HYACINTHOS 25141

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the cevian triangle of G.

Denote:

Nab, Nac = the NPC centers of GAB', GAC', resp.
Nbc, Nba = the NPC centers of GBC', GBA', resp.
Nca, Ncb = the NPC centers of GCA', GCB', resp.

Ma, Mb, Mc = the midpoints of NbcNcb, NcaNac, NabNba, resp.

M1, M2, M3 = the midpoints of NabNac, NbcNba, NcaNcb, resp.

Mi, Mii, Miii = the midpoints of NbaNca, NcbNab, NacNbc, resp.

 

ma, mb, mc = the medians of MaMbMc

m1, m2, m3 = the medians of M1M2M3

mi, mii, miii = the medians of MiMiiMiii

1. The parallels to ma, mb, mc through A, B, C are concurrent at a point on the circumcircle.

2. The parallels to m1, m2, m3 through A, B, C are concurrent

3. The parallels to mi, mii, miii through A, B, C are concurrent.

[César Lozada]:

1)      Z1=X(98)

 

2)      Z2 = 1/(3*a^4-7*(b^2+c^2)*a^2+4*b^ 4-6*b^2*c^2+4*c^4) : :  (barycentrics)

= On Kiepert hyperbola and these lines: {76,632}, {83,5070}, {114,8587}, {262,3054}, {547,598}, {671,5054}

= [ 7.336689063226488, 0.27204824918920, 0.066159203287322 ]

 

3)      Z3 = 1/(3*a^4-5*(b^2+c^2)*a^2+2*b^ 4-6*b^2*c^2+2*c^4) : :  (barycentrics)

= On Kiepert hyperbola and these lines: {2,5097}, {4,7603}, {76,3628}, {83,3526}, {262,3055}, {549,598}, {671,5055}, {1916,6721}, {2996,7486}, {3815,7607}, {6036,8587}, {9771,11167}

= [ 1.345482695685709, 1.18464523467115, 2.199533459895563 ]

 

César Lozada