[Antreas P. Hatzipolakis]:
Let ABC be a triangle and A'B'C' cevian triangle of H
Denote:
Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.
M1, M2, M3 = the midpoints of AH, BH, CH, resp.
ma, mb, mc = the midpoints of MaM1, MbM2, McM3, resp.
mab, mac = the orthogonal projections of ma on BB', CC', resp
mbc, mba = the orthogonal projections of mb on CC', AA', resp.
mca, mcb = the orthogonal projections of mc on AA', BB', resp.
La, Lb, Lc = the Euler lines of mamabmac, mbmbcmba, mcmcamcb, resp.
La, Lb, Lc are concurrent.
[Peter Moses]:
Hi Antreas,
At:
a^2 (a^12 b^2-4 a^10 b^4+5 a^8 b^6-5 a^4 b^10+4 a^2 b^12-b^14+a^12 c^2-8 a^10 b^2 c^2+8 a^8 b^4 c^2+2 a^6 b^6 c^2+11 a^4 b^8 c^2-26 a^2 b^10 c^2+12 b^12 c^2-4 a^10 c^4+8 a^8 b^2 c^4-16 a^6 b^4 c^4-6 a^4 b^6 c^4+48 a^2 b^8 c^4-30 b^10 c^4+5 a^8 c^6+2 a^6 b^2 c^6-6 a^4 b^4 c^6-52 a^2 b^6 c^6+19 b^8 c^6+11 a^4 b^2 c^8+48 a^2 b^4 c^8+19 b^6 c^8-5 a^4 c^10-26 a^2 b^2 c^10-30 b^4 c^10+4 a^2 c^12+12 b^2 c^12-c^14)::
Denote:
Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.
M1, M2, M3 = the midpoints of AH, BH, CH, resp.
ma, mb, mc = the midpoints of MaM1, MbM2, McM3, resp.
mab, mac = the orthogonal projections of ma on BB', CC', resp
mbc, mba = the orthogonal projections of mb on CC', AA', resp.
mca, mcb = the orthogonal projections of mc on AA', BB', resp.
La, Lb, Lc = the Euler lines of mamabmac, mbmbcmba, mcmcamcb, resp.
La, Lb, Lc are concurrent.
[Peter Moses]:
Hi Antreas,
At:
a^2 (a^12 b^2-4 a^10 b^4+5 a^8 b^6-5 a^4 b^10+4 a^2 b^12-b^14+a^12 c^2-8 a^10 b^2 c^2+8 a^8 b^4 c^2+2 a^6 b^6 c^2+11 a^4 b^8 c^2-26 a^2 b^10 c^2+12 b^12 c^2-4 a^10 c^4+8 a^8 b^2 c^4-16 a^6 b^4 c^4-6 a^4 b^6 c^4+48 a^2 b^8 c^4-30 b^10 c^4+5 a^8 c^6+2 a^6 b^2 c^6-6 a^4 b^4 c^6-52 a^2 b^6 c^6+19 b^8 c^6+11 a^4 b^2 c^8+48 a^2 b^4 c^8+19 b^6 c^8-5 a^4 c^10-26 a^2 b^2 c^10-30 b^4 c^10+4 a^2 c^12+12 b^2 c^12-c^14)::
on lines {{4,10293},{5,5181},{6,5609},{51,14448},{74,11403},{110,11426},{125,1595},{265,7528},{389,546},{974,11381},{1112,7507},{1192,12041},{1511,11425},{1598,5622},{2777,11566},{2781,10110},{3567,13148},{5640,15044},{5876,12236},{6241,12133},{6243,12358},{6677,15115},{6723,13348},{9815,9826},{10625,13416},{10733,15028},{11432,14094}}.
midpoint of X(i) and X(j) for these {i,j}: {{7687, 11746}, {9826, 10113}}.
X[389] + 3 X[7687], 5 X[1112] - X[7731], 3 X[1112] - 7 X[9781], 3 X[974] + X[11381], X[389] - 3 X[11746], X[4] + 3 X[12099], X[6241] + 3 X[12133], X[5876] + 3 X[12236], X[6243] + 3 X[12358], 5 X[3567] - X[13148], 3 X[6723] - X[13348], X[10625] - 3 X[13416], 9 X[51] - X[14448], X[1112] + 3 X[14644], 7 X[9781] + 9 X[14644], X[7731] + 15 X[14644], 3 X[9826] - 5 X[15026], 3 X[10113] + 5 X[15026], 3 X[10733] + 13 X[15028], 9 X[5640] + 7 X[15044].
Best regards,
Peter Moses.
midpoint of X(i) and X(j) for these {i,j}: {{7687, 11746}, {9826, 10113}}.
X[389] + 3 X[7687], 5 X[1112] - X[7731], 3 X[1112] - 7 X[9781], 3 X[974] + X[11381], X[389] - 3 X[11746], X[4] + 3 X[12099], X[6241] + 3 X[12133], X[5876] + 3 X[12236], X[6243] + 3 X[12358], 5 X[3567] - X[13148], 3 X[6723] - X[13348], X[10625] - 3 X[13416], 9 X[51] - X[14448], X[1112] + 3 X[14644], 7 X[9781] + 9 X[14644], X[7731] + 15 X[14644], 3 X[9826] - 5 X[15026], 3 X[10113] + 5 X[15026], 3 X[10733] + 13 X[15028], 9 X[5640] + 7 X[15044].
Best regards,
Peter Moses.
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου