Denote:
Ab, Ac = the orthogonal projections of A' on AB, AC, resp.
Bc, Ba = the orthogonal projections of B' on BC, BA, resp.
Ca, Cb = the orthogonal projections of C' on CA, CB, resp.
N1, N2 = the NPC centers of AbBcCa, AcCbBa, resp. (bicentric points)
The line N1N2 is perpendicular to Euler line of ABC.
Conjecture:
Let P1, P2 be same points on the Euler lines of AbBcCa, AcCbBa, resp.
The line P1P2 is perpendicular to Euler line of ABC
Conjecture proved.
If P is such that OP=t*OH, them the line P1P2 is perpendicular to Euler line of ABC at:
Zp = 2*S^4-((8*t-2)*SW^2+((-68*t+ 12)*R^2+3*SA)*SW+144*R^4*t-3* SA^2)*S^2+((12*t-3)*SW^2-34*( 3*t-1)*R^2*SW+72*(3*t-1)*R^4)* (SB+SC)*SA : : (barys)
Examples:
Zp(X(3)) = EULER LINE INTERCEPT OF X(389)X(523)
= 2*S^4-(-2*SW^2+(12*R^2+3*SA)* SW-3*SA^2)*S^2+(-3*SW^2+34*R^ 2*SW-72*R^4)*(SB+SC)*SA : : (barys)
= On lines: {2, 3}, {389, 523}, {2452, 11432}, {2453, 9786}, {3258, 3574}, {10095, 12052}
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (381, 10124, 7483), (6893, 11548, 3560), (7394, 8226, 11818), (7450, 13163, 4240)
= [ 3.462941932118642, 2.58428668096595, 0.253262041799351 ]
Zp(X(4)) = EULER LINE INTERCEPT OF X(523)X(12241)
= 2*S^4-(6*SW^2+(-56*R^2+3*SA)* SW+144*R^4-3*SA^2)*S^2+(144*R^ 4-68*R^2*SW+9*SW^2)*(SB+SC)*SA :: (barys)
= On lines: {2, 3}, {523, 12241}, {5462, 12052}
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1316, 7491, 452), (6936, 11292, 6940)
= [ -0.897767607407853, -1.76491919668880, 5.276885898418712 ]
Zp(X(5)) = 2*S^4-(2*SW^2+(-22*R^2+3*SA)* SW+72*R^4-3*SA^2)*S^2+(36*R^4- 17*R^2*SW+3*SW^2)*(SB+SC)*SA : : (barys)
= On lines: {2, 3}
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1314, 6840, 1981), (4199, 14093, 426), (4208, 7412, 1583), (4224, 11288, 3523), (6846, 7532, 7562), (6921, 11335, 471), (7401, 11007, 13747), (8368, 12057, 4204), (10128, 10565, 2676)
= [ 1.282587162355394, 0.40968374213857, 2.765073970109032 ]
César Lozada
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου