HYACINTHOS 24756

[Antreas P. Hatzipolakis]:

 

 

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

Denote:

A1, B1, C1 = the orthogonal projections of A,B,C on OA', OB', OC', resp.

A2, B2, C2 = the orthogonal projections of A',B',C' on OA, OB, OC, resp.

M1,M2, M3 = the midpoints of A1A2,B1B2,C1C2, resp.

The centroid of M1M2M3 lies on the Euler line of ABC.

[Peter Moses]:

Hi Antreas,

 

(a^2-b^2-c^2) (2 a^4+3 a^2 b^2+b^4+3 a^2 c^2-2 b^2 c^2+c^4)::

on lines {{2,3},{216,9300},{305,7767},{ 524,6665},{539,5447},{577, 5306},{1038,5434},{1040,3058}, {1503,3819},{1611,2549},{3564, 3917},{5268,7354},{5272,6284}} .

Anticomplement X[10128].

Complement X[428].

Midpoint of X(2) and X(7667).

Reflection of X(i) in X(j) for these {i,j}: {{2, 7734}, {428, 10128}, {6756, 10127}, {7576, 9825}, {10127, 140}}.

SA SB SC X[2] - S^2 SW X[3].

X[1885] + 5 X[3522], 7 X[3523] - X[3575], 4 X[140] - X[6756], 7 X[3526] - X[7553], 5 X[631] - X[7576], X[428] + 3 X[7667], X[428] - 6 X[7734], X[7667] + 2 X[7734], 5 X[631] - 2 X[9825], 3 X[7734] - X[10128], 3 X[7667] + 2 X[10128], X[6240] - 13 X[10299].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,20,7714),(2,376,9909),(2, 428,10128),(2,1370,5064),(2, 5064,5),(2,7714,5020),(3,1368, 6676),(3,6643,6823),(3,7386, 1368),(3,10300,5159),(427, 7485,140),(465,466,441),(548, 6677,22),(1368,6676,5159),( 1368,7386,10300).(1370,7484,5) ,(5064,7484,2),(6676,10300, 1368).

X(i)-complementary conjugate of X(j) for these (i,j): {{3108,226},{7953,8062}}.

 

Best regards,

Peter Moses.