HYACINTHOS 26123

[Antreas P. Hatzipolakis]:

 

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

 

Denote:

Ab, Ac = the orthogonal projections of A on BI, CI, resp.
Bc, Ba = the orthogonal projections of B on CI, AI, resp.
Ca, Cb = the orthogonal projections of C on AI, BI, resp.

Oa, Ob, Oc = the circumcenters of A'AbAc, B'BcBa, C'CaCb, resp.

ABC, OaObOc are orthologic
(and also A'B'C', OaObOc are orthologic)

[Peter Moses]:

 

 

Hi Antreas,

 

(ABC, OaObOc) orthology: X(3227).

 

(OaObOc,  ABC) orthology: X(946).

 

 

(A'B'C', OaObOc) orthology: 

 

(a b+a c-2 b c)^2:: 

on lines {{2,668},{10,537},{76, 4740},{115,1211},{536,6381},{ 599,2810},{812,4370},{891, 4728},{1017,3570},{1084,4755}, {1146,3452},{1500,4033},{2482, 2787},{3679,3789},{3762,6184}, {4482,8649}}.

complement X[3227]. 

midpoint X[2] and X[668]

reflection X[1015] in X[2].

2 X[668] + X[1015], 3 X[1015] - 2 X[3227], 3 X[668] + X[3227], 5 X[3227] - 3 X[9263], 5 X[1015] - 2 X[9263], 5 X[2] - X[9263], 5 X[668] + X[9263].

on Steiner inelipse.

X(i)-complementary conjugate of X(j) for these (i,j): {{6, 4871}, {31, 536}, {101, 891}, {536, 2887}, {692, 4763}, {890, 1086}, {891, 116}, {899, 141}, {1918, 2229}, {1919, 1646}, {3230, 10}, {3768, 11}, {4526, 124}, {6381, 626}}.

X(i)-Ceva conjugate of X(j) for these (i,j): {{2, 536}, {536, 8031}, {668, 891}}.

X(8031)-cross conjugate of X(536).

crosspoint of X(2) and X(536).

crossdifference of every pair of points on line  {739, 890}.

crosssum of X(6) and X(739).

barycentric product X(i)X(j) for these {i,j}: {{536, 536}, {899, 6381}, {3227, 8031}}..

barycentric quotient X(i)/X(j) for these {i,j}: {{536, 3227}, {3230, 739}, {8031, 536}}.

 

 

(OaObOc, A'B'C') orthology: X(946).

 

Best regards,

Peter Moses.