HYACINTHOS 26514

[Le Viet An]:

Let ABC be a triangle and P a point.

Denote:

Lab, Lac = the parallels through P to AB, AC, resp.

Ab, Ac = the orthogonal projections of A on Lab, Lac, resp.

Ea = the Euler line of AAbAc.
Similarly Eb, Ec.

If P lies on the Neuberg cubic then Ea, Eb, Ec are concurrent.

Which is the point of concurrence for P = X(14), X(15), X(16), X(30), X(74)... ?
And in general, which is the locus of the point of concurrence?

 

[César Lozada]:

Locus: A graphic sketch shows that locus of Q(P) (=point of concurrence)  has not a simple form.

 

ETC pais (P,Q(P) ): (1,1), (3,140), (4,3574), (30,30), (74,125)

 

Q( X(13) )  = midpoint of X(13) and X(61)

= sqrt(3)*(2*(SA-2*SW)*S^2+(SB+ SC)*SA*SW)-S*(2*S^2-3*SW*SA+3* SW^2) : : (barycentrics)

= On lines: {4, 13}, {6, 5617}, {17, 299}, {30, 10613}, {62, 618}, {114, 9300}, {115, 6783}, {396, 511}, {397, 8259}, {530, 11299}, {533, 5459}, {5613, 6034}, {6772, 13103}, {11129, 11289}

= midpoint of X(13) and X(61)

= reflection of X(i) in X(j) for these (i,j): (618, 6694), (635, 6669)

= {X(6), X(6115)}-Harmonic conjugate of X(6782)

= [ 0.998826716332080, 1.13614291648387, 2.393107055265354 ]

 

Q( X(14) )  = midpoint of X(14) and X(62)

= sqrt(3)*(2*(SA-2*SW)*S^2+(SB+ SC)*SA*SW)+S*(2*S^2-3*SW*SA+3* SW^2) : : (barycentrics)

= On lines: {4, 14}, {6, 5613}, {18, 298}, {30, 10614}, {61, 619}, {114, 9300}, {115, 6782}, {395, 511}, {398, 8260}, {531, 11300}, {532, 5460}, {5617, 6034}, {6775, 13102}, {11128, 11290}

= midpoint of X(14) and X(62)

= reflection of X(i) in X(j) for these (i,j): (619, 6695), (636, 6670)

= {X(6), X(6114)}-Harmonic conjugate of X(6783)

= [ 1.999365130350214, 3.57404492294152, 0.243541782786305 ]

 

Q( X(15) )  = midpoint of X(15) and X(17)

= (2*S^2+(SW+2*sqrt(3)*S)*(SB+ SC))*(SA+sqrt(3)*S) : : (barycentrics)

=  On lines: {4, 15}, {30, 10611}, {61, 302}, {114, 6109}, {193, 627}, {396, 13350}, {511, 8259}, {532, 5463}, {623, 6673}

= midpoint of X(15) and X(17)

= reflection of X(i) in X(j) for these (i,j): (623, 6673), (629, 6671)

= [ 2.244549285362310, 2.15401147881961, 1.113479941788197 ]

 

Q( X(16) )  = midpoint of X(16) and X(18)

= (2*S^2+(SW-2*sqrt(3)*S)*(SB+ SC))*(SA-sqrt(3)*S) : : (barycentrics)

=  On lines: {4, 16}, {30, 10612}, {62, 303}, {114, 6108}, {193, 628}, {395, 13349}, {511, 8260}, {533, 5464}, {624, 6674}

= midpoint of X(16) and X(18)

= reflection of X(i) in X(j) for these (i,j): (624, 6674), (630, 6672)

= [ -11.532442746808720, 4.58413798465905, 5.789696375670599 ]

 

César Lozada