HYACINTHOS 27179

[Antreas P. Hatzipolakis]:

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

Denote:

Bc = the reflection of B' in OC', 

Cb = the reflection of C' in OB'

 

Ca = the reflection of C' in OA', 

Ac = the reflection of A' in OC' 

 

Ab = the reflection of A' in OB', 

Ba = the reflection of B' in OA' 

 

A*B*C* = the triangle bounded by BcCb, CaAc, AbBa 

 

The Euler line of A*B*C* is the line X(110)X(3)X(74) of ABC.

The line X(110)X(3)X(74) of A*B*C* is parallel to the Euler line of ABC. 

 

1. Which point wrt triangle A*B*C* is the O = X(3) of ABC?

A'B'C', A*B*C* share the same X(110)

 

ABC, A*B*C* are parallelogic.
Parallelogic center (ABC, A*B*C*) = X(110) of ABC
Parallelogic center (A*B*C*, ABC) = X(110) of A*B*C* = X(110) of A'B'C' = complement of X(110) of ABC = X(125) of ABC

A'B'C', A*B*C* are parallelogic

Parallelogic center ( A*B*C*, A'B'C') = Parallelogic center (A'B'C', A*B*C*) = X(110) of A'B'C' = X(110) of A*B*C* = X(125)

2. ABC,AB*C* are orthologic
Orthologic center (ABC, A*B*C*) = X(74) of ABC
Orthologic center (A*B*C*, ABC) = X(74) of A*B*C* = U wrt triangle ABC
U = ???

3. A*B*C*, A'B'C' are orthologic
Orthologic center (A*B*C*, ABC) = orthologic center (A*B*C*, A'B'C') =  X(74) of A*B*C* = U
Which is the other orthologic center (A'B'C', A*B*C*) ?

4. The circumcircle of A*B*C* and the circumcircle of A'B'C' ( = NPC of ABC) intersect at two points.
The one is the X(110) of A*B*C* = X(110) of A'B'C'.= X(125) of ABC
Which is the other one?


[Peter Moses]:

Hi Antreas,

1). X(186).

2). U = 4 a^10-6 a^8 b^2-5 a^6 b^4+11 a^4 b^6-3 a^2 b^8-b^10-6 a^8 c^2+22 a^6 b^2 c^2-13 a^4 b^4 c^2-6 a^2 b^6 c^2+3 b^8 c^2-5 a^6 c^4-13 a^4 b^2 c^4+18 a^2 b^4 c^4-2 b^6 c^4+11 a^4 c^6-6 a^2 b^2 c^6-2 b^4 c^6-3 a^2 c^8+3 b^2 c^8-c^10:: 

on lines {{2,10721},{3,113},{4,6699},{5,13202},{20,68},{22,12893},{30,125},{52,974},{110,376},{140,1539},{146,3522},{165,12368},{247,7422},{265,1657},{378,7706},{381,6723},{382,7687},{399,15696},{516,11709},{542,1350},{548,1511},{550,5562},{569,15472},{631,12900},{1060,10118},{1112,9730},{1209,11598},{1216,12825},{1531,15122},{1533,7575},{1593,15473},{2778,9943},{2781,9967},{2931,11414},{2972,12113},{3024,15326},{3028,15338},{3070,8994},{3071,13969},{3098,5181},{3146,14644},{3528,15051},{3529,10733},{3576,11723},{3845,15088},{4299,10065},{4302,10081},{4324,12896},...}.
complement X(10721).
midpoint of X(i) and X(j) for these {i,j}: {{20,74},{110,12244},{265,1657},{550,14677},{3529,10733},{5925,11744},{6241,12219},{6361,7984},{7722,13201},{9140,11001},{10264,15704},{10620,12121},{12383,15054},{13445,13619}}.
reflection of X(i) in X(j) for these {i,j}: {{4,6699},{52,974},{113,3},{125,12041},{382,7687},{1511,548},{1531,15122},{1533,7575},{1539,140},{5181,3098},{5642,8703},{7728,5972},{10990,14677},{11693,15688},{11807,9729},{12162,12358},{12295,125},{12699,11735},{12825,1216},{13202,5},{13417,14708},{15063,1511},{16003,74},{16105,9826}}.
X[5972]}, X[110] - 3 X[376], 3 X[74] - X[3448], 3 X[20] + X[3448], X[146] - 5 X[3522], 3 X[113] - 4 X[5972], 3 X[3] - 2 X[5972], 3 X[381] - 4 X[6723], 3 X[113] - 2 X[7728], 3 X[3] - X[7728], 2 X[1112] - 3 X[9730], 3 X[125] - 2 X[10113], 3 X[3534] + X[10620], 2 X[550] + X[10990], 3 X[3576] - 2 X[11723], 5 X[10113] - 6 X[11801], 5 X[125] - 4 X[11801], 2 X[11801] - 5 X[12041], X[10113] - 3 X[12041], 3 X[3534] - X[12121], 3 X[376] + X[12244], 8 X[11801] - 5 X[12295], 4 X[10113] - 3 X[12295], 4 X[12041] - X[12295], 3 X[165] - X[12368], 5 X[631] - 4 X[12900], 10 X[5972] - 9 X[14643], 5 X[7728] - 9 X[14643], 5 X[113] - 6 X[14643], 5 X[3] - 3 X[14643], X[3146] - 3 X[14644], X[11562] - 3 X[14855], X[10733] - 5 X[15021], X[3529] + 5 X[15021], X[146] - 3 X[15035], 5 X[3522] - 3 X[15035], X[265] - 3 X[15041], X[1657] + 3 X[15041], 7 X[3528] - 5 X[15051], X[4] - 3 X[15055], 2 X[6699] - 3 X[15055], 3 X[4] - 5 X[15059], 6 X[6699] - 5 X[15059], 9 X[15055] - 5 X[15059], X[382] - 3 X[15061], 2 X[7687] - 3 X[15061], 4 X[548] - X[15063], 3 X[3845] - 4 X[15088], X[399] - 5 X[15696], 2 X[3448] - 3 X[16003], 2 X[20] + X[16003], ...
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7728, 5972), (4, 15055, 6699), (146, 3522, 15035), (376, 12244, 110), (382, 15061, 7687), (1657, 15041, 265), (3534, 10620, 12121), (5972, 7728, 113), ...

3). X(113). 

4). X(122).

Best regards,
Peter Moses.