)

= midpoint of X(i) and X(j) for these {i,j}: {5,1656}, {140,3859}, {631,3858}, {632,3091}

= reflection of X(i) in X(j) for these (i,j): (546,3091), (632,3628), (3522,3530), (3843,3850), (5071,10109)

= On lines: {2,3}, {373,5876}, {576,3630}, {3303,10593}, {3304,10592}, {3614,5563}, {3625,10175}, {3633,5886}, {3635,5901}, {3746,7173}, {4668,5844}, {4691,9956}, {5305,7603}, {5690,7988}, {5943,12046}, {6560,10148}, {6561,10147}, {10095,10170}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2,3627,12108), (3,11541,550), (4,5,11737), (5,3627,5072), (5,3845,5068), (140,546,12103), (546,547,3628), (546,548,3627), (546,3091,3859), (546,3628,140), (546,12103,3853), (1656,3843,2), (3091,5076,3858), (3627,3850,546), (3627,5072,3850), (3627,12108,548), (3628,3856,10303), (3843,5072,3091), (3857,12102,546), (5070,12101,140), (10303,11541,3)

= [ 1.175820856891049, 0.30319908888386, 2.888070871038149 ]

 

César Lozada