HYACINTHOS 26947

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the pedal triangle of I.

Denote:

 

A", B", C" = the antipodes of A', B', C' in the incircle, resp.

 

A* = AA" /\ BC
B* = BB" /\ CA
C* = CC" /\ AB

MaMbMc = the medial triangle of A"B"C"

M1, M2, M3 = the midpoints of AA*, BB*, CC*, resp.

 

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.

 

Cyclologic centers?

 

 

[Angel Montesdeoca[:

***  Cyclologic center of  M1M2M3 wrt MaMbMc is X(3035) = complement of Feuerbach point.

*** Cyclologic center of  MaMbMc wrt M1M2M3 is
 
 V = a (a^5 (b+c)    
     -a^4 (b^2+6 b c+c^2)
     +a^3 (-2 b^3+7 b^2 c+7 b c^2-2 c^3)
     +a (b-c)^2 (b^3-6 b^2 c-6 b c^2+c^3)
     +a^2 (2 b^4+5 b^3 c-18 b^2 c^2+5 b c^3+2 c^4)
     -(b^2-c^2)^2 (b^2-b c+c^2)) : ... : ....
  

V lies on lines X(i)X(j) for these {i, j}: {1,104}, {9,644}, {10,11}, {55,214}, {78,13278}, {80,497}, {100,1697}, ...   

V is the midpoint of X(i) and X(j), for these {i, j}: {1,12758}, {11,3057}, {10284,12619}.

V is the reflection of X(i) in X(j), for these {i, j}: {5083,1}, {5836,6667}, {12736,1387}, {14740,960}.

(6 - 9 - 13) - search numbers  of W: (2.74701291332711, 3.01069105643210, 0.288487790534261)

Angel Montesdeoca