HYACINTHOS 28467

 

[Antreas P. Hatzipolakis]:

 

1 and 2: Hyacinthos 28463 

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Let ABC be a triangle and P a point on the Euler line (such that PO/OH = t : number?).

3. Denote:

N1, N2, N3 = the NPC centers of GBC, GCA, GAB, resp.

Aa, Ab, Ac = the midpoints of AN1, BN1, CN1, resp.
Ba, Bb, Bc = the midpoints of AN2, BN2, CN2, resp.

Ca, Cb, Cc = the midpoints of AN3, BN3, CN3, resp.

Pa, Pb, Pc = same to P points of the triangles AaAbAc, BaBbBc, CaCbCc, resp.
(ie Pa lies on the Euler line of AaAbAc such that PaOa / OaHa = t and similarly Pb, Pc)

Conjecture:

The centroid of PaPbPc lies on the Euler line of ABC.

Which is it in terms of t?

 

 

[César Lozada]:

 

 

Your conjecture is true for any t.

 

If Gp is the centroid of PaPbPc, then

Gp(t) = (-2+4*t)*a^4-(b^2+c^2)*(-3+2*t)*a^2+(b^2-c^2)^2*(-2*t-1) : : (barys)

= midpoint of P and N

 

ie, OGp/OH = (2*t+1)/4

 

ETC pairs (P,Gp(P)):

(2,547), (3,140), (4,546), (20,548), (21,10021), (26,13383), (140,3628), (376,12100), (381,5066), (382,3853), (546,3850), (547,10109), (548,3530), (549,2), (632,1656), (1656,12812), (1657,12103), (1658,10020), (2070,10096), (3530,16239), (3543,12101), (3545,14892), (3651,11277), (3830,14893), (3843,3859), (3845,381), (3850,12811), (3853,3861), (3857,3851), (3858,3091), (3861,3856), (5066,11737), (5428,6675), (5499,442), (5500,10286), (5501,15957), (6644,6677), (7502,6676), (7575,468), (7715,13861), (8703,549), (10126,10289), (10127,10128), (10154,10201), (10205,10126), (10212,12043), (10226,5498), (10285,5501), (10295,18571), (11001,15691), (11250,23336), (11539,15699), (11563,403), (11819,6756), (12100,10124), (12107,18282), (13163,23409), (13371,10224), (13631,13362), (14891,11540), (14893,3860), (15122,5159), (15327,12056), (15331,10125), (15334,12057), (15681,15690), (15686,8703), (15687,3845), (15690,14891), (15691,15759), (15699,5055), (15704,550), (15711,15694), (15712,632), (15714,15713), (15761,13406), (16160,6841), (16340,3154), (17504,11539), (18282,12010), (18572,10297), (19710,376), (20030,19940), (20120,20030), (23335,13371)

 

Some others:

Gp( X(22) ) = midpoint of X(5) and X(22)

= 2*a^10-5*(b^2+c^2)*a^8+2*(b^2-c^2)^2*a^6+2*(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*a^4-2*(b^2-c^2)^2*(2*b^4+b^2*c^2+2*c^4)*a^2+(b^4-c^4)*(b^2-c^2)^3 : : (barys)

= as a point on the Euler line, this center has Shinagawa coefficients (-5*E-12*F, 7*E+4*F)

= on lines: {2, 3}, {523, 11619}, {2781, 10272}, {3564, 19127}, {3589, 13364}, {6689, 13598}, {8717, 23329}, {9019, 13451}, {9820, 10627}, {11591, 16252}, {13391, 23292}

= midpoint of X(i) and X(j) for these {i,j}: {5, 22}, {6676, 16618}, {7502, 15760}

= reflection of X(i) in X(j) for these (i,j): (140, 6676), (427, 3628), (18570, 3530)

= complement of the complement of X(12083)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10020, 140), (140, 10096, 6677), (550, 7542, 23336), (2937, 13160, 11819), (3530, 16238, 140), (3530, 18282, 16238), (3547, 10565, 18420), (6636, 7552, 2072), (6677, 13383, 10096), (7517, 7558, 5), (7542, 23336, 140), (10125, 16196, 140), (10565, 18420, 26), (13383, 16197, 140)

= [ -11.9052889476672700, -12.7434024177163300, 17.9577687469421200 ]

 

Gp( X(23) ) = midpoint of X(5) and X(23)

= 2*a^10-5*(b^2+c^2)*a^8+2*(b^4-b^2*c^2+c^4)*a^6+(b^2+c^2)*(4*b^4-5*b^2*c^2+4*c^4)*a^4-(b^2-c^2)^2*(4*b^4-3*b^2*c^2+4*c^4)*a^2+(b^4-c^4)*(b^2-c^2)^3 : : (barys)

= 2*X(12002)-3*X(13446), 3*X(14643)+X(15107)

= As a point on the Euler line, this center has Shinagawa coefficients (-3*E-24*F, 17*E+8*F)

= on lines: {2, 3}, {511, 10272}, {523, 6140}, {1154, 16534}, {1533, 12041}, {5305, 16308}, {5446, 15806}, {5462, 20193}, {7286, 15325}, {8157, 24305}, {8254, 10110}, {8705, 18583}, {11649, 13451}, {12002, 13446}, {14643, 15107}, {21660, 22051}

= midpoint of X(i) and X(j) for these {i,j}: {5, 23}, {468, 16619}, {550, 18325}, {1533, 12041}, {2070, 11563}, {7575, 11799}

= reflection of X(i) in X(j) for these (i,j): (3, 22249), (140, 468), (548, 18571), (858, 3628), (18323, 3861), (18572, 3850)

= orthoptic circle of Steiner inellipse-inverse-of X(7533)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 18282, 140), (23, 186, 2937), (140, 10096, 468), (186, 18325, 550), (403, 18572, 3850), (1656, 5899, 5189), (2070, 18378, 23), (7426, 11799, 7575), (7495, 7574, 15122), (7545, 7552, 5)

= [ -1.2503473881203710, -2.1165688630372750, 5.6830647969888210 ]

 

César Lozada