[Antreas P. Hatzipolakis]:
ORTHOGONAL PROJECTIONS:
Let ABC be a triangle.
Denote:
A', B', C' = the orthogonal ptojections of I on BC, CA, AB, resp. (ie A'B'C' = the pedal triangle of I)
MaMbMc = the medial triangle of A'B'C'
M1, M2, M3 = the midpoints of AA', BB', CC', resp.
Denote:
A', B', C' = the orthogonal ptojections of I on BC, CA, AB, resp. (ie A'B'C' = the pedal triangle of I)
MaMbMc = the medial triangle of A'B'C'
M1, M2, M3 = the midpoints of AA', BB', CC', resp.
1. The triangles M1M2M3, MaMbMc are cyclologic
2. The Euler lines of M1MbMc, M2McMa, M3MaMb are concurrent at the NPC center of A'B'C'
REFLECTIONS:
Let ABC be a triangle.
Denote:
A', B', C' = the reflections of I in BC, CA, AB, resp.
MaMbMc = the medial triangle of A'B'C'
M1, M2, M3 = the midpoints of AA', BB', CC', resp.
1. The triangles M1M2M3, MaMbMc are circumcyclologic
ie the circumcircles of M1MbMc, M2McMa, M3MaMb and M1M2M3 are concurrent
the circumcircles of MaM2M3, MbM3M1, McM1M2 and MaMbMc are concurrent.
2. The Euler lines of M1MbMc, M2McMa, M3MaMb are concurrent at the NPC center of A'B'C'
[Peter Moses]:
Hi Antreas,
Orthogonal projections.
1).
(M1M2M3, MaMbMc): X(1387).
(MaMbMc, M1M2M3):
a (a^8 b-2 a^7 b^2-2 a^6 b^3+6 a^5 b^4-6 a^3 b^6+2 a^2 b^7+2 a b^8-b^9+a^8 c+a^6 b^2 c-6 a^5 b^3 c-3 a^4 b^4 c+12 a^3 b^5 c-a^2 b^6 c-6 a b^7 c+2 b^8 c-2 a^7 c^2+a^6 b c^2+8 a^5 b^2 c^2+a^4 b^3 c^2-8 a^3 b^4 c^2-3 a^2 b^5 c^2+2 a b^6 c^2+b^7 c^2-2 a^6 c^3-6 a^5 b c^3+a^4 b^2 c^3+4 a^3 b^3 c^3+2 a^2 b^4 c^3+6 a b^5 c^3-5 b^6 c^3+6 a^5 c^4-3 a^4 b c^4-8 a^3 b^2 c^4+2 a^2 b^3 c^4-8 a b^4 c^4+3 b^5 c^4+12 a^3 b c^5-3 a^2 b^2 c^5+6 a b^3 c^5+3 b^4 c^5-6 a^3 c^6-a^2 b c^6+2 a b^2 c^6-5 b^3 c^6+2 a^2 c^7-6 a b c^7+b^2 c^7+2 a c^8+2 b c^8-c^9)::
on lines {{1,104},{2,12665},{11,1071},{40,13278},{80,12115},{116,119},{214,10269},{354,1537},{515,12736},{912,6713},{942,2829},{952,5836},{1387,6001},{1479,5553},{2077,3218},{2771,11281},{2802,5882},{3035,9940},{3873,12703},{3874,11248},{5554,9803},{5722,12761},{5768,6246},{5777,6667},{5840,13369},{6264,11919},{6684,14740},{6884,9964},{8096,13267},{8104,12685},{10679,12515},{10724,11220},{10805,12247},{10942,12619},{11219,12691},{11544,13374}}.
complement X(12665).
midpoint of X(i) and X(j) for these {i,j}: {{11, 1071}, {104, 11570}, {5884, 11715}, {9803, 12757}}.
reflection of X(i) in X(j) for these {i,j}: {{3035, 9940}, {5083, 12005}, {5777, 6667}, {11729, 13373}, {14740, 6684}}.
3 X[354] - X[1537], X[119] - 3 X[10202], X[10724] + 3 X[11220], 3 X[11219] - X[12691], X[12751] - 5 X[15016].
incircle inverse of X(1795).
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1768, 12775), (10805, 12247, 12749).
X(521)-gimel conjugate of X(12775).
2). X(942).
Reflections.
1).
(M1M2M3, MaMbMc):
a^9 b-a^8 b^2-3 a^7 b^3+2 a^6 b^4+3 a^5 b^5-a^3 b^7-2 a^2 b^8+b^10+a^9 c-2 a^8 b c+2 a^7 b^2 c+2 a^6 b^3 c-5 a^5 b^4 c+a^4 b^5 c+2 a b^8 c-b^9 c-a^8 c^2+2 a^7 b c^2-4 a^6 b^2 c^2+3 a^5 b^3 c^2+a^4 b^4 c^2-3 a^3 b^5 c^2+7 a^2 b^6 c^2-2 a b^7 c^2-3 b^8 c^2-3 a^7 c^3+2 a^6 b c^3+3 a^5 b^2 c^3-4 a^4 b^3 c^3+4 a^3 b^4 c^3-6 a b^6 c^3+4 b^7 c^3+2 a^6 c^4-5 a^5 b c^4+a^4 b^2 c^4+4 a^3 b^3 c^4-10 a^2 b^4 c^4+6 a b^5 c^4+2 b^6 c^4+3 a^5 c^5+a^4 b c^5-3 a^3 b^2 c^5+6 a b^4 c^5-6 b^5 c^5+7 a^2 b^2 c^6-6 a b^3 c^6+2 b^4 c^6-a^3 c^7-2 a b^2 c^7+4 b^3 c^7-2 a^2 c^8+2 a b c^8-3 b^2 c^8-b c^9+c^10::
complement X(12665).
midpoint of X(i) and X(j) for these {i,j}: {{11, 1071}, {104, 11570}, {5884, 11715}, {9803, 12757}}.
reflection of X(i) in X(j) for these {i,j}: {{3035, 9940}, {5083, 12005}, {5777, 6667}, {11729, 13373}, {14740, 6684}}.
3 X[354] - X[1537], X[119] - 3 X[10202], X[10724] + 3 X[11220], 3 X[11219] - X[12691], X[12751] - 5 X[15016].
incircle inverse of X(1795).
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1768, 12775), (10805, 12247, 12749).
X(521)-gimel conjugate of X(12775).
2). X(942).
Reflections.
1).
(M1M2M3, MaMbMc):
a^9 b-a^8 b^2-3 a^7 b^3+2 a^6 b^4+3 a^5 b^5-a^3 b^7-2 a^2 b^8+b^10+a^9 c-2 a^8 b c+2 a^7 b^2 c+2 a^6 b^3 c-5 a^5 b^4 c+a^4 b^5 c+2 a b^8 c-b^9 c-a^8 c^2+2 a^7 b c^2-4 a^6 b^2 c^2+3 a^5 b^3 c^2+a^4 b^4 c^2-3 a^3 b^5 c^2+7 a^2 b^6 c^2-2 a b^7 c^2-3 b^8 c^2-3 a^7 c^3+2 a^6 b c^3+3 a^5 b^2 c^3-4 a^4 b^3 c^3+4 a^3 b^4 c^3-6 a b^6 c^3+4 b^7 c^3+2 a^6 c^4-5 a^5 b c^4+a^4 b^2 c^4+4 a^3 b^3 c^4-10 a^2 b^4 c^4+6 a b^5 c^4+2 b^6 c^4+3 a^5 c^5+a^4 b c^5-3 a^3 b^2 c^5+6 a b^4 c^5-6 b^5 c^5+7 a^2 b^2 c^6-6 a b^3 c^6+2 b^4 c^6-a^3 c^7-2 a b^2 c^7+4 b^3 c^7-2 a^2 c^8+2 a b c^8-3 b^2 c^8-b c^9+c^10::
on lines {{225,403},{946,1319}}.
(MaMbMc, M1M2M3):
(MaMbMc, M1M2M3):
a (a^11 b-2 a^10 b^2-2 a^9 b^3+7 a^8 b^4-2 a^7 b^5-8 a^6 b^6+8 a^5 b^7+2 a^4 b^8-7 a^3 b^9+2 a^2 b^10+2 a b^11-b^12+a^11 c-2 a^10 b c+2 a^9 b^2 c-a^8 b^3 c-6 a^7 b^4 c+13 a^6 b^5 c-4 a^5 b^6 c-13 a^4 b^7 c+13 a^3 b^8 c+a^2 b^9 c-6 a b^10 c+2 b^11 c-2 a^10 c^2+2 a^9 b c^2+6 a^7 b^3 c^2-4 a^6 b^4 c^2-16 a^5 b^5 c^2+18 a^4 b^6 c^2+6 a^3 b^7 c^2-14 a^2 b^8 c^2+2 a b^9 c^2+2 b^10 c^2-2 a^9 c^3-a^8 b c^3+6 a^7 b^2 c^3-10 a^6 b^3 c^3+13 a^5 b^4 c^3+6 a^4 b^5 c^3-27 a^3 b^6 c^3+11 a^2 b^7 c^3+10 a b^8 c^3-6 b^9 c^3+7 a^8 c^4-6 a^7 b c^4-4 a^6 b^2 c^4+13 a^5 b^3 c^4-24 a^4 b^4 c^4+15 a^3 b^5 c^4+12 a^2 b^6 c^4-12 a b^7 c^4+b^8 c^4-2 a^7 c^5+13 a^6 b c^5-16 a^5 b^2 c^5+6 a^4 b^3 c^5+15 a^3 b^4 c^5-24 a^2 b^5 c^5+4 a b^6 c^5+4 b^7 c^5-8 a^6 c^6-4 a^5 b c^6+18 a^4 b^2 c^6-27 a^3 b^3 c^6+12 a^2 b^4 c^6+4 a b^5 c^6-4 b^6 c^6+8 a^5 c^7-13 a^4 b c^7+6 a^3 b^2 c^7+11 a^2 b^3 c^7-12 a b^4 c^7+4 b^5 c^7+2 a^4 c^8+13 a^3 b c^8-14 a^2 b^2 c^8+10 a b^3 c^8+b^4 c^8-7 a^3 c^9+a^2 b c^9+2 a b^2 c^9-6 b^3 c^9+2 a^2 c^10-6 a b c^10+2 b^2 c^10+2 a c^11+2 b c^11-c^12)::
on lines {{1,14987},{1511,2646}}.
2). X(65).
Best regards,
Peter Moses.
2). X(65).
Best regards,
Peter Moses.
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