HYACINTHOS 25306

[Antreas P. Hatzipolakis]:

 

 

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

Denote:

A1, B1, C1 = the reflections of P in BC, CA, AB, resp.

A2, B2, C2 = the reflections of A1, B1, C1 in B'C', C'A', A'B', resp.

The circumcircles of PA1A2, PB1B2, PC1C2 are coaxial.

[Tran Quang Hung]:

Dear Mr Antreas Hatzipolakis,

 

I see similar problem as following 

 

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

Denote:

A1, B1, C1 = the reflections of P in B'C', C'A', A'B', resp.

A2, B2, C2 = the reflections of A1, B1, C1 in BC, CA, AB, resp.

The circumcircles of PA1A2, PB1B2, PC1C2 are coaxial.

 

[César Lozada]:

 

Dear Mr. Hung,

 

They are coaxial. Moreover, if Q is the 2nd point of intersection the line (PQ) is parallel to the Euler line of ABC.

 

Q = reflection-of-P in the orthic axis {230,231}

 

For P = x:y:z (barycentrics), Q= (SA+SW-6*R^2)*S^2*x+(S^2-3*SB* SC)*(SB*y+SC*z) : :

 

ETC pairs (P,Q):

(2,7426), (4,10295), (5,7575), (23,858), (24,862), (30,30), (111,5913), (112,5523), (115,187), (186,403), (187,115), (230,230), (231,231), (232,232), (297,7473), (395,11549), (396,11537), (403,186), (468,468), (476,3580), (523,523), (647,647), (650,650), (676,676), (858,23), (862,24), (1316,5112), (1495,3258), (1637,1637), (1886,1886), (1990,1990), (2070,2072), (2072,2070), (2485,2485), (2489,2489), (2490,2490), (2491,2491), (2492,2492), (2493,2493), (2501,2501), (2835,6009), (2977,2977), (3003,3003), (3011,3011), (3012,3012), (3018,3018), (3064,3064), (3258,1495), (3290,3290), (3291,3291), (3310,3310), (3580,476), (3806,3806), (4874,4874), (5089,5089), (5112,1316), (5523,112), (5913,111), (6103,6103), (6104,6107), (6105,6106), (6106,6105), (6107,6104), (6108,6109), (6109,6108), (6110,6111), (6111,6110), (6129,6129), (6130,6130), (6131,6131), (6132,6132), (6133,6133), (6134,6134), (6586,6586), (6587,6587), (6588,6588), (6589,6589), (6590,6590), (6591,6591), (6753,6753), (7426,2), (7473,297), (7575,5), (7649,7649), (7662,7662), (8105,8105), (8106,8106), (8607,8607), (8608,8608), (8609,8609), (8610,8610), (8755,8755), (8756,8756), (8758,8758), (9125,9125), (9189,9189), (9209,9209), (10295,4), (10418,10418), (11062,11062), (11176,11176), (11537,396), (11549,395), (11657,11657)             

 

Others:

Q( I ) = (3*sin(A/2)-2*sin(3*A/2))*cos( (B-C)/2)-cos(A)*cos(B-C)+sin( A/2)*cos(3*(B-C)/2)+1/2 : :  (trilinears)

= On lines: {1,30}, {186,1068}, {225,403}, {523,10015},…

= [ -0.144865155952838, -0.14002392148596, 3.804464961068274 ]

 

Q( O ) = (cos(2*A)+3)*cos(B-C)-cos(A)* cos(2*(B-C))-4*cos(A) : : trilinears

= (3*R^2-SW)*X(3)+(6*R^2-SW)*X( 4)

= [E+4*F, -5*E+4*F]

= On lines: {2,3}, {11,4351}, {12,4354}, {94,10688}, {113,511}, {115,3003}, {125,1533}, {265,1177}, {399,3564}, {495,9642}, {524,5655}, …

= midpoint of X(i) and X(j) for these {i,j}: {4,23}, {125,1533}

= reflection of X(i) in X(j) for these (i,j): (3,468), (403,11563), (858,5)

=complement of X(7464)

=circumcircle-inverse-of-X( 6644)

=orthoptic circle of Steiner inellipse-inverse-of-X(1995)

=polar circle-inverse-of-X(378)

=Steiner circle-inverse-of-X(382)

=[ -4.365092964401092, -5.22309766184147, 9.271313462136668 ]

 

César Lozada