[Antreas P. Hatzipolakis]:
Denote:
MaMbMc = the midheight triangle
(ie Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.)
M1, M2, M3 = the midpoints of A'X(110), B'X(110), C'X(110), resp.
M1, M2, M3 = the midpoints of A'X(110), B'X(110), C'X(110), resp.
Mi, Mii, Miii = the midpoints of A'X(74), B'X(74), C'X(74), resp.
Mx, My, Mz = the midpoints of M1Mi, M2Mii, M3Miii, resp.
1. The perpendicular bisectors of MaM1, MbM2, McM3 are concurrent
2. MaMi, MbMii, McMiii are concurrent
3. MaMx, MbMy, McMz are concurrent.
4. M1Mi, M2Mii, M3Miii are concurrent (parallel)
5. Let (Na, Nb, Nc), (Oa, Ob, Oc) be the NPC centers, circumcenters, resp. of the (right angled) triangles MaM1Mi, MbM2Mii, McM3Miii, resp.
The circumcenter of NaNbNc is the N of ABC.
The circumcenter of NaNbNc is the N of ABC.
The circumcenter of OaObOc is the midpoint of ON = X(140)
[Angel Montesdeoca]:
**** 1. The perpendicular bisectors of MaM1, MbM2, McM3 are concurrent at
U1 = (a^2 (-a^12 (b^2+c^2)+4 a^10 (b^4+c^4)+a^8 (-5 b^6+2 b^4 c^2+2 b^2 c^4-5 c^6)+(b^2-c^2)^4 (b^6+2 b^4 c^2+2 b^2 c^4+c^6)+2 a^6 (3 b^6 c^2-8 b^4 c^4+3 b^2 c^6)-2 a^2 (b^2-c^2)^2 (2 b^8-b^6 c^2-3 b^4 c^4-b^2 c^6+2 c^8)+a^4 (5 b^10-15 b^8 c^2+12 b^6 c^4+12 b^4 c^6-15 b^2 c^8+5 c^10)): ... : ...)
U1 is the midpoint of X(i) and X(j), for these {i, j}: {3,1986}, {113,185}, {125,11562}, {1511,6102}, {1539,13491}, {7722,7723}, {10575,13202}, {11561,13630}, {12358,13148}
U1 is the reflection of X(i) in X(j), for these {i, j}: {5,9826}, {974,13630}, {5907,12900}, {6699,9729}, {7687,5462}, {10113,11746}, {12133,546}, {12236,389}, {12358,140}
(6 - 9 - 13) - search numbers of U1: (4.66529367578157, 4.43894811966114, -1.58566591283406)
**** 2. MaMi, MbMii, McMiii are concurrent at X(974) = 2nd EHRMANN POINT
see Hyacinthos message 3695, Sept. 1, 2001, and related messages.
**** 3. MaMx, MbMy, McMz are concurrent at X(389) = CENTER OF THE TAYLOR CIRCLE
**** 4. M1Mi, M2Mii, M3Miii are concurrent (parallel). Point on line at infinity: X(5663) = ISOGONAL CONJUGATE OF X(477) (X(477) = TIXIER ANTIPODE).
Angel Montesdeoca
[Angel Montesdeoca]:
**** 1. The perpendicular bisectors of MaM1, MbM2, McM3 are concurrent at
U1 = (a^2 (-a^12 (b^2+c^2)+4 a^10 (b^4+c^4)+a^8 (-5 b^6+2 b^4 c^2+2 b^2 c^4-5 c^6)+(b^2-c^2)^4 (b^6+2 b^4 c^2+2 b^2 c^4+c^6)+2 a^6 (3 b^6 c^2-8 b^4 c^4+3 b^2 c^6)-2 a^2 (b^2-c^2)^2 (2 b^8-b^6 c^2-3 b^4 c^4-b^2 c^6+2 c^8)+a^4 (5 b^10-15 b^8 c^2+12 b^6 c^4+12 b^4 c^6-15 b^2 c^8+5 c^10)): ... : ...)
U1 is the midpoint of X(i) and X(j), for these {i, j}: {3,1986}, {113,185}, {125,11562}, {1511,6102}, {1539,13491}, {7722,7723}, {10575,13202}, {11561,13630}, {12358,13148}
U1 is the reflection of X(i) in X(j), for these {i, j}: {5,9826}, {974,13630}, {5907,12900}, {6699,9729}, {7687,5462}, {10113,11746}, {12133,546}, {12236,389}, {12358,140}
(6 - 9 - 13) - search numbers of U1: (4.66529367578157, 4.43894811966114, -1.58566591283406)
**** 2. MaMi, MbMii, McMiii are concurrent at X(974) = 2nd EHRMANN POINT
see Hyacinthos message 3695, Sept. 1, 2001, and related messages.
**** 3. MaMx, MbMy, McMz are concurrent at X(389) = CENTER OF THE TAYLOR CIRCLE
**** 4. M1Mi, M2Mii, M3Miii are concurrent (parallel). Point on line at infinity: X(5663) = ISOGONAL CONJUGATE OF X(477) (X(477) = TIXIER ANTIPODE).
Angel Montesdeoca
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