[Antreas P. Hatzipolakis]
Let ABC be a triangle and A'B'C' the pedal triangle of I.
Denote:
(Na), (Nb), (Nc) = the NPCs of IBC, ICA, IAB, resp.
(I1), (I2), (I3) = the reflections of the incircle (I) in NbNc, NcNa, NaNb, resp.
Ra = the radical axis of (I1), (Na)
Rb = the radical axis of (I2), (Nb)
Rc = the radical axis of (I3), (Nc)
A*B*C* = the triangle bounded by Ra, Rb, Rc
1. ABC, A*B*C* are orthologic
2. A'B'C', A*B*C* are parallelogic.
Parallelogic center (A'B'C', A*B*C*) = the Feuerbach point.
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[Ercole Suppa]:
Hi Antreas,
1.
Orthology center (ABC, A*B*C*) = X(80) = reflection of incenter in Feuerbach point
Orthology center (A*B*C*, ABC) =
= 2 a^10-4 a^9 (b+c)+a (b-c)^4 (b+c)^3 (b^2-3 b c+c^2)-(b^2-c^2)^4 (b^2-b c+c^2)-a^8 (3 b^2-8 b c+3 c^2)+2 a^2 b c (b^2-c^2)^2 (3 b^2-4 b c+3 c^2)-a^7 (-11 b^3+3 b^2 c+3 b c^2-11 c^3)-a^6 (2 b^4+9 b^3 c-16 b^2 c^2+9 b c^3+2 c^4)-a^5 (9 b^5-13 b^4 c+10 b^3 c^2+10 b^2 c^3-13 b c^4+9 c^5)-a^4 (-4 b^6+6 b^5 c+8 b^4 c^2-22 b^3 c^3+8 b^2 c^4+6 b c^5-4 c^6)+a^3 (b^7-2 b^6 c+12 b^5 c^2-9 b^4 c^3-9 b^3 c^4+12 b^2 c^5-2 b c^6+c^7) : : (barys)
= on the line through X(6713) parallel to Euler line
= (6-9-13) search numbers [4.60537394290521204, 4.52675892935177332, -1.61880275052310029]
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2.
Parallelology center (A'B'C', A*B*C*) = X(11) = Feuerbach point
Parallelology center (A*B*C*, A'B'C') =
= 4 a^8 b c-b c (b^2-c^2)^4+3 a^2 b c (b^2-c^2)^2 (b^2-b c+c^2)-a^4 b (b-c)^2 c (b^2+5 b c+c^2)-a^6 b c (5 b^2-6 b c+5 c^2)+a^7 (b^3-3 b^2 c-3 b c^2+c^3)-a (b-c)^2 (b+c)^3 (b^4+b^3 c-3 b^2 c^2+b c^3+c^4)-a^5 (3 b^5-6 b^4 c+b^3 c^2+b^2 c^3-6 b c^4+3 c^5)-a^3 (-3 b^7+b^6 c+2 b^4 c^3+2 b^3 c^4+b c^6-3 c^7) : : (barys)
= on the line through X(6713) parallel to Euler line
= (6-9-13) search numbers [-0.393969957550176711, -0.459396575662429179, 4.14054055315845009]
Best regards,
Ercole Suppa
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