[Antreas P. Hatzipolakis]:
Let ABC be a triangle, P a point and A'B'C', A"B"C" the pedal, antipedal triangles of P, resp.
Denote:
Ab, Ac = the orthogonal projections of A' on AB, AC, resp.
[Peter Moses]:
Hi Antreas,
CC + Inf + Deg 5.
For P = O: At
Let ABC be a triangle, P a point and A'B'C', A"B"C" the pedal, antipedal triangles of P, resp.
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.
Ma, Mb, Mc = the midpoints of AbAc, BcBa, CaCb, resp.
Which is the locus of P such that A"B"C", MaMbMc are perspective?
Ca, Cb = the orthogonal projections of C' on CA, CB, resp.
Ma, Mb, Mc = the midpoints of AbAc, BcBa, CaCb, resp.
Which is the locus of P such that A"B"C", MaMbMc are perspective?
[Peter Moses]:
Hi Antreas,
CC + Inf + Deg 5.
For P = O: At
a^2 (a^8-4 a^6 b^2+6 a^4 b^4-4 a^2 b^6+b^8-4 a^6 c^2-4 a^4 b^2 c^2+16 a^2 b^4 c^2-8 b^6 c^2+6 a^4 c^4+16 a^2 b^2 c^4+14 b^4 c^4-4 a^2 c^6-8 b^2 c^6+c^8)::
on lines {{1,6883},{2,155},{3,51},{5,1498},{6,140},{22,15024},{24,2918},{25,13336},{26,2916},{30,9815},{52,7484},{125,399},{143,1350},{159,182},{195,394},{371,8943},{372,8939},{373,7529},{389,7393},{569,9937},{631,5422},{632,12161},{1147,5050},{1154,13154},{1199,3533},{1216,11432},{1351,5447},{1511,14528},{1899,7405},{1993,3525},{1995,11465},{2917,6644},{2929,11425},{2930,15462},{2931,15035},{2935,7526},{2948,15016},{3066,7517},{3357,9729},{3517,12017},{3523,15018},{3546,3618},{3567,7485},{3796,7506},{3800,10279},{5020,14530},{5054,15047},{5056,11456},{5067,11441},{5198,14845},{5544,11484},{5609,14924},{5640,10323},{5643,12082},{5898,15089},{5943,7387},{5946,7516},{6688,6759},{6815,12293},{7395,9730},{7399,14852},{7503,15045},{7509,7691},{7514,9786},{8053,11248},{9777,10625},{9833,10127},{10110,13347},{10170,12164},{10539,11284},{10541,12106},{10574,15054},{10594,11451},{10627,11477},{11403,14855},{11472,11479}}.
complement X(11487).
midpoint of X(3) and X(3527).
X[3527] - 3 X[5644], X[3] + 3 X[5644], X[11387] - 13 X[15028].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5644, 3527), (182, 11695, 6642), (373, 10984, 7529), (1199, 3533, 15066), (7395, 9730, 12163), (7514, 12006, 9786), (9827, 10610, 9935).
on the cubic K832.
on the Feuerbach hyperbola of of the tangential triangle.
X(i)-Ceva conjugate of X(j) for these (i,j): {{631, 3}, {5422, 6}}.
crosssum of X(523) and X(5522).
For P = H at X(11433)
Best regards,
Peter Moses.
on lines {{1,6883},{2,155},{3,51},{5,1498},{6,140},{22,15024},{24,2918},{25,13336},{26,2916},{30,9815},{52,7484},{125,399},{143,1350},{159,182},{195,394},{371,8943},{372,8939},{373,7529},{389,7393},{569,9937},{631,5422},{632,12161},{1147,5050},{1154,13154},{1199,3533},{1216,11432},{1351,5447},{1511,14528},{1899,7405},{1993,3525},{1995,11465},{2917,6644},{2929,11425},{2930,15462},{2931,15035},{2935,7526},{2948,15016},{3066,7517},{3357,9729},{3517,12017},{3523,15018},{3546,3618},{3567,7485},{3796,7506},{3800,10279},{5020,14530},{5054,15047},{5056,11456},{5067,11441},{5198,14845},{5544,11484},{5609,14924},{5640,10323},{5643,12082},{5898,15089},{5943,7387},{5946,7516},{6688,6759},{6815,12293},{7395,9730},{7399,14852},{7503,15045},{7509,7691},{7514,9786},{8053,11248},{9777,10625},{9833,10127},{10110,13347},{10170,12164},{10539,11284},{10541,12106},{10574,15054},{10594,11451},{10627,11477},{11403,14855},{11472,11479}}.
complement X(11487).
midpoint of X(3) and X(3527).
X[3527] - 3 X[5644], X[3] + 3 X[5644], X[11387] - 13 X[15028].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5644, 3527), (182, 11695, 6642), (373, 10984, 7529), (1199, 3533, 15066), (7395, 9730, 12163), (7514, 12006, 9786), (9827, 10610, 9935).
on the cubic K832.
on the Feuerbach hyperbola of of the tangential triangle.
X(i)-Ceva conjugate of X(j) for these (i,j): {{631, 3}, {5422, 6}}.
crosssum of X(523) and X(5522).
For P = H at X(11433)
Best regards,
Peter Moses.
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