HYACINTHOS 29466

[Antreas P. Hatzipolakis]:

 

= isotomic conjugate of the anticomplement of X(8965)
= X(i)-cross conjugate of X(j) for these (i,j): {8965, 2}, {8978, 81}
= X(i)-isoconjugate of X(j) for these (i,j): {55, 482}, {10252, 13456}, {30335, 32082}
= cevapoint of X(1) and X(13389)
= trilinear pole of line {650, 6364}
= barycentric product X(7)*X(15890)
= barycentric quotient X(i)/X(j) for these {i,j}: {57, 482}, {494, 26504}, {7968, 19029}, {13389, 31535}, {15890, 8}

Note: Just for reference the point A3 = {a (a - 2 R), (a + b - c) R - a b, (a - b + c) R - a c}
(the 'external' version R -> -R)
A3B3C3 is perspective to lots of triangles.

Let ABC be a triangle and P a point.

The circle (B, BP) intersects AB at Ac1 (between A,B) and Ac2 (on the extension of BA) AND BC at A1c (between B,C) and A2c (on the extension of BC). The circle (C, CP) intersects AC at Ab1 (between A,C) and Ab2 AND BC at A1b (between B,C) and A2b (on the extension of CB).
The lines Ab1A1b and Ac1A1c intersect at A3.
Similarly B3, C3

For P = O:
ABC, A3B3C3 are perspective.
Perspector?

[Peter Moses]:

Hi Antreas,

X(4)X(175)∩X(8)X(492) =

= a*(b (a - b + c) - 2*S)*(c (a + b - c) - 2*S) : :
= (a*b - (a + b - c)*R)*(a*c - (a - b + c)*R) : :

= lies on on Feuerbach circumhyperbola and these lines: {2, 13426}, {4, 175}, {8, 492}, {9, 3084}, {79, 481}, {80, 31539}, {176, 3296}, {354, 7269}, {482, 5557}, {1372, 5561}, {5551, 21169}, {5556, 31602}, {5558, 17805}, {5560, 17803}, {13388, 16441}, {13435, 13454}

.
= X(i)-isoconjugate of X(j) for these (i,j): {55, 481}, {10253, 13427}, {30336, 32083}
= cevapoint of X(1) and X(13388)
= trilinear pole of line {650, 6365}
= barycentric product X(7)*X(15889)
= barycentric quotient X(i)/X(j) for these {i,j}: {57, 481}, {493, 26495}, {7969, 19030}, {13388, 31534}, {15889, 8}

Presumably, there is another construction that gives:

X(4)X(176)∩X(8)X(491) =

= a*(b (a - b + c) + 2*S)*(c (a + b - c) + 2*S) : :
= (a*b + (a + b - c)*R)*(a*c + (a - b + c)*R) : :

= lies on Feuerbach circumhyperbola and these lines: {2, 13454}, {4, 176}, {8, 491}, {9, 3083}, {79, 482}, {80, 31538}, {175, 3296}, {354, 7269}, {481, 5557}, {1371, 5561}, {5556, 31601}, {5558, 17802}, {5560, 17806}, {7133, 13389}, {13424, 13426}

 

Best regards,
Peter Moses.