HYACINTHOS 28370

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote:

(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.

(Oa), (Ob), (Oc) = the concentric circles (P, PNa), (P, PNb), (P, PNc), resp.

Ra, Rb, Rc = the radical axes of ((Na), (Oa)),  ((Nb), (Ob)),  ((Nc), (Oc)), resp.

A*B*C* = the triangle bounded by Ra, Rb, Rc, resp.

A", B", C" = the intersections Ra /\ BC, Rb /\ CA, Rc /\ AB, resp.

1. P = O:

 

ABC, A*B*C* are perspective (homothetic).

1.1. Homothetic center?

 

A'B'C', A*B*C* are perspective (homothetic).

1.2. Homothetic center?

 

A'B'C' (and ABC), A*B*C* are orthologic.

Orthologic center (A'B'C', A*B*C*) = orthocenter of A'B'C' = O

1.3. Orthologic center (A*B*C*, A'B'C') [= orthologic center (A*B*C*, ABC)] = orthocenter of A*B*C* = ?

 

2. P = I:

 

A", B", C" are collinear.
2.1. Line (point it is the trilinear pole of)

 

ABC, A*B*C* are perspective.

2.2. Perspector?

 

A'B'C', A*B*C* are orthologic.

Orthologic center (A'B'C', A*B*C*) = antipode of Feuerbach point in the incircle = X(1317)

2.3. Orthologic center (A*B*C*, A'B'C') ?

 

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

Parallelogic center (A'B'C', A*B*C*) = Feuerbach point X(11)

2.4. Parallelogic center (A*B*C*, A'B'C') ?

 

[Peter Moses]:

Hi Antreas,

Call
t0 = {A, B, C}
t1 = {A',B',C'}
t2 = {A",B",C"}
t3 = {A*,B*,C*}

then

For P = O
Orthologics:
{0,1} at X(4).
{1,0} at X(3).
{0,3} at X(4).
{3,0} at 

2 a^16-6 a^14 b^2+4 a^12 b^4+2 a^10 b^6-2 a^6 b^10-4 a^4 b^12+6 a^2 b^14-2 b^16-6 a^14 c^2+16 a^12 b^2 c^2-11 a^10 b^4 c^2-9 a^8 b^6 c^2+13 a^6 b^8 c^2+11 a^4 b^10 c^2-24 a^2 b^12 c^2+10 b^14 c^2+4 a^12 c^4-11 a^10 b^2 c^4+20 a^8 b^4 c^4-10 a^6 b^6 c^4-19 a^4 b^8 c^4+36 a^2 b^10 c^4-20 b^12 c^4+2 a^10 c^6-9 a^8 b^2 c^6-10 a^6 b^4 c^6+24 a^4 b^6 c^6-18 a^2 b^8 c^6+22 b^10 c^6+13 a^6 b^2 c^8-19 a^4 b^4 c^8-18 a^2 b^6 c^8-20 b^8 c^8-2 a^6 c^10+11 a^4 b^2 c^10+36 a^2 b^4 c^10+22 b^6 c^10-4 a^4 c^12-24 a^2 b^2 c^12-20 b^4 c^12+6 a^2 c^14+10 b^2 c^14-2 c^16:: 

= lies on these lines: {4,94},{1511,22823}.
{1,3} at X(3).
{3,1} same as {3,0}.
-----------------------------------------------------------

For P = X(1)
Orthologics:
{0,1} at X(1).
{1,0} at X(1).
{0,3} at X(80).
{3,0} at

= X(1)X(4)∩X(3)X(20107)

Barycentrics 2 a^7-2 a^6 b-2 a^5 b^2+2 a^4 b^3-2 a^3 b^4+2 a^2 b^5+2 a b^6-2 b^7-2 a^6 c+4 a^5 b c-a^4 b^2 c+a^3 b^3 c+a^2 b^4 c-5 a b^5 c+2 b^6 c-2 a^5 c^2-a^4 b c^2+2 a^3 b^2 c^2-3 a^2 b^3 c^2-2 a b^4 c^2+6 b^5 c^2+2 a^4 c^3+a^3 b c^3-3 a^2 b^2 c^3+10 a b^3 c^3-6 b^4 c^3-2 a^3 c^4+a^2 b c^4-2 a b^2 c^4-6 b^3 c^4+2 a^2 c^5-5 a b c^5+6 b^2 c^5+2 a c^6+2 b c^6-2 c^7 :: 

= 3 X[4881] - 5 X[8227], 5 X[3843] - X[18524], 2 X[6681] - 3 X[23513].

= lies on these lines: {1,4},{3,20107},{30,6713},{382,10893},{517,6246},{519,10738},{1319,16174},{2077,10724},{3627,7681},{3667,21179},{3814,5840},{3830,22753},{3843,11496},{3845,7680},{4299,10598},{4302,6968},{4324,6949},{4881,8227},{5046,6684},{5176,14217},{5450,10896},{5806,16125},{6681,23513},{6796,12953},{6923,10165},{6929,10175},{6965,10172},{7704,21842},{7743,11715},{7956,15687},{8666,11928},{10265,12764},{10525,11362}.
= midpoint of X(i) and X(j) for these {i,j}: {{4, 3583}, {2077, 10724}, {5176, 14217}}.
= reflection of X(i) in X(j) for these {i,j}: {1319, 16174}, {1519, 18483}, {11715, 7743}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5225, 6256), (4, 5603, 18513).

{1,3} at X(1317).
{3,1} at a (2 a^6-6 a^5 b+12 a^3 b^3-6 a^2 b^4-6 a b^5+4 b^6-6 a^5 c+22 a^4 b c-19 a^3 b^2 c-14 a^2 b^3 c+25 a b^4 c-8 b^5 c-19 a^3 b c^2+42 a^2 b^2 c^2-19 a b^3 c^2-4 b^4 c^2+12 a^3 c^3-14 a^2 b c^3-19 a b^2 c^3+16 b^3 c^3-6 a^2 c^4+25 a b c^4-4 b^2 c^4-6 a c^5-8 b c^5+4 c^6):: 

= lies on these lines: {1,3},{3036,3814},{5690,20107},{6681,10283}.

Parallelogics:
{1,3} at X(11)
{3,1} at a^2 (2 a^5-2 a^4 b-4 a^3 b^2+4 a^2 b^3+2 a b^4-2 b^5-2 a^4 c+6 a^3 b c-a^2 b^2 c-6 a b^3 c+3 b^4 c-4 a^3 c^2-a^2 b c^2+6 a b^2 c^2-b^3 c^2+4 a^2 c^3-6 a b c^3-b^2 c^3+2 a c^4+3 b c^4-2 c^5):: 

= lies on these  lines: {1,3},{5,6681},{20,10598},{24,1878},{30,6713},{104,13587},{140,3814},{182,9037},{214,14988},{355,4188},{404,9956},{515,12619},{535,549},{631,5080},{912,22935},{953,8697},{993,5123},{2771,18861},{3523,20067},{3582,10738},{3838,6914},{4299,6958},{4881,6265},{4996,5440},{5057,6875},{5146,7501},{5303,6940},{5450,6924},{5587,18515},{5840,15325},{5886,6950},{6882,15326},{6906,9955},{6942,18481},{6961,10526},{6970,18516},{6971,10483},{7288,10525},{7508,10165},{7743,10058},{11499,19537},{15446,17606},{16371,22758},{22793,22835}.
--------------------------------------------------------------------

For X(5)
Orthologics:
{0,1} at X(54).
{1,0} at X(5).
{0,3} at X(1141).

{3,0} at 

= X(4)X(11538)∩X(30)X(137)

2 a^16-9 a^14 b^2+17 a^12 b^4-17 a^10 b^6+5 a^8 b^8+13 a^6 b^10-21 a^4 b^12+13 a^2 b^14-3 b^16-9 a^14 c^2+34 a^12 b^2 c^2-41 a^10 b^4 c^2+10 a^8 b^6 c^2-9 a^6 b^8 c^2+56 a^4 b^10 c^2-61 a^2 b^12 c^2+20 b^14 c^2+17 a^12 c^4-41 a^10 b^2 c^4+28 a^8 b^4 c^4-a^6 b^6 c^4-48 a^4 b^8 c^4+105 a^2 b^10 c^4-60 b^12 c^4-17 a^10 c^6+10 a^8 b^2 c^6-a^6 b^4 c^6+26 a^4 b^6 c^6-57 a^2 b^8 c^6+108 b^10 c^6+5 a^8 c^8-9 a^6 b^2 c^8-48 a^4 b^4 c^8-57 a^2 b^6 c^8-130 b^8 c^8+13 a^6 c^10+56 a^4 b^2 c^10+105 a^2 b^4 c^10+108 b^6 c^10-21 a^4 c^12-61 a^2 b^2 c^12-60 b^4 c^12+13 a^2 c^14+20 b^2 c^14-3 c^16:: 

= lies on these lines: {4,11538},{30,137}.

Parallelogics:
{0,3} at X(930).
{3,0} at (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^22 b^2-8 a^20 b^4+27 a^18 b^6-48 a^16 b^8+42 a^14 b^10-42 a^10 b^14+48 a^8 b^16-27 a^6 b^18+8 a^4 b^20-a^2 b^22+a^22 c^2-22 a^20 b^2 c^2+118 a^18 b^4 c^2-307 a^16 b^6 c^2+480 a^14 b^8 c^2-524 a^12 b^10 c^2+470 a^10 b^12 c^2-378 a^8 b^14 c^2+247 a^6 b^16 c^2-110 a^4 b^18 c^2+28 a^2 b^20 c^2-3 b^22 c^2-8 a^20 c^4+118 a^18 b^2 c^4-432 a^16 b^4 c^4+705 a^14 b^6 c^4-556 a^12 b^8 c^4+84 a^10 b^10 c^4+334 a^8 b^12 c^4-491 a^6 b^14 c^4+372 a^4 b^16 c^4-152 a^2 b^18 c^4+26 b^20 c^4+27 a^18 c^6-307 a^16 b^2 c^6+705 a^14 b^4 c^6-594 a^12 b^6 c^6+163 a^10 b^8 c^6-79 a^8 b^10 c^6+342 a^6 b^12 c^6-529 a^4 b^14 c^6+375 a^2 b^16 c^6-103 b^18 c^6-48 a^16 c^8+480 a^14 b^2 c^8-556 a^12 b^4 c^8+163 a^10 b^6 c^8-12 a^8 b^8 c^8-71 a^6 b^10 c^8+328 a^4 b^12 c^8-478 a^2 b^14 c^8+248 b^16 c^8+42 a^14 c^10-524 a^12 b^2 c^10+84 a^10 b^4 c^10-79 a^8 b^6 c^10-71 a^6 b^8 c^10-138 a^4 b^10 c^10+228 a^2 b^12 c^10-406 b^14 c^10+470 a^10 b^2 c^12+334 a^8 b^4 c^12+342 a^6 b^6 c^12+328 a^4 b^8 c^12+228 a^2 b^10 c^12+476 b^12 c^12-42 a^10 c^14-378 a^8 b^2 c^14-491 a^6 b^4 c^14-529 a^4 b^6 c^14-478 a^2 b^8 c^14-406 b^10 c^14+48 a^8 c^16+247 a^6 b^2 c^16+372 a^4 b^4 c^16+375 a^2 b^6 c^16+248 b^8 c^16-27 a^6 c^18-110 a^4 b^2 c^18-152 a^2 b^4 c^18-103 b^6 c^18+8 a^4 c^20+28 a^2 b^2 c^20+26 b^4 c^20-a^2 c^22-3 b^2 c^22):: 

= lies on these lines: {}.

Best regards,
Peter Moses.

Note (APH): In red only those new points which are not in Hyacinthos 28368