HYACINTHOS 25365

[Antreas P. Hatzipolakis]:

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote:

Aa, Ab, Ac = the orthogonal projections of A on PA', PB', PC', resp. = Aa, B', C', resp.

Bb = the orthogonal projection of B on PB'

Cc = the orthogonal projection of C on PC'

Na, Nb, Nc = the NPC centers of AaB'C', BbC'A', CcA'B', resp.

ABC, NaNbNc are orthologic.

Which are the loci of the orthologic centers as P moves on a line, the Euler line, for example?

 

[César Lozada]:
 

I couldn’t find a general expression for Na, but it seems that:

 

Za(P) =  orthologic center ABC->NaNbNc = is always X(54)

 

For P on the Euler line of ABC, Zn(P) =  orthologic center NaNbNc->ABC  lies on the line {140, 389 }. If OP=t*OH then X(140)Zn(P) = t*X(140)X(389).

 

Some Zn(P):

 

Zn(G) = X(5892)

Zn(O) = X(140)

Zn(H) = X(389)

 

Zn(I) = (b+c)*a^5-(b-c)^2*a^4-(b+c)*( 2*b^2-b*c+2*c^2)*a^3+(2*b^2+b* c+2*c^2)*(b-c)^2*a^2+(b^3-c^3) *(b^2-c^2)*a-(b^2-c^2)*(b-c)*( b^3+c^3) :: (trilinears)
= (3*R+2*r)*X(1)+(R-2*r)*X(104)

= on lines: {1,104}, {3,3874}, {4,5557}, {5,2801}, {10,10202}, {12,10265}, {30,6583}, {40,3873}, {48,1729}, {57,6796}, {65,4311}, {72,10165}, {84,11020}, {140,3678}, {354,946}, {355,5883}, {515,942}, {517,548}, {518,5771}, {551,5887}, {581,982}, {631,5904}, {758,1385}, {912,1125}, {938,6256}, {944,4317}, {950,5570}, {952,3754}, {1006,6763}, {1064,3953}, {1210,10958}, {1482,3892}, {1483,2802}, {1490,10980}, {2771,5901}, {3149,4860}, {3218,10902}, {3333,6261}, {3336,11491}, {3337,6905}, {3555,11362}, {3576,3868}, {3577,9845}, {3616,5693}, {3651,5536}, {3742,5777}, {3833,9956}, {3878,10246}, {3889,7982}, {3894,7987}, {4015,11231}, {5045,6001}, {5253,6326}, {5439,10175}, {5542,6245}, {5708,11500}, {5728,6260}, {5770,10198}, {6705,11018}, {6952,11219}, {9948,10569}, {10573,10805}, {11025,11372}

= midpoint of X(i) and X(j) for these {i,j}: {1,5884}, {3,3874}, {65,5882}, {3555,11362}, {11570,11715}

= reflection of X(i) in X(j) for these (i,j): (3678,140), (3754,5885), (6684,9940)

= [ 2.951749391576936, 3.26651517490852, 0.016885026242974 ]

 

Zn(N) = midpoint of X(140) and X(389)

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

= (4*R^2+OH^2)*X(5)-OH^2*X(113)

= on lines: {2,6102}, {3,143}, {5,113}, {30,5462}, {51,550}, {52,549}, {54,1511}, {140,389}, {156,6642}, {182,1658}, {186,6152}, {381,10574}, {382,5640}, {511,3530}, {546,5943}, {547,5907}, {548,5446}, {568,631}, {632,5562}, {1112,3520}, {1199,1493}, {1539,3521}, {1656,5876}, {1657,9781}, {1986,6143}, {3523,6243}, {3526,5889}, {3528,11002}, {3628,10219}, {3845,10575}, {3850,6000}, {3851,6241}, {3858,11381}, {5012,5944}, {5054,11412}, {5055,11465}, {5070,11459}, {6146,9827}, {7514,9786}, {7526,10601}, {9703,11423}, {10272,11806}, {11245,11264}

= midpoint of X(i) and X(j) for these {i,j}: {3,143}, {52,10627}, {125,11561}, {140,389}, {548,5446}, {5462,9729}, {6102,11591}, {8254,11802}, {10272,11806}

= reflection of X(i) in X(j) for these (i,j): (3628,11695), (10095,5462), (10627,11592)

= complement of X(11591)

= [ 3.343432698720069, 2.90988198927620, 0.083008012999289 ]

 

Zn(K) = (6*a^6-9*(b^2+c^2)*a^4+4*(b^2- c^2)^2*a^2-(-c^4+b^4)*(b^2-c^ 2))/a : : (trilinears)

= X(4)-5*X(6)

= On lines: {3,3629}, {4,6}, {5,6329}, {20,5102}, {30,5097}, {69,10303}, {98,9300}, {125,11245}, {140,3631}, {141,3526}, {182,524}, {193,5085}, {511,548}, {542,5066}, {575,3564}, {578,6696}, {597,1352}, {1350,1992}, {1351,3534}, {2854,9826}, {3398,7789}, {3523,11008}, {3567,9973}, {3618,7486}, {3815,9755}, {3818,3857}, {5306,9744}, {6144,10519}, {6247,11426}, {6279,11314}, {6280,11313}, {6676,11225}, {10168,11540}, {10192,11433}, {11064,11422}

= midpoint of X(i) and X(j) for these {i,j}: {3,3629}, {6,8550}, {182,1353}, {5480,6776}, {8584,11179}

= reflection of X(i) in X(j) for these (i,j): (5,6329), (3589,575), (3631,140)

= [ 2.660290471563721, 3.49449449536947, -0.006427309454666 ]

 

César Lozada