[Antreas P. Hatzipolakis]:
Let ABC be a triangle and P a point.
Denote:
Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.
MaMbMc = the midway triangle of P.
(ie Ma, Mb, Mc = the midpoints of AP, BP, CP, resp.)
N1, N2, N3 = the midpoints of NaMa, NbMb, NcMc, resp.
1. P = N
The NPC center of N1N2N3 lies on the Euler line of ABC
2. P = O
The Circumcenter of N1N2N3 lies on the Euler line of ABC.
The centroid of N1N2N3 lies on the Euler line of ABC.
Conjecture:
Let P be a point on the Euler line of ABC.
The P-point of N1N2N3 ( = same as the P point on the Euler line of N1N2N3) lies on the Euler line of ABC.
[César Lozada]:
> Conjecture:
> Let P be a point on the Euler line of ABC.lies on the Euler line of ABC.
> The P-point of N1N2N3 ( = same as the P point on the Euler line of N1N2N3)
Not for all P on the Euler line. It is true for P satisfying OP/OH=t= a constant invariant of (a,b,c). I published a list of such points In the preamble just before X(14226):
The appearance of (i, t) in the following list (complete up to I<=14213) means that X(I), on the Euler line, satisfies OX(i)/OH = t :
(2, 1/3), (3, 0), (4, 1), (5, 1/2), (20, -1), (140, 1/4), (376, -1/3), (381, 2/3), (382, 2), (546, 3/4), (547, 5/12), (548, -1/4), (549, 1/6), (550, -1/2), (631, 1/5), (632, 3/10), (1656, 2/5), (1657, -2), (2041, 2+sqrt(3)), (2042, 2-sqrt(3)), (2043, -sqrt(3)/3), (2044, sqrt(3)/3), (2045, (4-sqrt(3))/13), (2046, (4+sqrt(3))/13), (2675, (-15+24*sqrt(5))/59), (2676, (75-12*sqrt(5))/109), (3090, 3/7), (3091, 3/5), (3146, 3), (3522, -1/5), (3523, 1/7), (3524, 1/9), (3525, 3/11), (3526, 2/7), (3528, -1/7), (3529, -3), (3530, 1/8), (3533, 5/17), (3534, -2/3), (3543, 5/3), (3544, 9/17), (3545, 5/9), (3627, 3/2), (3628, 3/8), (3830, 4/3), (3832, 5/7), (3839, 7/9), (3843, 4/5), (3845, 5/6), (3850, 5/8), (3851, 4/7), (3853, 5/4), (3854, 11/17), (3855, 7/11), (3856, 11/16), (3857, 9/14), (3858, 7/10), (3859, 13/20), (3860, 17/24), (3861, 7/8), (5054, 2/9), (5055, 4/9), (5056, 5/11), (5059, -5), (5066, 7/12), (5067, 5/13), (5068, 7/13), (5070, 4/11), (5071, 7/15), (5072, 6/11), (5073, 4), (5076, 6/5), (5079, 6/13), (7486, 7/17), (8703, -1/6), (10109, 11/24), (10124, 7/24), (10299, 1/13), (10303, 3/13), (10304, -1/9), (11001, -5/3), (11539, 5/18), (11540, 13/48), (11541, 9), (11737, 13/24), (11812, 5/24), (12100, 1/12), (12101, 13/12), (12102, 9/8), (12103, -3/4), (12108, 3/16), (12811, 9/16), (12812, 9/20), (14093, -2/15)
Pairs (i,t) added on January 23, 2018, (for i<=15932):
(14269, 8/9), (14782, 3/7-1/7*sqrt(2)), (14783, 3/7+1/7*sqrt(2)), (14784, sqrt(2)-1), (14785, -1-sqrt(2)), (14813, -1/2+1/2*sqrt(3)), (14814, -1/2*sqrt(3)-1/2), (14869, 3/14), (14890, 17/72), (14891, 1/24), (14892, 19/36), (14893, 11/12), (15022, 9/19), (15640, 13/3), (15681, -4/3), (15682, 7/3), (15683, -7/3), (15684, 8/3), (15685, -8/3), (15686, -5/6), (15687, 7/6), (15688, -2/9), (15689, -4/9), (15690, -5/12), (15691, -7/12), (15692, 1/15), (15693, 2/15), (15694, 4/15), (15695, -4/15), (15696, -2/5), (15697, -7/15), (15698, 1/21), (15699, 7/18), (15700, 2/21), (15701, 4/21), (15702, 5/21), (15703, 8/21), (15704, -3/2), (15705, 1/27), (15706, 2/27), (15707, 4/27), (15708, 5/27), (15709, 7/27), (15710, -1/27), (15711, 1/30), (15712, 1/10), (15713, 7/30), (15714, -1/30), (15715, 1/33), (15716, 2/33), (15717, 1/11), (15718, 4/33), (15719, 5/33), (15720, 2/11), (15721, 7/33), (15722, 8/51), (15723, 10/33), (15759, -1/24), (15764, 1/6*sqrt(3)-1/6), (15765, 1/2-1/6*sqrt(3))
For example, P=X(21) is not in the list because for P=X(21), OP/OH=R/(3*R+2*r), whose value depends upon the triangle.
For P such that OP/OH=t=invariant constant, the P-point of N1N2N3=P’ has a long expression (or maybe I could not simplify it):
P’ = 3*(2*a^4-(b^2+c^2)*a^2-(b^2-c^ 2)^2)*(a^6-(b^2+c^2)*a^4-(b^4- 3*b^2*c^2+c^4)*a^2+(b^4-c^4)*( b^2-c^2))^2*a^6*b^6*c^6/(b^3- c*b^2-a*b^2-a*b*c-b*a^2-b*c^2- a*c^2-c*a^2+a^3+c^3)^6+4*(2*a^ 4-(b^2+c^2)*a^2-(b^2-c^2)^2)*( a^6-(b^2+c^2)*a^4-(b^4-3*b^2* c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^ 2))^2*a^5*b^5*c^5/(b^3-c*b^2- a*b^2-a*b*c-b*a^2-b*c^2-a*c^2- c*a^2+a^3+c^3)^5+(2*a^4-(b^2+ c^2)*a^2-(b^2-c^2)^2)*(5*a^12- 8*(b^2+c^2)*a^10-(13*b^4-38*b^ 2*c^2+13*c^4)*a^8+2*(b^2+c^2)* (16*b^4-31*b^2*c^2+16*c^4)*a^ 6-(13*b^8+13*c^8+3*b^2*c^2*( 10*b^4-27*b^2*c^2+10*c^4))*a^ 4-2*(b^4-c^4)*(b^2-c^2)*(4*b^ 4-15*b^2*c^2+4*c^4)*a^2+(5*b^ 4+12*b^2*c^2+5*c^4)*(b^2-c^2)^ 4)*a^4*b^4*c^4/(b^3-c*b^2-a*b^ 2-a*b*c-b*a^2-b*c^2-a*c^2-c*a^ 2+a^3+c^3)^4+(2*a^4-(b^2+c^2)* a^2-(b^2-c^2)^2)*(4*a^12-5*(b^ 2+c^2)*a^10-20*(b^2-c^2)^2*a^ 8+(b^2+c^2)*(50*b^4-93*b^2*c^ 2+50*c^4)*a^6-(40*b^8+40*c^8+ b^2*c^2*(11*b^4-94*b^2*c^2+11* c^4))*a^4+(b^4-c^4)*(b^2-c^2)* (11*b^4+27*b^2*c^2+11*c^4)*a^ 2+3*(b^2-c^2)^4*b^2*c^2)*a^3* b^3*c^3/(b^3-c*b^2-a*b^2-a*b* c-b*a^2-b*c^2-a*c^2-c*a^2+a^3+ c^3)^3+(2*a^16+(b^2+c^2)*a^14- (35*b^4-36*b^2*c^2+35*c^4)*a^ 12+(b^2+c^2)*(89*b^4-149*b^2* c^2+89*c^4)*a^10-(95*b^8+95*c^ 8+2*b^2*c^2*(21*b^4-124*b^2*c^ 2+21*c^4))*a^8+(b^2+c^2)*(43* b^8+43*c^8+3*b^2*c^2*(24*b^4- 77*b^2*c^2+24*c^4))*a^6-(b^2- c^2)^2*(b^8+c^8+b^2*c^2*(56*b^ 4+115*b^2*c^2+56*c^4))*a^4-(b^ 4-c^4)*(b^2-c^2)^3*(5*b^4+7*b^ 2*c^2+5*c^4)*a^2+(b^2+c^2)^2*( b^2-c^2)^6)*a^2*b^2*c^2/(b^3- c*b^2-a*b^2-a*b*c-b*a^2-b*c^2- a*c^2-c*a^2+a^3+c^3)^2-(-2*(b^ 2+c^2)*a^14+(11*b^4-6*b^2*c^2+ 11*c^4)*a^12-(b^2+c^2)*(25*b^ 4-42*b^2*c^2+25*c^4)*a^10+(30* b^8+30*c^8+b^2*c^2*(5*b^4-74* b^2*c^2+5*c^4))*a^8-2*(b^2+c^ 2)*(10*b^8+10*c^8+b^2*c^2*(3* b^4-28*b^2*c^2+3*c^4))*a^6+(b^ 4-c^4)^2*(7*b^4+10*b^2*c^2+7* c^4)*a^4-(b^8-c^8)*a^2*(b^2-c^ 2)^3-(b^2-c^2)^6*b^2*c^2)*a*b* c/(b^3-c*b^2-a*b^2-a*b*c-b*a^ 2-b*c^2-a*c^2-c*a^2+a^3+c^3)+ a^2*b^2*c^2*(2*a^10-5*(b^2+c^ 2)*a^8+2*(b^4+5*b^2*c^2+c^4)* a^6+(b^2+c^2)*(2*b^2+b*c-2*c^ 2)*(2*b^2-b*c-2*c^2)*a^4-(b^2- c^2)^2*(4*b^4+5*b^2*c^2+4*c^4) *a^2+(b^4-c^4)*(b^2-c^2)^3) : : (barys)
ETC pairs (P,P’): (2,547), (3,5498), (4,546), (5,13469)
Examples:
P’(X(20)) = midpoint of X(7568) and X(14010)
= 5*S^4+(16*R^2*(48*R^2-19*SW)- 7*SB*SC+28*SW^2)*S^2-4*(4*R^2- SW)*(80*R^2-13*SW)*SB*SC : : (barys)
= on line {2,3}
= midpoint of X(7568) and X(14010)
= [ 11.4735776557899400, 10.5737901453322900, -8.9751491521487150 ]
P’(X(550)) = reflection of X(15078) in X(4220)
= 288*S^4+(125*R^2*(259*R^2-92* SW)-288*SB*SC+860*SW^2)*S^2-5* (5*R^2*(1325*R^2-484*SW)+196* SW^2)*SB*SC : : (barys)
= on line {2, 3}
= reflection of X(15078) in X(4220)
= [ 6.7832418350278460, 5.8958275488947970, -3.5717892834941020 ]
César Lozada
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