HYACINTHOS 25561

[Antreas P. Hatzipolakis]:

Let ABC be a triangle, P, Q two points on the Euler line and PaPbPc, QaQbQc the pedal triangles of P,Q, resp.

Denote:

Na, Nb, Nc = the NPC centers of PPbPc, PPcPa, PPaPb, resp.

N1, N2, N3 = the reflections of Na, Nb, Nc in QQa QQb, QQc, resp.

The circumcenter O* of N1N2N3 lies on the Euler line

Which is the ratio O*P/O*Q ?

For P = H, Q = N, the O* is the midpoint of HN.

[César Lozada]:

 

Let P, Q such that OP=p*OH and Q=q*OH. Then O* satisfies OO*=(p/4+q)*OH and has barycentric coordinates:

O*(P,Q) = (p/4+q)*(2*a^4-(b^2+c^2)*a^2-( b^2-c^2)^2)+4*a^2*(-a^2+b^2+c^ 2)  : :  (Barycentrics)

= (p/4+q)*(S^2-3*SB*SC)-S^2+SB* SC : : (barycentrics)
= Shinagawa coefficients: (-p-4*q+4, 3*p+12*q-4)

 

O*P/O*Q = (3*p-4*q)/p 

 

It can be deduced that Q*=Q for p=0, i.e., for P=O.

 

Some ETC triads (P, Q, O*(P,Q) ): (2, 2, 547) , (2, 5, 5066), (4, 2, 5066), (4, 3, 140), (4, 4, 3853), (4, 5, 546), (5, 2, 10109), (5, 3, 3530), (5, 5, 3850), (20, 3, 548), (20, 4, 546), (20, 5, 140)

 

Other:

 

O*(G, O) = X(395)X(10645) /\ X(396)X(10646)

= 10*a^4-11*(b^2+c^2)*a^2+(b^2- c^2)^2 : : (barycentrics)

= X(2)+3*X(3) = 11*X(2)-3*X(4) = 5*X(2)-3*X(5) = 13*X(2)+3*X(20) = 2*X(2)-3*X(140) = 5*X(2)+3*X(376) = 7*X(2)-3*X(381) = 23*X(2)-3*X(382) = X(40)+3*X(3653) = 3*X(165)+X(3656)

= Shinagawa coefficients: (11, -9)

= On lines: {2,3}, {35,5298}, {36,4995}, {40,3653}, {165,3656}, {182,8584}, {187,9300}, {230,8589}, {395,10645}, {396,10646}, {524,5092}, {539,10213}, {541,10272}, {551,3579}, {553,5122}, {574,5306}, {597,3098}, {952,4669}, {1216,11592}, {1327,8253}, {1328,8252}, {1503,10193}, {1587,6497}, {1588,6496}, {1992,12017}, {2482,12042}, {3055,6781}, {3058,5010}, {3068,6452}, {3069,6451}, {3576,3654}, {3655,4677}, {3793,7837}, {3815,8588}, {3819,5663}, {4316,5326}, {4324,7294}, {4745,6684}, {5204,10056}, {5217,10072}, {5434,7280}, {5442,10543}, {5585,7737}, {5609,11693}, {5642,12041}, {6390,7771}, {6410,8981}, {6445,7586}, {6446,7585}, {6456,9540}, {7288,10386}, {7618,8667}, {7767,7799}, {7811,7871}, {8182,9766}, {9729,10627}, {9774,11149}, {10192,11204}

= midpoint of X(i) and X(j) for these {i,j}: {2,8703}, {3,549}, {5,376}, {381,550}, {547,548}, {551,3579}, {597,3098}, {2482,12042}, {3534,3845}, {3655,5690}, {5642,12041}, {8182,12040}, {10192,11204}, {10304,11539}

= reflection of X(i) in X(j) for these (i,j): (2,11812), (4,11737), (5,10124), (140,549), (381,3628), (546,547), (547,140), (549,3530), (3543,3861), (3830,3860), (3845,10109), (3853,381), (5066,2), (10109,11540)

= anticomplement of X(10109)

= complement of X(3845)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2,3830,5), (2,3845,10109), (2,5066,547), (3,381,10304), (5,3830,3860), (5,3860,5066), (140,3853,3628), (140,5066,2), (376,3839,1657), (381,631,11539), (381,5055,3544), (381,10304,550), (381,11539,3628), (3146,3523,631), (3146,10304,376), (3830,5054,2), (3845,8703,3534), (3845,10109,5066), (3853,11539,547), (3860,11812,10124), (5067,5073,3857)

= [ 5.744114368769439, 4.85944132713001, -2.374693837845366 ]

 

O*(G, H) = X(1327)X(6441) /\ X(1328)X(6442)

= 14*a^4-(b^2+c^2)*a^2-13*(b^2- c^2)^2 : :  (barycentrics)

= 13*X(2)-9*X(3) = X(2)-9*X(4) = 7*X(2)-9*X(5) = 25*X(2)-9*X(20) = 10*X(2)-9*X(140) = 17*X(2)-9*X(376) = 5*X(2)-9*X(381) = 11*X(2)+9*X(382)

= Shinagawa coefficients: (1, -27)

= On lines: {2,3}, {1327,6441}, {1328,6442}

= midpoint of X(i) and X(j) for these {i,j}: {5,3543}, {381,3627}, {382,549}, {3830,3845}

= reflection of X(i) in X(j) for these (i,j): (3,11737), (140,381), (376,3628), (381,3861), (547,546), (549,3850), (550,10124), (3534,11812), (5066,3845), (8703,10109), (10124,3856)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2,3845,3860), (4,3627,3861), (4,3830,3845), (4,3853,546), (4,5076,5), (5,3627,5073), (140,3853,3627), (140,3861,546), (381,5073,3524), (382,3839,549), (3146,3858,3530), (3524,3543,5073), (3534,3830,3543), (3534,5076,3830), (3543,3839,3522), (3543,3845,11812), (3627,3845,8703), (3627,3861,140), (3845,5066,546), (3856,11541,140), (5070,11541,550)

= [ -6.714867936353443, -7.56667386808677, 11.978300822746040 ]

 

O*(N, H) = midpoint of X(140) and X(382)

= 10*a^4-(b^2+c^2)*a^2-9*(b^2-c^ 2)^2 : : (barycentrics)

= 27*X(2)-19*X(3) = 3*X(2)-19*X(4) = 15*X(2)-19*X(5) = 21*X(2)-19*X(140) = 35*X(2)-19*X(376) = 11*X(2)-19*X(381) = 9*X(2)-19*X(546) = 17*X(2)-19*X(547)

= Shinagawa coefficients: (1, -19)

= On lines: {2,3}, {517,4536}, {5447,11017}, {11565,11645}

= midpoint of X(i) and X(j) for these {i,j}: {4,3853}, {140,382}, {546,3627}, {3543,5066}

= reflection of X(i) in X(j) for these (i,j): (140,3856), (3530,3850), (3628,546), (3850,3861), (3861,4), (5447,11017), (11737,3845), (11812,381)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3,382,11541), (4,382,3845), (4,3543,3843), (4,3627,546), (4,3830,5), (4,5076,3627), (5,3627,3146), (382,3845,140), (382,5055,5059), (546,3853,3627), (3091,3146,376), (3091,11541,3), (3146,3523,3529), (3146,3525,1657), (3146,3830,3627), (3146,3839,3525), (3543,3843,550), (3627,5076,3853), (3628,3861,546), (3830,5054,3543), (3832,5073,549)

= [ -7.233992199066897, -8.08442866788747, 12.576342266937350 ]

 

César Lozada